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Statistical analysis of turbulent pipe flow : a numericalapproachCitation for published version (APA):Veenman, M. P. B. (2004). Statistical analysis of turbulent pipe flow : a numerical approach. TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR576404

DOI:10.6100/IR576404

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https://research.tue.nl/en/publications/statistical-analysis-of-turbulent-pipe-flow--a-numerical-approach(751f3213-ae14-4369-96c5-29ad1e6f9d49).html

Statistical Analysis of Turbulent Pipe Flow:

A Numerical Approach

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr. R.A. van Santen, voor eencommissie aangewezen door het College voor

Promoties in het openbaar te verdedigenop woensdag 16 juni 2004 om 16.00 uur

door

Maurice Petrus Bernardus Veenman

geboren te Nijmegen

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. J.J.H. Brouwersenprof.dr.ir. W. van de Water

Copromotor:dr. J.G.M. Kuerten

Copyright c© 2004 by M.P.B. VeenmanAll rights reserved. No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form, or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the prior permission of the author.

Printed by the Eindhoven University Press.

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Veenman, Maurice P.B.

Statistical analysis of turbulent pipe flow : a numerical approach / by Maurice P.B.Veenman. – Eindhoven : Technische Universiteit Eindhoven, 2004.Proefschrift. – ISBN 90-386-3025-5NUR 978

Subject headings: turbulent flow / pipe flow / direct numerical simulation /Lagrangian stochastic model

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Large and small scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Numerical and experimental techniques . . . . . . . . . . . . . . . . . 4

1.4 Statistical description of turbulence . . . . . . . . . . . . . . . . . . . . 5

1.5 Stochastic modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5.1 Diffusion theory or Markov process for particle position . . . . 6

1.5.2 Markov process for particle velocity . . . . . . . . . . . . . . . 7

1.6 Purpose and outline of this thesis . . . . . . . . . . . . . . . . . . . . . 9

2 Numerical Techniques 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Equations of motion and numerical approach . . . . . . . . . . . . . . 14

2.2.1 Fluid equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Lagrangian statistics . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2 Eulerian statistics . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3 Statistical accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Lagrangian-Eulerian Connection 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Connection between Lagrangian and Eulerian statistics . . . . . . . . . 37

3.2.1 Single-time statistics . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.2 Structure- and moment-functions . . . . . . . . . . . . . . . . . 39

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Single-time statistics . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.2 Structure functions . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.3 Moment functions . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

vi Contents

4 Acceleration Statistics 51

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Single-point variances . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Geometrical statistical properties . . . . . . . . . . . . . . . . . . . . . 544.4 Pressure statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.5 Tennekes’ hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 The Small Scale Structure of Pipe Flow 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Local Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Dissipation rate as a discriminator for local isotropy . . . . . . . . . . 765.4 Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 Lagrangian Stochastic Model for Pipe Flow 85

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Linear stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.1 Model properties . . . . . . . . . . . . . . . . . . . . . . . . . . 866.2.2 Kolmogorov constant . . . . . . . . . . . . . . . . . . . . . . . . 886.2.3 Damping coefficients . . . . . . . . . . . . . . . . . . . . . . . . 916.2.4 Well-mixed condition . . . . . . . . . . . . . . . . . . . . . . . . 936.2.5 Numerical technique . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.3.1 Lagrangian statistics . . . . . . . . . . . . . . . . . . . . . . . . 966.3.2 Point source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Bibliography 107

Summary 115

Samenvatting 117

Dankwoord 119

Curriculum Vitae 121

Chapter 1

Introduction

1.1 Background

The influence of turbulence is noticeable all around us. A few examples are theboundary layer of the earth’s atmosphere, the flow around your car, the smoke raisingfrom your cigarette and the flow in your cup when you stir your coffee. The importanceof understanding turbulence, from an engineering point of view, originates from thedesire to be able to predict e.g. the drag force of an airplane, the resistance in apiping system or the heat transfer in a heat exchanger, with the intention to developnew or more efficient machines. Also from a more fundamental viewpoint turbulenceis of great interest.

In general Newtonian fluid flow in motion can manifest itself in three states. Inthe first state the flow is smooth and regular, which is known as ”laminar”. Theflow moves in layers and there are no fluctuations of the physical properties. In thesecond state there can locally be small fluctuations in the flow field and the flow iscalled ”transitional”. It means that the flow is between laminar and the third andfinal state which is called ”turbulent”. In this turbulent state, the subject of thisthesis, physical properties like e.g. the velocity and pressure fluctuate in both timeand space. The word ”turbulence” originates from the Latin word ”turbulentus”which means ”restless”. The complicated structure of turbulent flows greatly affectsproperties compared to the laminar case. For example, turbulent flows are much moreefficient in transferring momentum, heat and propagating chemical reactions. It isdue to properties like these that turbulent flows are significant for many problems intechnological applications and in natural science.

The chaotic state of fluid motion arises when the speed of the fluid exceeds aspecific threshold, below which viscous forces damp out the chaotic behaviour. Per-haps the simplest way to define turbulence is by reference to the Reynolds number,a parameter that characterises a flow. Named after the British engineer OsborneReynolds, this number indicates the ratio, or relative importance, of the flow’s iner-tial and viscous forces:

Re =UD

ν.

2 Introduction

Here U and D are characteristic velocity and length scales of the flow and ν is thekinematic viscosity. If the Reynolds number is low, viscous forces damp out anyfluctuation and the flow remains smooth and stable. If the Reynolds number isincreased inertial forces start to dominate the viscous forces and eventually the flowbecomes irregular and chaotic and we end up with what we call a turbulent flow.

The opinions on an exact definition of turbulence are divided. When people de-fine turbulence they tend to do so by describing characteristic features of such a flow.Chaotic behaviour, irregularity and fluctuating properties are the most important gen-erally accepted phenomena of a turbulent flow (see e.g. [61] or [94]). Other featuresthat have been addressed are a wide range of interacting scales (which is a conse-quence of the nonlinearity of the equations governing the flow), loss of predictability,dissipative, diffusive and turbulent flows are three dimensional and rotational.

The dynamics of fluid flow are governed by a form of Newton’s law includingviscous effects and has the form

∂u

∂t+ u · ∇u = −1

ρ∇p + ν∇2u + f . (1.1)

In this equation u is the velocity vector, p is the pressure and f is an externalforce which are all a function of space and time. The density, which we will assumeto be constant, is denoted by ρ. This set of equations is called the Navier-Stokesequations, derived independently by the French engineer Claude Navier in 1827 andthe Irish mathematician George Stokes in 1845. In this thesis we will only considerincompressible flow for which the continuity equation is given by

∇ · u = 0. (1.2)

The flow is said to be divergence free or solenoidal. The combination of the Navier-Stokes and continuity equations describes laminar, transitional and turbulent flows.So even for turbulent flows it suggests that the fluid flow is deterministic given aninitial condition of the flow field including the boundaries. However, the evolving flowfield is extremely sensitive to small fluctuations in the initial field. So small differencesin the initial field lead to completely different solutions within a finite time period.It is then said that the set of equations are ”ill posed” and the solution is called”deterministic chaos” [61].

For several laminar flows analytic solutions are readily available. For transitionaland turbulent flows, however, this is not the case. Many renowned researchers haveillustrated the difficulty of dealing with turbulence in catchy, but also rather frustratedquotes. For example Richard Feynman called turbulence ”the most important unsolvedproblem of classical physics.” The British physicist Horace Lamb said, ”when I dieand go to heaven there are two matters on which I hope for enlightenment. One isquantum electrodynamics, and the other is the turbulent motion of fluids. And aboutthe former I am rather optimistic.” Bradshaw called it the ”invention of the Devilon the seventh day of creation.” This is inherent to the complexity of the problem.Since no analytical solutions are possible for turbulence most research restricts itselfto a phenomenological description of turbulence. Figure 1.1 shows the cross sectionsthrough a pipe flow for three different Reynolds numbers. The Reynolds number

1.2 Large and small scales 3

Figure 1.1. Streamlines at a cross-section through pipe flow at three different, increasing(from left to right) Reynolds numbers. The streamlines have been calculated from the radialand tangential velocity components only. These results have been obtained from directnumerical simulations. The Reynolds numbers based on the bulk velocity and diameter areReb = 5300, Reb = 10312 and Reb = 15469.

increases from left to right. In this figure the streamlines in the two-dimensionalplane are shown for a frozen turbulence field. Observing these curves we can seethat the flow behaves chaotic in an ”eddy-like” pattern. With increasing Reynoldsnumber the range of scales (or eddy-sizes) present in the flow also increases. Sincethe macro-structures, the large scales, are confined by the geometry of the flow, inthis case the diameter of the pipe, it means that the broadening of the scales can onlybe achieved by introducing smaller scales.

1.2 Large and small scales

One of the most persistent hypotheses concerns the small scales of turbulence andstems from the observation of the break-up of eddy’s. In 1922 Lewis Richardson madethe assumption that turbulence is organised as an hierarchy of eddies of various scales.The biggest eddies, which are anisotropic and have the size of the flow geometry, breakup in smaller eddies which on their turn break up in even smaller eddies, etc. Thiscascade process continues until viscous forces can no longer be neglected. At thesmallest scales the incoming energy is dissipated into heat. The flux of energy thatgoes trough this cascade process is called the dissipation rate ε.

It was Kolmogorov who derived the now famous two similarity hypotheses forthese small scales in 1941 [45]. To get to his hypotheses he made an importantaddition to the ideas of Richardson. He assumed that during the cascade process theanisotropy is gradually lost and from a certain eddy size the scales are homogenousand isotropic. So, in spite of the fact that the mean flow and large-scale eddiesfor any real flow are nonhomogeneous and anisotropic, there is a range of scaleswhere the flow is locally homogeneous and locally isotropic. With these ingredients,Kolmogorov proposed in the first hypothesis that the regime with sufficiently small-scale components of the velocity of any flow with sufficiently high Reynolds numberwill be universal, that means independent of the type of flow, and determined by two

4 Introduction

parameters only: 〈ε〉, the mean dissipation rate, and ν, the coefficient of viscosity.Using dimensional arguments the smallest scale where the viscosity plays an important

role is called the Kolmogorov scale and is given by η =(ν3/ 〈ε〉

)1/4. The second

hypothesis assumes that the universal statistical regime extends to scales which aremuch larger than η but much smaller then the external length scale L0. This universalregime he called the ”inertial subrange”, where the viscosity will no longer play anyrole but is fully determined by the average dissipation rate 〈ε〉. From these hypothesesseveral statistical properties of the small-scale turbulence can be deduced and hishypotheses created an avalanche of follow-up research even until today. The conceptof universality of the small scales has been a great leap forward.

This might be an interesting view point from a physicist, it will however notsolve any problem encountered in practice. From an engineering point of view, theproblem of turbulence would be solved if a statistical description (or statistical model)is capable of predicting the most interesting features of the flow under consideration.For example, if a pollutant is emitted from an industrial chimney, the engineer wouldbe satisfied if the statistical model would predict the dispersion of the pollutant inthe correct way. And industry is much helped if the drag force on a flying jetlineror motorcar could be accurately determined. In general we can say that for practicalapplications the large-scale components of the turbulence play the most importantrole (although these are influenced by the smaller scales). These components are thekey factor in the transfer of momentum and heat. They depend considerably on thegeometry of the boundaries of the flow and will be different for different types of flow.It is for these large-scale components that better theoretical models are needed.

1.3 Numerical and experimental techniques

For turbulent motion, an enormous number of degrees of freedom are exited. Soeven when the initial solution is known with infinite accuracy, it is unlikely that amathematical expression for the time-dependence of the velocity and pressure fieldsis found for any kind of turbulent flow. Then there remain two possible approaches tocollect information from the flow, the first option being experimental techniques likeHot Wire Anemometry (HWA), Laser Doppler Anemometry (LDA) or Particle ImageVelocimetry (PIV), where HWA is generally still the most used method applicablefrom low to very large Reynolds numbers. The remainder of this thesis, however,will focus on the second option: numerical solution of the flow field. Turbulence isa phenomenon occurring in solutions of the Navier-Stokes equation and therefore itshould be possible to solve a turbulent flow numerically. If the geometry, e.g. a circularpipe, is discretised in space and time using a computational grid, the Navier-Stokesequations reduce to a set of algebraic equations. Furthermore, if an appropriate initialcondition and boundary conditions are chosen we can obtain a numerical simulationwithout any modelling assumptions. The result is a turbulent field evolving in time,provided that the Reynolds number is high enough, from which statistical propertiescan be calculated. This approach is called ’direct numerical simulation’ (DNS). Thismethod has the advantage that a complete three dimensional field of the velocity andpressure is available. Moreover, statistical properties can be calculated that are up

1.4 Statistical description of turbulence 5

till now impossible to measure in an experiment. A nice review paper on DNS isfrom Moin and Mahesh [56]. Although this approach may seem the solution to all(engineering) problems there is one serious drawback connected to this method. The

range of scales that has to be solved increases proportional to (L0/η)3, where L0 is

an external length scale and η is the Kolmogorov length scale. A scaling expression

for this ratio is easily found. For the Kolmogorov scale we can write η =(ν3/ε

)1/4,

where ν is the kinematic viscosity, and based on the cascade assumption we know thatthe dissipation rate should scale like ε ∼ σ3/L0, where σ is the standard deviationof velocity fluctuations. This means that (η/L0) = Re−3/4, where the Reynoldsnumber is based on σ and L0: Re = σL0/ν. From this derivation we see thatthe number of grid points Nc needed in the simulation depends on the Reynoldsnumber: Nc ∼ (L0/η)

3 ∼ Re9/4. This means that with increasing Reynolds numbersthe number of grid points needed to fully resolve all turbulent motions grows veryfast. Using nowadays supercomputers the maximum Reynolds number that can bereached is of the order 104, which is still relatively low. And even at these relativelylow Reynolds numbers the computational times are of the order of weeks/months onthese supercomputers. Moreover, the complexity of the flow geometry that can besolved has increased during the past decades but is still relatively limited. All theselimitations combined have made DNS a research tool rather than an engineering tool.

1.4 Statistical description of turbulence

Since a single realization from a DNS calculation is not very useful, the most usedmethod to study turbulence is by means of statistical averages. These averages canbe obtained either by DNS or by performing measurements. In this case the phe-nomenology of turbulence is characterised by statistical quantities like structure- andcorrelation functions and probability density functions. With averaging however, in-formation is lost and the full process of all realizations cannot be recovered.

As we saw in the previous section, large-scale components are responsible for thetransport in turbulent flow. Important work on the theories of the large-scale com-ponents of turbulence has been done by e.g. Taylor [90] [92], Prandtl [69] and VonKarman [43]. These theories are based on the assumed resemblance between turbulentfluctuations and molecular chaos. From this early work originates the idea to describethe transport of turbulence as a diffusion process. Expressions for transport of matter,momentum and energy are formulated in a similar way as the phenomenological de-scriptions of diffusion of matter: the viscous transport of momentum and conductionof heat by the random motion of molecules. However, these theories of large-scaleturbulence are semi-empirical. The phenomenological descriptions are postulated andtheir validity has yet to be proven.

Taylor introduced a mathematical idealization called homogeneous isotropic tur-bulence, for which it is possible to derive expressions for statistical quantities whichexplain important aspects of turbulent flow. Taylor [91] concluded that the diffu-sion of matter in homogeneous isotropic turbulence is related to the time correlationof the fluctuating part of the velocity of marked fluid particles. The hypotheses ofKolmogorov discussed in the previous section certainly helped developing the idea of

6 Introduction

studying homogeneous isotropic turbulence. A comprehensive treatment of homoge-neous isotropic flow is given by Batchelor [10].

Another important classical work on statistical fluid mechanics is by Monin andYaglom [58] and [59]. In this book also the first probability density techniques ofMonin [57] and Lundgren [55] are discussed. Despite the numerous efforts, funda-mental questions on the theory of statistical turbulence have remained unanswered.Nevertheless, it has been widely applied in numerical methods, the so called ”ReynoldsAveraged Navier-Stokes” (RANS) models. In this case not all the turbulent motionsare resolved, as with a DNS, but only average properties, using empirical informationto provide a closure for this description.

1.5 Stochastic modelling

The most often used method to describe fluid properties is an Eulerian method. In anEulerian description of the flow, properties like the velocity, pressure and temperatureare evaluated at fixed points in space. Examples of such a description are HWA orLDA measurements at a fixed point or the grid points of a DNS as discussed in theprevious section. A different way to describe properties is not in a fixed coordinatesystem, but moving along with a finite size fluid parcel (also called a fluid particle) thatmoves with the flow. This is called a Lagrangian description. This way of describingturbulent flows has some physical advantages that are especially important in studyingmixing and dispersion, but is of course still hampered by the same limitation of havingno analytical solutions. A review article on Lagrangian investigations of turbulencehas been written by Yeung [114]. In a Lagrangian framework the position of a fluidparticle x(t) is a function of the initial position and t only, which is the time aftermarking the fluid particle. The velocity of a fluid particle is equal to the Eulerianvelocity u(x, t) at the current position of the particle so that

dx

dt= u(x, t). (1.3)

An interesting way to model Lagrangian statistics is by means of a stochastic equation.This is interesting compared to DNS, since for this approach there is no need to solvethe flow explicitly which reduces the computational effort drastically.

1.5.1 Diffusion theory or Markov process for particle position

Probably the first who used stochastic equations for turbulent flows was Taylor [91].To investigate the dispersion in turbulent flows, he introduced a stochastic model forthe position of a fluid particle x(t). The equation that is solved for this model is

dxµ

dt= Aµjxj + Bµjwj(t), (1.4)

where x is the position of a fluid particle, Aµj is called the drift tensor or dampingtensor, Bµj is the diffusion tensor and wj(t) is Gaussian white noise. In 1908 Langevinused this equation to model the velocity of particles undergoing Brownian motion.

1.5 Stochastic modelling 7

That is why nowadays stochastic models are sometimes also called Langevin mod-els. These stochastic models treat the position and velocity of a fluid particle jointlyas continuous Markov processes. A Markov process is a stochastic process in whichthe distribution of future states depends only on the present state and not on howit arrived in the present state. Under the assumption that evolution times are largecompared to the correlation time of the underlying stochastic process, the variables ofturbulent flow are approximated by those of a Markov process. These processes canbe represented in two equivalent forms: as a stochastic differential equation for theparticle velocity and position or as a Fokker-Planck equation for particle concentra-tion. In case the Markov assumption is applied to the displacement of a marked fluidparticle in homogeneous uniform turbulent flow, as Taylor did, the Fokker-Planckequation is equivalent to a diffusion equation. This is the underlying process onwhich most of the numerical RANS models, discussed in the previous section, arebased. Nonetheless, there is great uncertainly on the application of diffusion theory.It is unclear how and under which limit conditions the general stochastic process ofturbulent flow might reduce to that of diffusion theory or the Markov approximationof particle displacement and what the precise relationship of these theories is withphenomenological theories.

Instead of postulating the Fokker-Planck equation Brouwers [16] derived it fromthe general kinematic relationship between particle velocity and displacement andapplying exact asymptotical analysis. In this way he was able to derive the Fokker-Planck equation for fluid particle position, which is equivalent to the diffusion equa-tion, for general forms of turbulence including inhomogeneous flow. The conclusionis that the diffusion approximation in case of incompressible flow is valid when ε � 1,where ε = tcU/L0 with tc the Lagrangian correlation time, U a typical large-scalevelocity and L0 the external length scale. This means that for the diffusion limit tohold displacements due to random fluid velocity measured over time periods wherethere exists correlation between these velocities are small compared to the externallength scale of the flow. However, for practical cases of turbulent flow ε = O(1), sothat the limit process ε → 0 by which the diffusion approximation becomes exact doesnot exist. This has the consequence that all models based on the diffusion processresort to approximate methods of analysis and their predictive capacity is limited asε is not small. Since ε = O(1) and hence does not grow with Re, the results arenot completely wrong either. However, far better results can be expected from theMarkov approximation for the velocity discussed in the next section.

1.5.2 Markov process for particle velocity

As mentioned in the previous section, the assumption of a δ-correlated velocity processwill not become exact with increasing Reynolds number. Apart from a model for theposition of a fluid particle as proposed by Taylor, there are stochastic models forthe velocity of a fluid particle. If we restrict ourselves to homogeneous, isotropicturbulence for the moment, the general Langevin equation for the velocity of a fluidparticle is given by

dv′µ

dt= Aµjv

′j + Bµjwj(t), (1.5)

8 Introduction

where Aµj , Bµj and wj(t) are essentially the same as in equation (1.4) but are basedon different statistical properties and v′ is the fluctuating velocity of a fluid particle.Equation (1.5) is referred to as a Markov process for particle velocity and in thiscase the acceleration of a fluid particle is assumed to be δ-correlated. Since the cor-relation time of the acceleration corresponds to the Kolmogorov time tη, physicallythis means that the Kolmogorov time should go to zero. In the limiting process ofRe → ∞ this is true according to Kolmogorov’s scaling [45] since tη ∼ Re−1/2tc. Theproblem is the actual form of the functions Aµj and Bµj in equation (1.5). Equa-tion (1.5) is essentially a Lagrangian description of the flow. In case all relevantLagrangian statistics are known, for example from a DNS calculation, all Langevinparameters in equation (1.5) can be derived from these statistics. But DNS is limitedto moderate values of the Reynolds number. So, information obtained in this wayis only partially helpful in identifying the right form of the Langevin equation beingan asymptotic description of the true stochastic process at large Reynolds numbers.Another way to obtain information if from measurements. But here the problemis the Lagrangian nature of the Langevin equation. Considering the complex mea-surement techniques required to obtain Lagrangian information of a turbulent flow,it seems interesting to look for connections of the Langevin equation with Eulerianbased statistics. These follow from two consistency conditions usually applied. Thefirst of these is consistency with Kolmogorov’s similarity theory, which reduces theBµj function to isotropic fluctuations, i.e. Bµj = (C0ε)

1/2δµj . Here, C0 is calledKolmogorov’s constant and ε is the dissipation rate. The second consistency condi-tion is known as the well-mixed condition. This requirement stems from work doneby Thomson [96] where he showed how to ensure that the damping coefficients Aµj

in the stochastic model are consistent with prescribed, single-time Eulerian statis-tics. A model that satisfies this condition is called a model from the ’well-mixed’class. For homogeneous, isotropic turbulence the only sensible choice of coefficients

is Aµj = −δµj/tc and Bµj =(2σ2

v/tc)1/2

δµj where tc is the Lagrangian correlationtime (or integral time scale) and σ2

v is the variance of the velocity. For this type offlow tc = 2σ2

v/(C0ε), so that Bµj = (C0ε)1/2δµj (see [91] and [67]). Several proper-

ties can be derived from the result for the Lagrangian autocorrelation obtained fromthis Langevin model: ρ(τ) = 〈v′

i(t)v′i(t + τ)〉 = exp (−|τ |/tc). The first property is

that it contains only one time scale tc characteristic for the large, energy containingscales. Secondly, there is no Reynolds number dependence. And finally the slopeof ρ(τ) is discontinuous at τ = 0, indicating that the velocity from the Lagrangianstochastic model is continuous, but not differentiable. These properties also reveal theshortcomings of the Langevin model when finite Reynolds numbers are considered.

In the past, the velocity models have been compared with homogeneous andisotropic turbulence, usually with good results. This is due to the property thatfor this type of flow analytical relations can be derived for the damping coefficientsAµj (sometimes also referred to as the ’drift’-term) in the stochastic model. WhileThomson’s approach ensures consistency with the prescribed, single-time Eulerianstatistics, it does not give a unique stochastic model in two or three dimensions whenthe flow is not isotropic and homogeneous. This is called the ’non-uniqueness’ prob-lem, which is related to inhomogeneous flow, and means that the form of the damping

1.6 Purpose and outline of this thesis 9

functions Aµj in equation (1.5) is not known. For real turbulence they are generallyanisotropic, inhomogeneous (position dependent) and possibly nonlinear in the veloc-ity (as opposed to the linear form assumed in equation (1.5)). The non-uniquenessproblem reduced many investigations to finding the ’best performing’ model out ofthe many that are consistent with the Eulerian statistics (see e.g. [96] and [112]). Asa consequence, models were tested on their outcome by comparing the results withexperimental data or Lagrangian data from a DNS.

Another question is whether the non-uniqueness problem can be solved by a con-nection between Eulerian and Lagrangian velocity correlations in time. If such aconnection exists, the damping coefficients could be derived from the slope of theEulerian velocity correlation functions. Unfortunately, an effect which is known assweeping of small scales by large scales prevents this. The effect of sweeping wasfirst shown by Tennekes [95] for the case of homogeneous, isotropic turbulence. Ithas recently been generalised to all forms of turbulence (see Brouwers [17]). Oneof its implications is that the derivatives of the Eulerian correlation functions dif-fer fundamentally from those of Lagrangian data. This prohibits the determinationof the damping coefficients in the Langevin equation from information of Euleriancorrelation or structure functions.

1.6 Purpose and outline of this thesis

The main objective of this thesis is to make a contribution in the understandingof the stochastic nature of inhomogeneous turbulence. More specifically, we willstudy Eulerian and Lagrangian statistics following a numerical approach based ondirect numerical simulations of developed pipe flow. For this type of flow manyexperimental results are available to validate the results. Moreover, the flow is thesimplest inhomogeneous flow, which is realizable in experiments. With the numericalresults we then attempt to

• assess statistical properties of fully developed turbulent pipe flow

• attempt to reach Reynolds numbers as high as nowadays possible

• focuss on the physical parameters that are important for developing stochasticmodels

• check important theoretical hypotheses of inhomogeneous turbulence, i.e. Kol-mogorov’s similarity and Tennekes sweeping hypotheses

The DNS that has been developed is based on a pseudo-spectral method. In Chapter2 this method to solve the Navier-Stokes equations is extensively discussed. With thisnumerical code large databases are generated at two different Reynolds numbers.

Recently, Brouwers [16] derived an asymptotic expansion, which describes La-grangian statistical properties in terms of Eulerian velocity statistics in a frame whichmoves with the local average fluid velocity. To get to this result he made use of theexpansion briefly discussed in section 1.5.1. With the statistics calculated in Chapter

10 Introduction

2, the validity of the relation between Lagrangian and Eulerian statistics is tested forseveral statistical properties of interest. This will be the subject of Chapter 3.

Directly related to Lagrangian stochastic models is the acceleration of a fluid par-ticle. Stochastic models try to simulate the evolution of fluid particles under theassumption that the Reynolds number is high enough so that there is a distinctseparation between the smallest scales of the flow and the large, energy containingscales. In this case the fluid particle acceleration can be modelled by a δ-correlatedMarkov process. The assumption that the acceleration becomes uncorrelated if theRe−number is high enough is a simplification and resulted in attempts to quantifyacceleration statistics using both numerical and experimental techniques. The La-grangian acceleration is defined by the material derivative of the velocity vector:

a ≡ Du

Dt=

∂u

∂t+ (u · ∇)u, (1.6)

which can be written in a different form using the Navier-Stokes equations. Thereis discussion whether the acceleration variance as a function of the Reynolds numberobeys Kolmogorov scaling. Tennekes’ hypothesis [95] assumes that the total (or La-grangian) acceleration a is small compared to the local and convective accelerationsin equation (1.6). Due to the increase of computer power, these findings were laterconfirmed by DNS results of incompressible, forced, isotropic turbulence (Tsinober etal. [101], Vedula and Yeung [103]). Tennekes’ sweeping effect was recently generalisedto inhomogeneous turbulence by Brouwers. But these results have yet to be confirmedby DNS. Were velocity statistics the subject of Chapter 3, the acceleration statisticsare the subject of Chapter 4. Our primary interest in this chapter is to investigate thevarious contributions to the acceleration term obtained from DNS of inhomogeneouspipe flow and to compare the results with those obtained from isotropic turbulence.Brouwers’ [17] derived theoretical predictions of these acceleration statistics will beverified as well.

Stochastic models rely on the Kolmogorov hypotheses. For the Kolmogorov hy-potheses to hold, the break up of large eddies into smaller eddies should be accompa-nied by local isotropy of the flow although the flow itself can be strongly anisotropic.Kolmogorov’s hypotheses from 1941 predict that spatial structure functions of orderp, defined by Dp(r) = 〈(u(x + r) − u(x))

p〉, are related to the average dissipationrate and separation distance r = |r| by

Dp(r) = Cp [〈ε〉 r]p3 . (1.7)

In this equation the Cp are universal constants. The structure functions scale withr as Dp(r) ∼ rζp , with ζp = p/3. Nevertheless, there are strong indications thatthere are deviations from Kolmogorov’s theory so that ζp 6= p/3. It was first objectedby Landau, shortly after the presentation of the Kolmogorov hypotheses in 1941,that this scaling could not be correct due to intermittency: the effects of stronglyfluctuating values of the local energy dissipation. In Chapter 5 we will first quantifythe ’level’ of local isotropy, after that we will investigate the intermittent behaviourin further detail.

In the final chapter, a stochastic model as discussed in section 1.5 is derived forpipe flow. In this chapter only a linear model will be considered. To calculate the

1.6 Purpose and outline of this thesis 11

unknown damping coefficients we will make use of the Lagrangian data available. Inthis model also the assumed universal Kolmogorov constant appears for which we willdetermine a value using the Lagrangian second order structure function. To test themodel we will compare the results with DNS results of a point source in turbulentpipe flow as calculated by Brethouwer [15].

12 Introduction

Chapter 2

Numerical Techniques

2.1 Introduction

In this chapter the numerical methods we used to perform a Direct Numerical Simu-lation (DNS) of pipe flow are highlighted. This is certainly not the first attempt toperform a DNS of such a flow. The first DNS results ever presented for pipe flow byEggels [27] (and Eggels et al. [28]) were obtained with a finite volume method. As wesaw in the previous chapter, DNS is always restricted to a low Reynolds number. InEggels’ case this was 5300 based on the bulk velocity and diameter. Wagner et al. [106]used the same numerical method, but also performed simulations at higher Reynoldsnumbers. Loulou [54] introduced a hybrid B-Spline/Fourier expansion method andused the same Reynolds number as Eggels. Shan et al. [82] used a pseudo-spectralmethod to study transition. Furthermore, there are numerous experiments performedfor pipe flow. In the low Reynolds regime, the regime of the current DNS results,there are PIV (Particle Image Velocimetry) and LDA (Laser Doppler Anemometry)measurements available of Westerweel et al. [109], Den Toonder [99] and Durst [26].

The goal of this chapter is to explain the functioning of the numerical method andto demonstrate its appropriateness to calculate the desired statistical quantities. Thepseudo-spectral approach used in this case has a great resemblance with the methodof Shan et al. [82]. In section 2.2 the spatial and temporal discretisation are discussed,including the treatment at the boundaries and selection of an initial condition. Insection 2.3 some standard statistical properties are compared with other DNS resultsand experiments when possible. With the current numerical code fluid particles aretracked. The selection of interpolation and time integration methods are discussed aswell as the statistical accuracy of quantities of interest in section 2.4. In this samesection we will also discuss the calculation and accuracy of variables in the Eulerianmoving frame.

14 Numerical Techniques

2.2 Equations of motion and numerical approach

2.2.1 Fluid equations

In this section the numerical methods which are used to solve the Navier-Stokes equa-tions and conservation of mass equation are discussed. Throughout the whole thesiswe are only concerned with incompressible turbulent pipe flow, which means that thedensity is constant. Because of the cylindrical geometry the choice for cylindrical co-ordinates and cylindrical velocity components is natural. We use ωr, ωφ and ωz for thecylindrical components of the vorticity, which are defined as ωr = 1

r ∂uz/∂φ−∂uφ/∂z,ωφ = ∂ur/∂z−∂uz/∂r and ωz = 1

r ∂/∂r(ruφ)− 1r ∂ur/∂φ respectively. ∆ is the Laplace

operator and P is total pressure P = p + 12u2 with p the static pressure. Then the

governing equations can be written in the form

∂ur

∂t+ ωφuz − ωzuφ +

1

ρ

∂P

∂r=

η

ρ

(

∆ur −ur

r2− 2

r2

∂uφ

∂φ

)

∂uφ

∂t+ ωzur − ωruz +

1

ρ

1

r

∂P

∂φ=

η

ρ

(

∆uφ − uφ

r2+

2

r2

∂ur

∂φ

)

∂uz

∂t+ ωruφ − ωφur +

1

ρ

∂P

∂z=

η

ρ∆uz + f

(2.1)

ρ

(1

r

∂

∂r(rur) +

1

r

∂uφ

∂φ+

∂uz

∂z

)

= 0. (2.2)

For the numerical simulation these equations are made dimensionless. In this man-ner the results can be easily compared with other numerical results and experiments,provided that the Reynolds number is the same in both cases. We introduce a dimen-sionless velocity, time, density and pressure: ui = ui/u∗, where i is one of the threedirections in cylindrical coordinates and u∗ is some reference velocity, t = tu∗/R,ρ = ρ/ρ∗, where ρ∗ is a reference density and p = p/(ρ∗u∗2). Substituting this intothe Navier-Stokes (2.1) and continuity (2.2) equations we get

∂(uru∗)

∂(tR/u∗)+

u∗2

R(ωφuz−ωzuφ)+

1

ρ

∂(P u∗2)

∂(rR)=

u∗

R2

η

ρρ∗

(

∆ur−ur

r2− 2

r2

∂uφ

∂φ

)

∂(uφu∗)

∂(tR/u∗)+

u∗2

R(ωzur−ωruz)+

1

ρ

∂(P u∗2)

rR∂φ=

u∗

R2

η

ρρ∗

(

∆uφ−uφ

r2+

2

r2

∂ur

∂φ

)

∂(uzu∗)

∂(tR/u∗)+

u∗2

R(ωruφ−ωφur)+

1

ρ

∂(P u∗2)

∂(zR)=

u∗

R2

η

ρρ∗∆uz+f

(2.3)

1

rR

∂

∂(rR)(rurRu∗) +

1

rR

∂(uφu∗)

∂φ+

∂(uzu∗)

∂(zR)= 0. (2.4)

We assume that the dimensionless density ρ equals unity. If we then multiply theNavier-Stokes equations on both sides with R/u∗2, the term on the r.h.s outside the

2.2 Equations of motion and numerical approach 15

brackets becomes (R/u∗2)(u∗/R2)(η/ρ∗) = ν/(u∗R) = 2/Re, where ν = η/ρ∗. TheReynolds number is defined as Re = u∗D/ν where D is the diameter of the pipe.For convenience we now omit all tildes in the dimensionless quantities and choose theradius R equal to unity, hence equations (2.3) and (2.4) reduce to

∂ur

∂t+ ωφuz − ωzuφ +

∂P

∂r=

2

Re

(

∆ur −ur

r2− 2

r2

∂uφ

∂φ

)

∂uφ

∂t+ ωzur − ωruz +

1

r

∂P

∂φ=

2

Re

(

∆uφ − uφ

r2+

2

r2

∂ur

∂φ

)

∂uz

∂t+ ωruφ − ωφur +

∂P

∂z=

2

Re∆uz + f

(2.5)

1

r

∂

∂r(rur) +

1

r

∂uφ

∂φ+

∂uz

∂z= 0. (2.6)

The mean pressure gradient in the axial direction, −f , is not included in p. Withoutadding this term to the Navier-Stokes equations the turbulence would decay, the flowwould become laminar and finally seize to exist. It is the driving force through whichenergy is put into the system. We are free to choose how to implement this terminto our program, but there are two possible approaches. One is keeping the pressuregradient constant over time. This would result in a slightly fluctuating volumetricflow through the pipe. An alternative would be to keep this volumetric flow constantover time. It is preferable to implement the second method in the numerical codebecause of its faster convergence to fully developed turbulence in comparison to thefirst method. In order to obtain a constant total flow rate through the pipe we take

f = − 4

Re

∂〈uz〉∂r

∣∣∣∣r=R

,

where the brackets denote averaging over the homogeneous directions. This lastrelation follows from taking the volumetric integral of the axial component of theNavier-Stokes equations and requiring

∫ ∫

V

∫∂uz

∂t dV = 0. The forcing term is closely

related to the wall shear velocity, which in our case is equal to uτ =√

−f/2u∗.To solve equations (2.5) and (2.6) numerically, essentially the same method is used

as discussed by Shan et al. [82]. A finite part of the pipe of length L is consideredusing periodic boundary conditions in the axial direction. Combined with the naturalperiodicity in the tangential direction there are two periodic directions and the choicefor a spectral method is obvious. In this case the spatially fluctuating velocity com-ponents are described by a Fourier expansion. In the radial direction, where there isno periodicity, an expansion based on Chebyshev polynomials is used.

Each velocity component and the pressure are thus expanded as

u(r, φ, z) =

Mφ/2−1∑

kφ=−Mφ/2+1

Mz/2−1∑

kz=−Mz/2+1

ukφ,kz(r)e(ikφφ+2πikzz/L). (2.7)


xN x0

Figure 2.1. Chebyshev distribution of the collocation points. They are the projection ofequally spaced points on the unit circle projected on the x-axis.

In this way a hybrid method appears: Fourier-Galerkin in the two periodic directionsand Chebyshev-collocation in the radial direction. Derivatives in the periodic direc-tions can easily be calculated in spectral space, whereas derivatives with respect to rfollow from the Chebyshev derivative matrix (see Canuto et al. [19]). We will addressthis feature in more detail in the next subsection.

To illustrate the ease of calculation of spatial derivatives we will find an expression

for ∂u/∂φ. Using equation (2.7) we can write (using∑Mφ/2−1

kφ=−Mφ/2+1

∑Mz/2−1kz=−Mz/2+1 =

∑∗for convenience)

∂u(r, φ, z)

∂φ=

∑∗ikφukφ,kz

(r)e(ikφφ+2πikzz/L)

=∑∗

ikφ

{<(ukφ,kz

(r)) + i=(ukφ,kz(r))

}e(ikφφ+2πikzz/L)

=∑∗

kφ

{−=(ukφ,kz

(r)) + i<(ukφ,kz(r))

}e(ikφφ+2πikzz/L). (2.8)

The original solution was given by∑∗ {<(ukφ,kz

(r)) + i=(ukφ,kz(r))

}e(ikφφ+2πikzz/L).

So taking a spatial derivative with respect to a periodic direction simply reduces tomultiplication of the original solution with the correct wave number and reversingreal and imaginary parts, where the real part of the derivative gets an extra minussign. The derivative calculated in this way is exact.

Chebyshev polynomials

To discretise the radial direction, a Chebyshev expansion is used. Classical referenceson these polynomials are Fox and Parker [29], and also Canuto et al. [19] give anextensive introduction including many examples. In our case we use a collocation


method. The Chebyshev-Gaus-Lobatto collocation points are given by xj = cos( jπN )

for j = 0, . . . , N , where N +1 is the number of collocation points. These points xj arethe extrema of the Nth-order Chebyshev polynomial TN (x) = cos(N cos−1 x) and aredefined on the interval x ∈ [−1, 1]. The points are the projection of equally spacedpoints on the unit circle projected on the x-axis. Figure 2.1 shows the distribution ofthese points for N = 20.

The first derivative of a function u(x) at the collocation points xj can be calculatedin many different ways. In the collocation method usually the original solution ismultiplied by a matrix to obtain the derivative. This ”derivative” matrix (denotedD1 in case of the first derivative) is computed once during the calculation. The entriesof the derivative matrix D originate from differentiating Lagrange polynomials. Thisis very well documented in Canuto et al. [19]. In a similar way as for the first derivativea second derivative matrix can be constructed (D2).

Differentiation with respect to the radial direction for the mode combination (k, l)is then obtained by

∂uk,l

∂r(rj) =

N∑

i=0

D(1)ji uk,l(ri). (2.9)

If we would use this discretisation directly by projecting it on the radius r ∈ [0, 1]a dense cluster of collocation points would be the result near r = 0 and r = R.This would lead to very small cells near the pipe axis and thus necessitate the use ofvery small time steps. Therefore, the radial direction is divided into several elementsand in each element an expansion into Chebyshev polynomials is adopted. At theinterfaces between the elements the solution is required to be C1, i.e. the velocityand its derivative with respect to r should be continuous. The division of the radialdirection into elements makes it possible to reduce the number of Fourier modes inthe tangential direction in the element containing the axis of the pipe. This modereduction does not influence the global accuracy of the method and the correspondingincrease in grid size alleviates the time step restriction (see Loulou [54]). The left sideof figure 2.2, where the grid is shown, clearly illustrates the subdivision of the radialdirection into 5 elements. The radius of the pipe r ∈ [0, 1] is divided into elements[rp

2 , rp1 ] for p = 1, . . . , Nel, with r1

1 = 1, rNel

2 = 0 and rp2 = rp+1

1 . Here Nel is thenumber of elements and rp

2 and rp1 are the end and starting points of each element

p. In each element a Chebyshev expansion is adopted. Since Chebyshev polynomialsare only defined on x ∈ [−1, 1] we transform from r ∈ [rp

2 , rp1 ] to x ∈ [−1, 1] through

x = 2r/(rp1 − rp

2) − (rp1 + rp

2)/(rp1 − rp

2) or r = x(rp1 − rp

2)/2 + (rp1 + rp

2)/2.

Time integration

The Navier-Stokes equations are integrated in time using a time-splitting method byKarniadakis et al. [44]. Schematically the Navier-Stokes equation can be written as

∂u

∂t+ N(u) + ∇P = L(u) + f ,

where N denotes the nonlinear terms on the l.h.s. of (2.5), L the viscous terms onthe r.h.s. and f the forcing term. A second-order accurate time-splitting method


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−0.2 −0.1 0 0.1 0.2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Figure 2.2. Physical grid. Right: close up near the pipe axis where mode reduction isapplied. For clarity only half the number of grid lines in tangential direction is plotted.

with constant time step ∆t is then given by

un+1/3 = 2un − 12un−1 − ∆t(2N(un) − N(un−1)) + ∆tf

∆Pn+1 = 1∆t∇ · un+1/3

un+2/3 = un+1/3 − ∆t∇Pn+1

un+1 = 23un+2/3 + 2

3∆tL(un+1).

(2.10)

In these formulas the superscript denotes the time level. In the first step the nonlinearterms are treated explicitly. The products of vorticity and velocity are calculated ina pseudo-spectral way, where fast-Fourier transforms (FFT) are used to transformfrom spectral to physical space and back. A common problem with transformingthe solution between physical and spectral space is the possibility of aliasing errors:wrongly interpreted frequencies by transforming back from physical to spectral space.To prevent aliasing the 3/2-rule is applied, which is also known as ”padding”. Thisimplies that before transforming the solution to physical space extra Fourier modesequal to zero are added in both directions. In each dimension the total number ofextra modes is the original number divided by two, hence 3/2-rule. After transformingthe products back to Fourier space the extra modes are disregarded.

In the second step a Poisson equation for the pressure is solved to ensure thatthe solution after the third step satisfies the continuity equation. Due to the globalcharacteristics of the Chebyshev collocation method very often a matrix has to besolved. This is a full matrix with a block-like structure because of the division of theradius into elements. If we look for example at the Poisson equation for the pressurein cylindrical coordinates

∂2p

∂r2+

1

r

∂p

∂r+

1

r2

∂2p

∂φ2+

∂2p

∂z2=

1

r

∂

∂r(rur) +

1

r

∂uφ

∂φ+

∂uz

∂z, (2.11)


with the Fourier-Chebyshev discretisation it reduces to a problem which can be solvedfor each individual combination of Fourier modes using

4

(rp1 − rp

2)2

Np∑

i=0

Dp(2)ji pk,l(ri) +

2

(rp1 − rp

2)21

ri

Np∑

i=0

Dp(1)ji pk,l(ri)

−k2

r2j

pk,l(rj) −(

2πl

L

)2

pk,l(rj) = fk,l(rj), (2.12)

where fk,l(rj) is the r.h.s. of equation (2.11). So in each element p the Poisson equa-tion is solved individually and this is done for each Fourier combination separately. Ateach element interface we have the continuity conditions up

k,l(rp2)−up+1

k,l (rp+11 ) = 0 and

2/(rp1−rp

2)∑Np

i=0 Dp(1)Npi uk,l(r

pi )−2/(rp+1

1 −rp+12 )

∑Np+1

i=0 Dp+1(1)0i uk,l(r

p+1i ) = 0, so that

the velocity itself and its derivative are continuous at the element boundaries. Thisproblem can be written as a matrix-vector equation with N + 1 rows and columns.Assuming we have 3 elements (for convenience, in reality there are five elements) thematrix has the following structure

1 0 . . . 0 0∗ ∗ ∗ ∗ ∗∗ ∗ A1 ∗ ∗ 0 0∗ ∗ ∗ ∗ ∗0 0 . . . 0 1 −1

∗ ∗ D1(1)N1

∗ ∗ ∗ ∗ −D2(1)0 ∗ ∗

∗ ∗ ∗ ∗ ∗0 ∗ ∗ A2 ∗ ∗ 0

∗ ∗ ∗ ∗ ∗0 0 . . . 0 1 −1

∗ ∗ D2(1)N2

∗ ∗ ∗ ∗ −D3(1)0 ∗ ∗

∗ ∗ ∗ ∗ ∗0 0 ∗ ∗ A3 ∗ ∗

∗ ∗ ∗ ∗ ∗∗ ∗ D

3(1)N3

∗ ∗

.

Not all rows are used to solve the Poisson equation. Some rows of the matrixare required to supply the boundary conditions at the wall, pipe axis and elementboundaries. The first row of the matrix imposes the boundary condition at r = R,p0,0(r0) = 0. For (kφ, kz) 6= (0, 0) the derivative of the pressure should equal 0 andthe first row of the matrix will contain a part of the derivative matrix D. In a sim-ilar way the last row of the matrix contains the boundary condition at r = 0. Theelements create a matrix which has a block-like structure. An algorithm which treatsthis matrix as a band-matrix is used to solve this set of equations.

In the final step the linear, viscous terms in the Navier-Stokes equations are treatedimplicitly. The viscous parts of the equations for ur and uφ are coupled, but they canbe decoupled by introducing u± = ur ± iuφ (see [62]). Moreover, the equations for


different Fourier modes are completely decoupled, so that a one-dimensional equationfor each Fourier mode and each velocity component results. These one-dimensionalproblems are solved by a direct method as indicated with the Poisson equation exam-ple on page 18. The boundary conditions for the velocity and pressure are discussedin the next section.

2.2.2 Boundary conditions

Equations (2.5) and (2.6) need to be supplied with appropriate boundary conditionsfor the three velocity components and the pressure. There are two boundaries in thepipe flow simulation: one physical boundary at the wall of the pipe at r = R = 1 andone computational boundary at the pipe axis at r = 0. Inflow and outflow boundaryconditions are not needed because of the periodicity of the flow. First, we look at thephysical boundary at r = R. The no-slip boundary condition for the velocity is used:u = 0. So at j = 0, ur = uφ = uz = 0 for all kφ and kz. A point that requires specialattention is the boundary condition for the pressure at the wall of the pipe. A suitableboundary condition can be derived from the radial component of the Navier-Stokesequation (2.5). Since ur = uφ = uz = 0, all nonlinear terms, the time derivative andthe curvature terms in the linear part of the equation ( 1

r2 ur and − 2r2 ∂uφ/∂φ) vanish.

Also ∂2ur/∂φ2 = 0, ∂2ur/∂z2 = 0 and ∂uφ/∂φ = 0 at the wall. The Navier-Stokesequation at r = R reduces to

∂P

∂r

∣∣∣∣r=R

=2

Re

(∂2ur

∂r2+

1

r

∂ur

∂r

)∣∣∣∣r=R

.

Furthermore, from the continuity equation (2.6) follows that ∂ur/∂r = 0 since

1

r

∂uφ

∂φ=

∂uz

∂z= ur = 0,

hence the boundary condition for the pressure at r = R can be obtained from

∂P

∂r

∣∣∣∣r=R

=2

Re

(∂2ur

∂r2

)∣∣∣∣r=R

.

To apply this boundary condition the velocity has to be known at the next time step.Since this is not possible with the current time stepping scheme, the solution at thenew time level is found by linear extrapolation from the solution at the two previoustime levels, i.e.

∂Pn+1

∂r=

2

Re

(

2∂2un

r

∂r2− ∂2un−1

r

∂r2

)

.

Karniadakis et al. [44] pointed out that the boundary condition in this form can causenumerical instability. It can be improved by invoking the continuity equation, whichleads to

∂Pn+1

∂r= − 2

Re

(

2

[1

r

∂2uφ

∂r∂φ+

∂2uz

∂r∂z

]n

−[1

r

∂2uφ

∂r∂φ+

∂2uz

∂r∂z

]n−1)

.


In order to obtain a unique solution this boundary condition cannot be used for the(kφ, kz) = (0, 0) mode. Instead the mean pressure at the wall of the pipe is prescribed.The boundary condition presented in this way is no longer exact due to the linearextrapolation. This is the main reason that ∇ · u = 0 is not exactly satisfied. Theerror is however very small as proven by Karniadakis et al. [44].

Now we will focus on the numerical boundary condition at r = 0, which gives riseto some problems. The most important issue is the fact that the velocity componentsand pressure cannot depend on the angle φ at r = 0. In most cases this meansthat for kφ 6= 0 the Fourier modes are set to zero. However, we will discuss allvelocity components and the pressure in more detail separately, starting with themost simple one. The boundary condition for the axial velocity component followsfrom the regularity of the velocity, i.e. uz should be continuous and differentiable.This means that uz = 0 for kφ 6= 0 ∧ ∀kz, ∂uz/∂r = 0 for kφ = 0 ∧ ∀kz. Theboundary condition for the pressure is exactly the same as for the axial velocitycomponent: P = 0 for kφ 6= 0 ∧ ∀kz and ∂P /∂r = 0 for kφ = 0 ∧ ∀kz. For ur and uφ

the situation is more difficult, since they are not defined at r = 0. It is required thatux and uy must behave well near r = 0

{ux = ur cos φ − uφ sin φuy = ur sinφ + uφ cos φ.

(2.13)

In Fourier space this becomes (where we omit the summation over kz for clarity)

ux =1

2

Mφ/2−1∑

kφ=−Mφ/2+1

[(

urkφ+ iuφkφ

)

ei(kφ+1)φ+ +(

urkφ− iuφkφ

)

ei(kφ−1)φ]

=1

2

Mφ/2−1∑

kφ=−Mφ/2+1

[

u+kφei(kφ+1)φ + u−kφ

ei(kφ−1)φ]

(2.14)

uy =1

2i

Mφ/2−1∑

kφ=−Mφ/2+1

[(

urkφ+ iuφkφ

)

ei(kφ+1)φ+ +(

urkφ− iuφkφ

)

ei(kφ−1)φ]

=1

2i

Mφ/2−1∑

kφ=−Mφ/2+1

[

u+kφei(kφ+1)φ + u−kφ

ei(kφ−1)φ]

. (2.15)

Here we have introduced u± which are defined u± = ur ± iuφ. Now ux and uy aresingle-valued at r = 0 if u+kφ

= 0 if kφ 6= −1 and u−kφ= 0 if kφ 6= 1. The remaining

boundary conditions are similar as for uz: ∂u+/∂r = 0 for kφ = −1 ∧ ∀kz and∂u−/∂r = 0 for kφ = 1 ∧ ∀kz. In the implicit, viscous step of scheme (2.10) u+ andu− are also used, so that these boundary conditions are easily implemented. However,in the first and second step of this scheme ur and uφ are used. Then we can use

urkφ+ iuφkφ

= 0, kφ ≥ 0 ∧ ∀kz,

u∗rkφ

+ iu∗φkφ

= 0, kφ 6= 1 ∧ ∀kz,∂∂r u∗

rkφ+ i ∂

∂r u∗φkφ

= 0, kφ = 1 ∧ ∀kz,

(2.16)


which gives u∗rkφ

= u∗φkφ

= 0 for kφ 6= 1. For kφ = 1 we get

<(ur) − =(uφ) = 0=(ur) − <(uφ) = 0<(∂ur/∂r) − =(∂uφ/∂r) = 0

− =(∂ur/∂r) + <(∂uφ/∂r) = 0

(2.17)

Initial solution

The calculation is started from a random field superposed on an approximate meanfield with the axial velocity component given by a logarithmic velocity profile. Therandom field is chosen in such a way that it satisfies the continuity equation and thatthe lowest four Fourier modes in both periodic directions are unequal to zero. In thefirst time step, scheme (2.10) cannot be applied since only one field is available. Afirst-order time-splitting scheme is used instead. After a large number of time steps astate of fully-developed turbulence is reached. From that time onwards averaged flowquantities can be calculated.

2.3 Results

In this section results are shown for simulations at two different Reynolds numberswhich will be denoted by DNS1 and DNS2. The results presented in this sectionhave no other purpose than verifying the numerical method discussed in the previoussections. Results from previous researches are used to compare the present results.The DNS results by Eggels [27] (and Eggels et al. [28]), Wagner et al. [106] andLoulou [54] are included. Furthermore, there are numerous experiments performedfor pipe flow. In the low Reynolds regime, the regime of the current DNS results,Westerweel et al. [109] performed PIV (Particle Image Velocimetry) and LDA (LaserDoppler Anemometry) measurements. The Reynolds numbers based on the bulkvelocity, Ub, are set to 5300 and 10312 respectively for the two simulations DNS1and DNS2. The length L of the pipe is taken L = 10 which corresponds to 5D. Thenumber of Chebychev points and Fourier modes used in the simulation are (109×128×128) and (151×256×384) for respectively the radial, tangential and axial directions atthe two different Reynolds numbers. The largest grid spacings in tangential directionare at the wall due to the cylindrical coordinate system. In the radial direction thelargest spacings are found in the element containing the pipe axis. The simulations arerun for approximately 10D/uτ , with dimensionless fixed time steps of ∆t = 2.0×10−4

(≈ 1.0×10−4D/uτ ) and ∆t = 1.0×10−4 (≈ 5.0×10−5D/uτ ) for runs DNS1 and DNS2respectively, which leads to more than 100 statistically independent fields. This is farmore than the 46 fields of Loulou [54] and 41 fields of Eggels et al. [28]. Averaging isperformed over the periodic directions and time.

Table 2.1 summarises some simulation and experimental characteristics, includingthe grid spacing values expressed in wall units. Our axial resolution for run DNS1seems somewhat low. However, the same simulation was also performed on a (172 ×192× 192) grid and did not reveal significant differences. The reason that we will not

2.3 Results 23

Table 2.1. Overview of numerical and experimental results for various mean flow propertiesand grid resolution.

DNS1 DNS2 Eggelsa Wagnerb Loulouc PIVd LDAd

Nr 109 152 96 70 72 - -Nφ 128 256 128 240 160 - -Nz 128 394 256 486 192 - -

∆r+min 0.11 0.09 0.94 0.64 0.39 - -

∆r+max 4.03 4.47 1.88 7.68 5.70 - -

R∆φ+max 8.89 7.95 8.84 8.36 7.50 - -

∆z+ 14.10 8.43 7.00 6.58 9.90 - -Rec = UcD/ν 6954 13181 6950 13210 7248 7100 7200Reb = UbD/ν 5299 10312 5300 10300 5600 5450 5450Reτ = uτD/ν 362 647 360 640 380 366 371

Uc/uτ 19.19 20.38 19.31 20.64 19.12 19.38 19.39Ub/uτ 14.63 15.94 14.73 16.09 14.77 14.88 14.68Uc/Ub 1.31 1.28 1.31 1.28 1.29 1.30 1.32

Cf (×10−3)e 9.35 7.87 9.22 7.70 9.16 9.03 9.28δ∗/Rf 0.127 0.115 0.127 0.115 0.121 0.124 0.130θ∗/Rg 0.069 0.070 0.068 0.071 0.066 0.068 0.071

H = δ∗/θ∗ 1.85 1.65 1.86 1.63 1.84 1.83 1.83

aEggels et al. [28].bWagner et al. [106].cLoulou [54].dWesterweel et al. [109].eThe skin friction is defined as Cf = 2τw/U2

b , where the wall shear stress is given byτw = νduz(r)/dr|r=R.

fThe displacement thickness is defined by δ∗(2R − δ∗) = 2R∫

0

r(1 − uz(r)/uc)dr.

gThe momentum thickness is defined by θ∗(2R − θ∗) = 2R∫

0

ruz(r)/uc(1 − uz(r)/uc)dr.

use this grid resolution here to validate the results is that for this case no Lagrangianparticles were tracked. Values for the skin friction, Cf , displacement thickness, δ,momentum thickness, θ and the shape factor, H, show good resemblance for bothnumerical and experimental results.

Figure 2.3 shows the mean velocity normalised by the wall shear stress velocityas a function of the distance to the wall in wall units y+. A wall unit is defined asy+ = uτ (R − r)/ν. There is a good agreement between the current result at thelower Reynolds number with experimental results and with Eggels et al.. At higherReynolds number the agreement with Wagner et al. [106] is also very good, althoughfor large values of y+ the difference becomes somewhat larger. Also the law-of-the-wall, 〈uz〉+ = y+, and the logarithmic law, 〈uz〉+ = 2.5ln(y+) + C, are included inthe figure. When the constant in this logarithmic law is changed from the generallyaccepted 5.0 for high Reynolds numbers to 5.5, channel flow follows the logarithmicprofile perfectly (see e.g. Moin and Mahesh [56]), even at low Reynolds numbers.


100

101

102

0

5

10

15

20

<uz>+=y+→

<uz>+=2.5ln(y+)+5.5→

y+

〈uz〉/

uτ

Figure 2.3. Mean velocity normalised by uτ as a function of y+. Solid lines: current DNSat Reτ = 362 and Reτ = 647, dotted: Eggels [28], dash-dot: Wagner [106], (◦): PIV [109](Reb = 5450), (H): LDA [109] (Reb = 5450), (+): Durst [26] (Reb = 13500).

100

101

102

0

5

10

15

20

<uz>+=y+→

<uz>+=2.5ln(y+)+5.5→

Barenblatt1→Barenblatt2→

y+

〈uz〉/

uτ

Figure 2.4. Same as figure 2.3. Solid line: current DNS at Reτ = 647, dotted lines:Barenblatt [7], (�): Den Toonder [99] (Reb = 24580).

2.3 Results 25

However, pipe flow does not show this behaviour. Den Toonder [99] performed mea-surements at Reτ = 1382. This data is shown in figure 2.4 using the squares. Ata lower Reynolds number of Reb = 13500 which is more or less comparable to ourDNS2 run, Durst [26] performed LDA measurements. Barenblatt’s [7] and Baren-blatt’s and Prostokishin’s [8] power law incorporates a dependency on the Reynolds

number, 〈uz〉+ =(

1/√

(3)lnRe + 5/2)

y+3/(2lnRe). This is shown in the figure for

two Reynolds numbers, i.e. Reτ = 647 and Reτ = 1382, denoted by the numbers 1and 2 respectively. Den Toonder’s [99] data collapses quite well with this theoreticalprofile (Barenblatt2), whereas the DNS at Reτ = 647 does not coincide with curve1. The Reynolds number is based on the bulk velocity in both cases. Assuming DenToonder’s data is correct, apparently the theoretical profile only holds in case theReynolds number is sufficiently high.

If the radial component of the Navier-Stokes equations is averaged over time equa-tion (2.5) becomes

1

r

∂

∂r

(

ru2r

)

− 1

ru2

φ +∂P

∂r= 0. (2.18)

So there should be a balance between three separate terms and especially near thepipe axis this is a critical quantity due to the 1/r-terms. Figure 2.5 shows the resultfor Reτ = 362 (Reτ = 647 gives similar results). The sum of the three componentsis close to zero apart from some statistical inaccuracy. The first and second term inequation (2.18) balance each other in the region near the pipe axis where the pressureterm becomes zero. This figure illustrates the correctness of the boundary conditionsat r = 0.

Eggels et al. [28] extensively studied the various contributions for the turbulentkinetic energy. Here we will restrict ourselves to the total kinetic energy, i.e. summedover all three velocity components. The equation in cylindrical coordinates is givenby

1

r

∂

∂r

[

r{

u′rE

′k

}]

︸︷︷︸

T: Turbulent Transport

+1

r

∂

∂r

[

r

{1

ρu′

rp′}]

︸︷︷︸

Π: Velocity-Pressure Gradient

− 1

r

∂

∂r

[

r{

2ν(

u′rσ

′rr + u′

φσ′φr + u′

zσ′zr

)}]

︸︷︷︸

D: Viscous Diffusion

= − u′ru

′z

∂uz

∂r︸︷︷︸

P: Production

−ε. (2.19)

In this equation the turbulent kinetic energy is based on the fluctuating part of the

velocities: E′k = 1

2

(

u′r2

+ u′φ2

+ u′z2)

. Furthermore, σ′rr, σ′

φr and σ′zr are compo-

nents of the strain rate tensor in cylindrical coordinates. There is a difference in ourdefinition of the dissipation rate ε. We use ε = 2νS2

ij , where Sij is the strain tensorin cylindrical coordinates, whereas Eggels et al. [28] and Wagner et al. [106] usedε = 2νu′

i,ku′j,k. The difference turns out to be a transport term which Eggels and also

Wagner add to the viscous diffusion term D. The sum of D and ε should be identicalin both approaches. The different contributions of the equation are shown in figure 2.6as a function of wall coordinates. Here the DNS results of Wagner [106] are comparedwith the current DNS results. The results are at Reτ = 362 (left) and at Reτ = 647


0 0.2 0.4 0.6 0.8 1−10

−5

0

5

10

15

20

→

→→

r/R

1r

∂∂r

(

ru2r

)

1r

u2φ

∂P∂r

Figure 2.5. The three terms in the averaged radial component of the Navier-Stokes equa-

tions as a function of the radius. Solid line: ∂P/∂r, dashed: 1r

∂∂r

(

ru2r

)

, dash-dotted: 1ru2

φ,

dotted: the sum of the 3 components.

(right), where the data is normalised by uτ4/ν. There is a very close agreement for

all budgets between both numerical approaches in the lower Reynolds number case.At the higher Reynolds number there are differences mainly for the dissipation rateand viscous diffusion, for the reason explained earlier. Their sum, D + (−ε), is inmuch better agreement. Note that near the pipe axis, i.e. for large values of y+, thecurrent DNS still behaves correctly despite the 1/r-terms in equation (2.19). Thefinite volume method has some problems here, but these do not seem to affect theresults in regions further away from r = 0 (only visible for Reτ = 362, for Reτ = 647y+ continues until 322).

In Chapter 3 moment functions will be discussed. These are statistics evaluatedat different times or with a time delay, e.g. the correlation function 〈u′

r (0) u′r (t)〉. In

Chapter 4 acceleration statistics are discussed, which are closely related to these mo-ment functions. To investigate the suitability of the numerical code to predict thesequantities, in figure 2.7 the autocorrelation (ACF) for radial and axial velocity com-ponents is shown. Here the DNS data is compared with experimental data obtainedwith LDA [86] and HWA (Hot Wire Anemometry) [48]. The results are for a radialposition of r/R = 0.5, are normalised to unity at t = 0 and plotted as a functionof dimensionless time tuz(r)/D, where uz(r) is the average axial velocity. There isa very good agreement between the numerical and experimental results. In an ex-periment measurements take place at one, fixed point, so the averaging procedure isquite trivial in this case. From the (long) time record the ACF is calculated by time

2.4 Particles 27

0 50 100 150

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

LOS

S

TK

E

GA

INP

−εTD

Π

D−ε

y+0 50 100 150

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

LOS

S

TK

E

GA

IN

P

−εTD

Π

D−ε

y+

Figure 2.6. The components of the turbulent kinetic energy equation normalised withuτ

4/ν as a function of y+. D: Viscous Diffusion, T: Turbulent Diffusion, Π: Velocity-Pressure Gradient, P: Production, ε: Dissipation. Solid lines are results of the current DNS,dashed lines are results of Wagner et al. [106]. Left: Reτ = 362, right: Reτ = 647.

averaging. This approach is not feasible in a direct numerical simulation. Due to thehigh computational costs, very long time records cannot be generated. However, in aDNS one can store these records for several points in space, provided that those pointsare separated far enough so that they can be considered as uncorrelated. The appro-priate measure for this is the correlation length, i.e. the length over which a frozenturbulent field is correlated. The separation between the points should be larger thanthis correlation length. An alternative way to calculate the auto correlation functionin the DNS is to calculate spatial correlation functions at fixed moments in time anduse the Taylor hypothesis to transform these spatial correlation functions to temporalcorrelations. These two approaches give similar results, with differences smaller thanthe statistical errors.

2.4 Particles

2.4.1 Lagrangian statistics

Yeung [115] remarked that in order to obtain high-quality Lagrangian data from nu-merical simulations several requirements must be met. The first being a well resolvedinstantaneous velocity field evolving according to the Navier-Stokes equations. Inpractice this means that the DNS code should use a grid spacing sufficiently smallto resolve the small scales. A second requirement involves the accuracy of an inter-polation scheme. In general it is accepted that linear interpolation between the gridpoint values in not accurate enough. Later on we will discuss various schemes andshow that the method we use, a fourth order accurate method, is adequately capableof producing accurate particle statistics. The third and last requirement relates tostatistical accuracy. Whereas we can use time averaging for Eulerian statistics, this


0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

tuz (r) /D0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

tuz (r) /D

Figure 2.7. Autocorrelation functions (ACF) normalised to unity as a function of dimen-sionless time tuz(r)/D. Solid lines: current DNS, (◦, •): LDA by Smit [86], (�): HWA byKovalev [48]. Left: ACF for the radial velocity component at r/R = 0.5. Results are shownfor Reτ = 362 (bottom line and markers) and Reτ = 647 (top line and markers). Right:ACF for axial velocity components at r/R = 0.5 (top line and markers). For comparison theradial component is also shown (bottom line). In this case results are shown for Reτ = 362only.

is certainly not true for Lagrangian statistics. This means that in order to obtainparticle statistics always ensemble averages have to be used, with the restriction thatparticles over which is averaged should be statistically independent from each other.The latter can only be achieved to a certain extent, by tracking particles which aresufficiently separated in space. Lagrangian statistics also impose requirements on theinitial positions of the particles. It is for example not useful to average statisticsof two particles of which particle one started in a near wall region and particle twonear the pipe axis. So, for inhomogeneous turbulence one can only take ensembleaverages over particles which started from the same plane at a certain distance fromthe wall. The particles can be homogenously distributed over homogenous directions.This poses extra requirements on Lagrangian statistics in inhomogeneous simulations,compared to homogenous, isotropic simulations where Lagrangian statistics can bedetermined by ensemble averaging over the particles which are distributed over theentire computational domain.

To obtain the Lagrangian statistics, fluid-particles, e.g. passive particles that haveno mass and which move with the local fluid velocity, are modelled using

dx

dt= u(x, t), (2.20)

where x is the position of a fluid particle and u(x, t) is the fluid velocity at the currentparticle position. This equation can easily be transformed to cylindrical coordinates.To determine the fluid velocity u at the particle position the solution in physicalspace is used. This velocity field is calculated each time step since it is needed

2.4 Particles 29

to determine the nonlinear terms in the Navier-Stokes equations in scheme (2.10).Although the velocity field is known on physical grid points, a fluid particle is almostcertainly located somewhere between grid points. To determine u at the particleposition, an interpolation method is needed. In the radial direction we can make useof properties of the Chebyshev collocation discretisation, hence in the radial directioncubic Hermite interpolation is adopted and in the periodic directions cubic Lagrangeinterpolation. In this way a hybrid method appears which is fourth order accurateoverall. Trilinear interpolation is accurate enough if only mean particle properties aredesired (Kuerten [50]). For higher-order statistics however, this method appeared tobe inaccurate. The same conclusion was drawn very recently by Choi et al. [23] whofound that a higher order method is needed for an accurate description of Lagrangianstatistics in channel flow. An alternative for the fourth order interpolation is touse Direct Summation (see e.g. Balachandar et al. [6], Kontomaris et al. [47] andYeung et al. [116]) in combination with Hermite interpolation. In the two periodicdirections one could directly sum over all Fourier modes to obtain the velocity at anydesired spatial point, without the introduction of interpolation errors. In the radialdirection, an interpolation method remains. This method is only applicable in case offew particles due to the large CPU time needed, but the overall accuracy is certainlyhigher.

To integrate equation (2.20) over time the first order Euler Forward method is ap-plied. For many statistical properties of interest fluid-particles are only tracked untilthe Lagrangian correlation time, tc, has passed. This integration time, in combina-tion with the small time step used, is sufficiently small to keep truncation errors verysmall. A second order 2-stage Runge-Kutta resulted in negligible differences comparedto the first order method. The accuracy of the combination of interpolation methodand time integration is illustrated in figure 2.8. Here, the radial velocity of a singleparticle is shown for Euler Forward time integration in combination with the fourthorder accurate interpolation scheme, 2-stage Runge-Kutta in combination with thefourth order accurate interpolation scheme and Euler Forward in combination withdirect summation of the Fourier series in the periodic directions and Hermite inter-polation in the radial direction. The r.h.s. of the figure shows a zoomed-in version.Some small fluctuations due to the fourth order interpolation compared with directsummation can be observed, but the deviation is less than two percent. This indicatesthat the simple Euler forward method in combination with the fourth order accurateinterpolation scheme is accurate enough for our purposes. In conclusion, the errordue to the simple Euler forward method is small. The Lagrange interpolation schemecauses wiggles, but these have little to no influence on the low order statistics likecorrelation- and structure functions.

In order to get Lagrangian statistics, time averaging over one particle cannot beapplied, since a particle drifts to other radial positions. Instead, ensemble averag-ing is performed over particle records. 256 particles are released at a certain radialposition, homogeneously distributed over tangential and axial directions. Releasingmore particles will not improve the results, since additional particles will not be sta-tistically independent of neighboring particles. Velocity records are generated forapproximately 0.1D/uτ after which the Lagrangian correlation time has passed and


new particles are released to generate a new record. With a total modelling time ofapproximately 10D/uτ , this results in 100 independent records for 256 particles. Inaddition to this the number of independent samples within one record is approximately20. This number follows from spatial correlation functions of fluctuating velocities.The total number of statistically independent samples obtained is then equal to 2000.This number decreases near the axis of the pipe, since the number of statisticallyindependent points in the tangential direction decreases. The different approachesof numerical time integration and spatial interpolation were of minor influence onsingle particle properties but even less on ensemble averaged particle statistics, thequantities we are mainly interested in. Figure 2.9 gives the structure- and correla-tion function for the radial velocity component for several numerical methods. Thedifferences are negligible, indicating the applicability of our numerical discretisationmethod.

0 0.02 0.04 0.06 0.08 0.1−1

−0.5

0

0.5

tuτ/D

v′ (

t)u−

1τ

0.03 0.035 0.04 0.045 0.05 0.055 0.060.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

tuτ/D

v′ (

t)u−

1τ

Figure 2.8. Fluctuating radial velocity component of a single particle normalised withuτ as a function of dimensionless time obtained with various combinations of interpolationmethods and time integration schemes. Solid lines: Euler forward + 4th order interpolation,dashed lines: Euler forward + Direct Summation/Hermite scheme, dash-dotted lines: 2 stageRunge-Kutta + 4th order interpolation. On the r.h.s. a close up, where some wiggles areapparent due to the 4th order interpolation scheme. The fluid particle initially started fromr/R = 0.5.

2.4.2 Eulerian statistics

The Eulerian frame moves according to

dz

dt= uz(r)

dφ

dt=

dr

dt= 0.

(2.21)

So each time step the frame is moved in streamwise direction over a distance ∆tuz(r),where the value of uz(r), the mean axial velocity as a function of the radius, is

2.4 Particles 31

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

1.2

1.4

tuτ/D

⟨

(v′ r(t

)−

v′ r(0

))2⟩

u−

2τ

0 0.02 0.04 0.06 0.08 0.1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

tuτ/D

〈v′ r(t

)v′ r(0

)〉u−

2τ

Figure 2.9. Second order structure- (left) and correlation-function (right) normalised withu2

τ as a function of dimensionless time for the fluctuating part of the radial particle velocity.The data is calculated for 1 record and averaged over 256 particles. The results for thevarious time integration and interpolation methods coincide. The symbols are the same asin figure 2.8 and the fluid particles initially started from r/R = 0.5.

obtained from a previous calculation. At the current position of the frame directsummation of the Fourier series is used to evaluate the current components of thevelocity and their derivatives in space. Since there is no need for interpolation inthe radial and tangential directions (the marked fluid particles start on collocationpoints), the calculation time of this method is acceptable.

2.4.3 Statistical accuracy

For the moving frame statistics, like for the Lagrangian statistics, time records con-taining the components of the velocity and its spatial derivatives have been generated.In contrast to the Lagrangian statistics, time averaging can be applied in this casesince the radial position of a particle is fixed. Again, 256 homogeneously distributedpoints are used as starting points. In this way 256 records of length ≈ 10D/uτ aregenerated. Averaging is then performed over time and the homogeneously distributedpoints. The total number of statistically independent samples in this way is equalto the Lagrangian case and thus equals 2000. In both cases, Lagrangian and Eule-rian, the averaging time is sufficiently long. This is confirmed by the convergenceof statistical properties of interest as a function of averaging time (in the Euleriancase) or number of records (Lagrangian case). To illustrate this, figure 2.10 shows theaveraged Lagrangian velocity components and structure functions at tuτ/D = 0.048as functions of the number of records. In figure 2.11 the correlation and structurefunctions for the Eulerian case are shown as functions of the averaging time. Similarbehaviour is observed for other statistical properties. If independence between eachindividual Lagrangian record is assumed, and if the distribution between the recordsis Gaussian at each time t, an estimate of the remaining statistical error E(t) can be


made. Under these assumptions it is equal to, in percentage,

E(t) =σrec(Frec(t))√

N

1

F (t)100, (2.22)

where σrec is the standard deviation of the results from each individual record, N is thenumber of records (N =100 in this case) and F (t) is the statistical property of interest.The same approach can be used for the Eulerian data if the long signal of length≈ 10D/uτ is subdivided into 100 records of ≈ 0.1D/uτ . This error estimate is shownin figure 2.12 for the radial correlation function. On the l.h.s. the radial Lagrangiancorrelation function for all records is plotted. Within each record an average over 256particles is taken. There is a more or less Gaussian distribution, although the numberof records should be increased to give a more elaborate conclusion. Nevertheless,it can be used to give a reasonable error estimate of the correlation function. Onthe r.h.s. this error is plotted as a function of tuτ/D for both the Lagrangian andEulerian data. Due to the time averaging the error is smaller for the Eulerian case.For large times the correlation becomes so small that the statistical error increases.In the region t < tc (tc is the Lagrangian integral time scale), which is the region ofinterest for our purposes, the statistical error is smaller then 1.5 %. This holds for allstatistical data of interest.

0 20 40 60 80 100−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

N

⟨v′ µ(t

)⟩u−

1τ

0 20 40 60 80 1000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

N

⟨(v′ µ(t

)−

v′ ν(0

))2⟩

u−

2τ

Figure 2.10. Convergence of average Lagrangian velocity (left) and structure functions(right) normalised with uτ and uτ

2 respectively as functions of the number of records Nfor tuτ/D = 0.048. Solid: radial component (µ = ν = r), dashed: tangential component(µ = ν = φ), dash-dotted: axial component (µ = ν = z).

2.5 Summary

In this section the numerical procedures of the direct numerical simulation (DNS) codehave been explained. The method combines a Chebyshev collocation method in radialdirection with Fourier expansions in the periodic tangential and axial directions. Withthis code large databases have been generated for two different Reynolds numbers,

2.5 Summary 33

0 2 4 6 8 100.2

0.4

0.6

0.8

1

1.2

1.4

1.6

tuτ/D

⟨u′ µ(t

)u′ ν(0

)⟩u−

2τ

0 2 4 6 8 100.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

tuτ/D

⟨(u

′ µ(t

)−

u′ ν(0

))2⟩u−

2τ

Figure 2.11. Convergence of Eulerian correlation (left) and structure functions (right)normalised with uτ

2 as functions of the averaging time for tuτ/D = 0.048. Symbols are thesame as in figure 2.10: solid: radial component (µ = ν = r), dashed: tangential component(µ = ν = φ), dash-dotted: axial component (µ = ν = z).

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

tuτ/D

〈v′ r(t

)v′ r(0

)〉σ−

1rr

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

tuτ/D

E(t

)

Figure 2.12. Estimate of the error E(t). On the l.h.s. the Lagrangian correlation functionfor the radial component for all individual records. Each line corresponds to an average over256 particles and is normalised with its starting value. On the r.h.s. the statistical error inpercentage as a function of tuτ/D for the Lagrangian (solid) and Eulerian (dashed) data atr/R = 0.5.


i.e. Reτ = 362 and Reτ = 647 where the Reynolds number is based on the wall shearvelocity and the diameter. For various statistical properties the current DNS resultsare in good agreement with DNS results from others and experimental results. Basedon these findings we are confident that further results, which are difficult to validateby experiment, are also an accurate representation of reality. Lagrangian statisticshave been calculated as well at the two Reynolds numbers. The databases will beused to study velocity and acceleration statistics in Chapters 3 and 4 respectively andto study local isotropy in Chapter 5. In Chapter 6 the Lagrangian data will be usedto construct a linear stochastic model.

Chapter 3

Lagrangian-Eulerian

Connection

3.1 Introduction

During the last decade much effort has been spent to describe turbulent transportin a Lagrangian framework (see e.g. Yeung [114] for a recent overview). Such adescription could be of great help to understand for example turbulent dispersionand mixing of passive and reactive scalars or contaminants. In the Lagrangian casethe time evolution of marked fluid elements (viz. fluid-particles) which follow themotion of the fluid is recorded. This is in contrast with an Eulerian descriptionwhere the turbulent flow is described at fixed points in space and time. Langevinmodels, or stochastic models, try to simulate this evolution of fluid particles underthe assumption that the Reynolds number is high enough so that there is a distinctseparation between the smallest scales of the flow and the large, energy containingscales. In this case the smallest scales can be modelled using a δ-correlated Markovprocess ∗ incorporating Kolmogorov’s similarity hypothesis. The first attempts in thisdirection were taken by Taylor [91] with a stochastic model for the position of a fluidparticle. Later on other researchers followed, see e.g. [13], [32] and [77], where alsostochastic models for the velocity of a fluid particle were considered.

Several methods have been tried to relate the large-scale behaviour in the Lange-vin equation to Eulerian flow statistics in order to enable the solution of the Langevinequation without simultaneously solving the flow equations, see e.g. [75] and [111].However, Langevin-based Markov models suffer from what is called the non-uniquenessproblem. The well-mixed condition and Kolmogorov’s similarity hypothesis alone arenot sufficient to formulate the equations in a unique way. It appeared that this is onlypossible for homogeneous turbulence as shown by Borgas et al. [13]. Solutions to thenon-uniqueness problem resulted in various proposals, e.g. the inclusion of rotation ofparticles and Eulerian acceleration statistics (see [76], [77], [78] and [112]). All these

∗A Markov process is a stochastic process in which the distribution of future states depends onlyon the present state and not on how it arrived in the present state. See e.g. [42].

36 Lagrangian-Eulerian Connection

r)0()0( vu =

)( tvtru z )( )( tu

Figure 3.1. Schematic representation of a Lagrangian particle and the Eulerian frame thatmoves in space. At t=0 particles are marked. At time t the Eulerian frame moved in spacewith uz(r)t.

models are chosen ad hoc: the results are compared with Lagrangian data availablefrom DNS and the best performing model is selected.

This raises the question how we can construct a unique, asymptotically correctmodel. The most obvious method seems to just use Lagrangian data obtained fromeither a DNS or experiments at a certain Reynolds number and fit the stochasticmodel to these results. However, Lagrangian measurements are extremely tedious,and are currently in the stage of development and DNS calculations suffer from thealways low Reynolds number regime. If one would be able to connect, in a mathe-matical way, Lagrangian statistics to Eulerian statistics, more accurate measurementtechniques could be used, since we do not have to measure in a Lagrangian frameworkanymore. In that case real Lagrangian measurements become redundant. Recently,Brouwers [16] derived an asymptotic connection between Lagrangian and Eulerianquantities for times smaller than the Lagrangian correlation time. In this connectionLagrangian quantities can be expanded in a power series in t/tc in terms of higherorder Eulerian velocity correlations, where tc is the Lagrangian correlation time. Ifthe higher order terms in this series become smaller with increasing order, Lagrangianstatistics become measurable in an Eulerian frame, and there is an Eulerian equivalentof the Lagrangian similarity hypothesis of Kolmogorov. This would enable an experi-mental verification of this hypothesis and specification of the deterministic componentof the Langevin equation and therefore the solution of the non-uniqueness problem.The Eulerian frame is not fixed in space but moves with the local average velocity asindicated in figure 3.1.

The expansion is applicable to any Lagrangian property of interest. There are how-ever questions on the validity in connection with the random sweeping hypothesis ofTennekes [95]. This hypothesis states that the Taylor hypothesis for frozen turbulencealso applies for the case of random convection velocity in a frame of zero-mean-flow.The consequences of this hypothesis are far reaching with respect to Eulerian andLagrangian statistics and therefore also on the proposed expansion. The goal for thischapter is the validation of the expansion by means of the available DNS results.

In the next section a brief outline of the theoretical connection is addressed. Fora more extensive description the reader is referred to [16]. To assess the applicabilityof Brouwers’ theoretical connection statistics in an Eulerian moving frame as well asLagrangian statistics, discussed in the previous chapter, are calculated. In section 3.3,the results are shown and discussed for several statistical quantities of interest at

3.2 Connection between Lagrangian and Eulerian statistics 37

two different Reynolds numbers. Whether or not this theoretical connection gives aworkable method to describe Lagrangian properties is argued in section 3.4. In thefinal section a summary and perspectives are given.

3.2 Connection between Lagrangian and Eulerian

statistics

In search for an Eulerian description of Lagrangian properties, Brouwers investigatedan expansion based on Eulerian statistics which in the limit of high Reynolds numbersshould become equal to Lagrangian statistics. This would mean that with standardmeasurement techniques, like PIV or LDA, one could obtain Lagrangian informationof the flow of interest. In this section we will highlight some of the properties of thisexpansion, without attempting to cover the whole derivation.

In general, a turbulent flow field can be written as:

u(x, t) = u0(x, t) + u′(x, t),

where u0 is the mean velocity and u′ the fluctuating part. We will use vi for thevelocity in a Lagrangian frame and ui for the velocity in an Eulerian frame. Theposition of a passive particle, x(t), is governed by the differential equation

dx

dt= u0(x(t), t) + u′(x(t), t).

It can be shown [16] that for times small compared to the Lagrangian correlationtime tc, i.e. in the inertial range, a particle property z(x(t), t) is related to thecorresponding Eulerian property through

z(x(t), t) = z(xt0, t) +

∫ t

0

dt1∂z(xt

0, t)

∂xt1i0

u′i(x

t10 , t1) + O((t/tc)

2). (3.1a)

In this equation xt0 is the position of an imaginary particle at time t which moves

with the mean velocity u0 and which was at time t = 0 at the same position as thereal particle as illustrated in figure 3.1. Further, summation over repeated indices isapplied. Also the third term in the expansion of z(x(t), t) has been determined:

t∫

t0

dt1

t1∫

t0

dt2∂

∂xt2j0

{∂z(xt

0, t)

∂xt1i0

u′i(x

t10 , t1)u

′j(x

t20 , t2)

}

+ O((t/tc)3). (3.1b)

It replaces the O((t/tc)2) term on the r.h.s. of equation (3.1a) and has in the presented

form been obtained under the assumption of incompressible flow. Expansion (3.1a)can be used to relate Lagrangian statistical quantities to Eulerian statistical quanti-ties. In section 3.3 results for the drift velocity, standard deviation, structure- andcorrelation functions are discussed. In the DNS all these quantities are obtained in aLagrangian frame as well as in the Eulerian moving frame. The form of the expansionterms of the Eulerian statistics is discussed here.


3.2.1 Single-time statistics

If we take for z a fluctuating velocity component and take an ensemble average, theresult is

〈v′µ(t)〉 = 〈u′

µ(xt0, t)〉 +

∫ t

0

dt1

⟨∂u′

µ(xt0, t)

∂xt1i0

u′i(x

t10 , t1)

⟩

+ O((t/tc)2). (3.2)

Here 〈v′µ(t)〉 is the average fluctuating part of the velocity component of a passive

particle, which is called the drift velocity. Equation (3.2) shows that the drift velocityis not equal to zero in inhomogeneous turbulent flow, even though the first term on ther.h.s. equals zero. If this equation is applied to cylindrical pipe flow, where statisticalproperties depend only on the radial coordinate, it can be simplified to

〈v′µ(t)〉 =

1

r

∂

∂r

(

r

∫ t

0

dt1⟨u′

µ(r, φ, z + u0z(r)t, t)u

′r(r, φ, z + u0

z(r)t1, t1)⟩)

+ O((t/tc)2), (3.3)

where r, φ and z denote the radial, tangential and axial coordinates and ur the radialvelocity component. The initial position of the passive particle is (r, φ, z).The next expansion term of O((t/tc)

2) in equation (3.2) can also be derived. For thedrift velocity it has the general form

t∫

t0

dt1

t1∫

t0

dt2

⟨

∂

∂xt2j0

∂u′i(x

t0, t)

∂xt1i0

u′i(x

t10 , t1)u

′j(x

t20 , t2)

⟩

+ O((t/tc)3).

This term replaces the O((t/tc)2) term in equation (3.2). Using a Taylor series in

t, the double integral term can be reduced to a more simple form. As an example,the expansion till O((t/tc)

2) for the radial drift velocity for incompressible flow in acylindrical coordinate system is given by

〈v′r(t)〉 =

1

r

∂

∂r

r

t∫

t0

⟨u′

r(xt0, t)u

′r(x

t10 , t1)

⟩dt1

+1

4t2

1

r

∂2

∂r2

(

r⟨

u′r3⟩)

+ O((t/tc)3). (3.4)

For the radial velocity component the derivative operators in the double integral canbe taken out of the averaging brackets. Since for stationary and incompressible flowderivatives in the tangential and axial directions of averaged properties are zero, itreduces to a simple expression. The Taylor series reduces the simplified expression ofthird moments even further to fixed-frame statistics. For the other velocity compo-nents the O((t/tc)

2) term becomes more complicated.

Higher order moments of Lagrangian velocities can also be expressed in Eulerianstatistics. Applying equation (3.1a) to v′

µn(t)v′

νm

(t) and ensemble averaging of the

3.2 Connection between Lagrangian and Eulerian statistics 39

left and right hand side yields

⟨v′

µn(t)v′

νm

(t)⟩

=⟨u′

µn(xt

0, t)u′ν

m(xt

0, t)⟩

+

t∫

t0

dt1

⟨∂u′

µn(xt

0, t)u′ν

m(xt

0, t)

∂xt1i0

u′i(x

t10 , t1)

⟩

+ O((t/tc)2). (3.5)

In a similar way as for the drift velocity the O((t/tc)2) term can be calculated again.

In section 3.3 we will only discuss the results for n=m=1, viz. the standard deviationof fluctuating velocity components. For the radial component (µ=ν=r) the completeexpansion, accurate up till O((t/tc)

2), is described by

⟨v′

r(t)2⟩

=⟨u′

r(xt0, t)

2⟩

+1

r

∂

∂r

r

t∫

t0

⟨u′

r(xt0, t)

2u′r(x

t10 , t1)

⟩dt1

+1

6t2

1

r

∂2

∂r2

(

r⟨

u′r4⟩)

+ O((t/tc)3). (3.6)

3.2.2 Structure- and moment-functions

A third statistical quantity of interest is the structure function. This property con-tains information about the Lagrangian version of Kolmogorov’s similarity hypothesis(see [16] and [59]). An Eulerian equivalent would open the possibilities to verify thishypothesis by experiment. If we use Dµν for the structure function of order 2 betweencomponents µ and ν we can write

Dµν(t0, t) =⟨(v′

µ(t) − v′µ(t0))(v

′ν(t) − v′

ν(t0))⟩

=⟨(u′

µ(x(t), t) − u′µ(x0, t0))(u

′ν(x(t), t) − u′

ν(x0, t0))⟩

=⟨(u′

µ(xt0, t) − u′

µ(x0, t0))(u′ν(xt

0, t) − u′ν(x0, t0))

⟩

+O((t/tc)2). (3.7)

In this case the next expansion term is directly of O((t/tc)2) instead of O(t/tc). The

form of this term is not shown here, but can be derived from equation (3.1b).

The final quantity we are interested in, is the correlation function. In a similarway as for all other quantities, Lagrangian correlation functions can be expressed inEulerian correlation functions. Using equation (3.1a) the result is:

Bµν(t0, t) = 〈v′µ(t0)v

′ν(t)〉 =

= 〈u′µ(x0, t0)u

′ν(xt

0, t)〉

+

∫ t

0

dt1

⟨

u′µ(xt

0, t)∂u′

ν(xt0, t)

∂xt1i0

u′i(x

t10 , t1)

⟩

+ O((t/tc)2). (3.8)


For cylindrical pipe flow equation (3.8) can be simplified to

Bµν(t0, t) = 〈v′µ(t0)v

′ν(t)〉 = 〈u′

µ(x0, t0)u′ν(xt

0, t)〉

+1

2t(T inh

µν + T aniµν ) + O((t/tc)

2), (3.9)

where

T inhµν =

1

r

∂

∂r

(r〈u′

µ(x0, t0)u′ν(x0, t0)u

′r(x0, t0)〉

)

and

T aniµν =

⟨{

u′µ(x0, t0)

∂u′ν(x0, t0)

∂xi− u′

ν(x0, t0)∂u′

µ(x0, t0)

∂xi

}

u′i(x0, t0)

⟩

.

In order to arrive at this result, the integral over the time in equation (3.8) has beenexpanded in a Taylor series in t, so that up to first order all terms are evaluated atthe initial particle position x0. The Taylor expansion can only be used in restrictedsituations. In general, the integral term has to be evaluated in its original form.We see that in this first order expansion the Lagrangian correlation function can beexpressed as the Eulerian correlation function in a moving frame plus fixed-point thirdmoments of Eulerian velocity components. It should be noted that the first correctionterm T inh

µν equals zero in homogeneous parts of the flow and the second term T aniµν

equals zero for isotropic turbulence. This shows that small scales do not contributeto these terms, since they are according to the Kolmogorov hypothesis homogeneousand isotropic [59]. This expansion of the Lagrangian correlation function remainsaccurate for times small compared to the Lagrangian correlation time. In section 3.3also the O((t/tc)

2) term will be calculated and evaluated. Once the structure functionDµν(t0, t) and standard deviation σµν(t) are calculated up till O((t/tc)

2) it can bederived from the relation Bµν(t0, t) = −(Dµν(t0, t) − σµν(t) − σµν(t0))/2.

3.3 Results

In the next subsections various statistical results are presented and discussed. Acomparison is made between the Lagrangian and Eulerian statistics at the Reynoldsnumbers Reτ = 362 and Reτ = 647. This is done for two radial positions, viz.r/R = 0.5 and r/R = 0.9. In the region around r/R = 0.5 the turbulence tends tobe isotropic, whereas at r/R = 0.9 the turbulence is strongly inhomogeneous. TheEulerian statistics are calculated up to the first and second order term(s) in t/tc,which involves first- and second derivatives with respect to spatial coordinates of thevelocity components in the moving frame. For all Lagrangian data the statistical erroris included in the figures using equation (2.22). This is not done for the Eulerian datafor the sake of simplicity. The statistical error in this case is even smaller than inthe Lagrangian data as we saw in section 2.4.2. For all of the results we will restrictourselves to the radial component. In all cases the other two components give similarresults and do not give additional information.

3.3 Results 41

3.3.1 Single-time statistics

In this section single-time statistics are discussed. By this we mean statistical proper-ties which are evaluated at a single time, although the property can depend on time.The drift velocity 〈v′

i(t)〉 and the variance 〈v′i(t)v

′i(t)〉 are examples of single-time

statistics: they depend on t only. In the next sections we will consider structure- andmoment functions, which are properties evaluated at different times.

Figure 3.2 shows the results for the O((t/tc)) and O((t/tc)2) expansions of the

radial drift velocity at the positions r/R = 0.5 and r/R = 0.9 (see equation (3.4)) atReτ = 362. In the figures also the results for 〈u′

r(xt0, t)〉 without any expansion terms

are included (denoted by O(1)), which is close to zero as expected. A value unequalto zero would mean that there is a net flow through the cylindrical surface, which isin contradiction with conservation of mass. The small deviation from zero observedin this case can be attributed to the limited averaging time. The Lagrangian particleshave a positive drift velocity for r/R = 0.5 and a negative drift velocity for r/R = 0.9.This is due to the preferential velocity of the fluid particles. There is a bigger chancefor a particle which started at r/R = 0.9 to move to the inner region than in thedirection of the wall. The starting values between the Lagrangian and Eulerian dataare slightly different due to the different approach of averaging. This difference is ameasure for the statistical error. If the number of records would go to infinity theEulerian and Lagrangian starting point would be equal and exactly zero. From thefigure it can be seen that the expansion correctly predicts the Lagrangian behaviour,up to tuτ/R ≈ 0.05. The Kolmogorov time tη is approx. 0.03R/uτ , whereas theLagrangian correlation time for the radial component tc is approx. 0.15R/uτ atr/R = 0.5. This value decreases by approx. a factor 2 at the position r/R = 0.9. Socloser to the wall the distinction between large- and small-scales of the turbulence isless well defined. Correct behaviour for times t ∼ tc cannot be expected, since theexpansion is only valid for times tη � t � tc. Compared to the O((t/tc)) term, thecontribution of the O((t/tc)

2) term is small.

In figure 3.3 results are shown for the standard deviation (µ=ν=r), e.g. n=m=1 inequation (3.5). Again the starting values between the Lagrangian and Eulerian dataare different. The variation of the standard deviation over time is relatively small.The fluctuation of the standard deviation over time is of the same order of magnitudeas the statistical error. The O(1) expansion, which is simply

⟨u′

µ(xt0, t)u

′ν(xt

0, t)⟩, is

constant over time: it is identical to the standard deviation evaluated at a fixed pointin space. The next terms in the expansion improve the O(1) result in the correctmanner. The slope of the Eulerian O(t/tc) expansion at t = 0 corresponds to theLagrangian data. Adding the O((t/tc)

2) term affects the results only very little ifwe consider t � tc. Noticeable is that the Lagrangian standard deviation varies onlyvery little over time, which makes the expansion look as if it is working not as goodcompared to the drift velocity. These same results, for the drift velocity and standarddeviation, are repeated for Reτ = 647 (see figures 3.4 and 3.5). In dimensionless timeTuτ/D, where T is the total modelling time, the runs at the two Reynolds numbersare approximately equally long (Tuτ/D = 9.79 and Tuτ/D = 8.10 respectively forReτ = 362 and Reτ = 647), so we expect the statistical error to be of the same order.The drift velocity is again accurately described by the Eulerian expansion at the two


0 0.05 0.1 0.15 0.2−0.05

0

0.05

0.1

0.15

0.2

0.25

tuτR−1

〈u′ r(t

)〉,〈

v′ r(t

)〉

0 0.05 0.1 0.15 0.2

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

tuτR−1

〈u′ r(t

)〉,〈

v′ r(t

)〉

Figure 3.2. Radial drift velocity at r/R = 0.5 (left) and r/R = 0.9 (right) as a function ofnormalised time. The results are for Reτ = 362. Solid: Lagrangian, dotted: Eulerian O(1),dashed: Eulerian O(t/tc), dash-dotted: Eulerian O((t/tc)

2).

0 0.05 0.1 0.15 0.20.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

tuτR−1

〈σrr(t

)〉

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

tuτR−1

〈σrr(t

)〉

Figure 3.3. Standard deviation of the radial velocity at r/R = 0.5 (left) and r/R = 0.9(right) as a function of normalised time. The results are for Reτ = 362. Symbols as infigure 3.2: solid: Lagrangian, dotted: Eulerian O(1), dashed: Eulerian O(t/tc), dash-dotted:Eulerian O((t/tc)

2).

3.3 Results 43

radial positions, indicated in figure 3.4. The Lagrangian correlation time is not verymuch affected by the increase in Reynolds number (we will discuss this in Chapter 6),so the expansion should hold to approximately the same dimensionless time as inthe Reτ = 362 case. Again, for the standard deviation only the result for the radialcomponent is shown. Similar behaviour is observed for the other components σzz andσzr. The results indicate that for single time statistics the expansion is accuratelycapable of describing Lagrangian properties. In the next section we will investigatethe validity of the expansion for moment functions, where statistics no longer dependsolely on one time level, but on a time difference.

0 0.05 0.1 0.15 0.2−0.05

0

0.05

0.1

0.15

0.2

0.25

tuτR−1

〈u′ r(t

)〉,〈

v′ r(t

)〉

0 0.05 0.1 0.15 0.2

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

tuτR−1

〈u′ r(t

)〉,〈

v′ r(t

)〉

Figure 3.4. Same as figure 3.2 but for Reτ = 647. Radial drift velocity at r/R = 0.5(left) and r/R = 0.9 (right) as a function of normalised time. Solid: Lagrangian, dotted:Eulerian O(1), dashed: Eulerian O(t/tc), dash-dotted: Eulerian O((t/tc)

2).

0 0.05 0.1 0.15 0.20.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

tuτR−1

〈σrr(t

)〉

0 0.05 0.1 0.15 0.20.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

tuτR−1

〈σrr(t

)〉

Figure 3.5. Same as figure 3.3 but for Reτ = 647. Standard deviation of the radial velocityat r/R = 0.5 (left) and r/R = 0.9 (right) as a function of normalised time. Solid: Lagrangian,dotted: Eulerian O(1), dashed: Eulerian O(t/tc), dash-dotted: Eulerian O((t/tc)

2).


3.3.2 Structure functions

From equation (3.7) it can be seen that for t � tc the Lagrangian structure func-tion can be based on Eulerian velocities measured in a frame that moves with themean Eulerian velocity. Figure 3.6 shows Eulerian descriptions accurate to O(t/tc)and O((t/tc)

2) only for the radial direction. According to Kolmogorov the structurefunctions in the three principal directions should be identical and in the inertial rangebe equal to C0 〈ε〉 (t − t0), where C0 is the Kolmogorov constant and 〈ε〉 is the meanenergy dissipation rate. An estimate of the Kolmogorov constant will be given inChapter 6.

Although only the result for the radial component is shown, we remark here thatfor regions away from the wall the structure functions in the principal directions arenearly identical. According to Kolmogorov’s hypotheses [45] (K41), the result for allthe cross-components should be zero. We find that cross terms with φ are all equal tozero and that the r − z component is relatively small compared to the results in theprincipal directions (about 20 to 25 percent over the whole time interval 0 ≤ t ≤ tc).Deviations from zero can most likely be attributed to the moderate Re−number.Closer to the wall the cross-component r − z is of the same order of magnitude asthe three diagonal components which also deviate from each other. In this regionseparation between large- and small-scale components of the turbulence is absent dueto the moderate Re−number. In general, the Eulerian approximations are in goodagreement with the Lagrangian data. Based on these results one would conclude thatthe Lagrangian representation of Kolmogorov’s similarity hypothesis can be explicitlytested by experiment, since the Eulerian velocities in equation (3.7) are measurable. If

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(t − t0)uτR−1

Drru−

2τ

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(t − t0)uτR−1

Drru−

2τ

Figure 3.6. Structure function of the radial component normalised by u2τ at r/R = 0.5

(left) and r/R = 0.9 (right) as a function of normalised time. The results are for Reτ = 362.Solid Lagrangian, dotted: Eulerian O(1), dashed: Eulerian O((t/tc)

2).

we then consider the radial structure function at the higher Reynolds number, shownin figure 3.7, it appears that the Eulerian expansion gives a less accurate result thanin the lower Reynolds number case. At the higher Reynolds number the O((t/tc)

2)result is even worse compared to the O(t/tc) result. These are the first indications

3.3 Results 45

that with increasing Reynolds number, the higher order terms are equally importantin case two-time statistical properties are considered. In the next section momentfunction are investigated, which belong to this same class of statistics.

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(t − t0)uτR−1

Drru−

2τ

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(t − t0)uτR−1

Drru−

2τ

Figure 3.7. Same as figure 3.6 but for Reτ = 647. Structure function of the radial compo-nent normalised by u2

τ at r/R = 0.5 (left) and r/R = 0.9 (right) as a function of normalisedtime. Solid Lagrangian, dotted: Eulerian O(1), dashed: Eulerian O((t/tc)

2).

3.3.3 Moment functions

Moment functions are moments of random variables at different times. More fre-quently they are called correlation functions. Equation (3.9) relates Lagrangian andEulerian correlation functions up till O(t/tc). The Lagrangian correlation tensor isdetermined by Eulerian time correlations in the moving frame and fixed point thirdmoments of Eulerian velocities, where these third moments can be split into a symme-trical and anti-symmetrical part. For the radial component of the correlation functionEulerian approximations accurate till O((t/tc)

2) are calculated as well. The resultsfor the radial correlation functions are presented in figure 3.8. For the range t � tcthe Eulerian approximations are in satisfactory agreement with the Lagrangian data.The O((t/tc)

2) term has a relatively large contribution.

The results for Reτ = 647, plotted in figure 3.9, show again that the expansiongives a less accurate result with increasing Reynolds number. This holds especiallyfor the position r/R = 0.5, whereas at r/R = 0.9 the expansion still works relativelywell. In the next section we will investigate why the expansion gives less accurateresults for the structure- and moment functions if the Reynolds number increases.


0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(t − t0)uτR−1

Brru−

2τ

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(t − t0)uτR−1

Brru−

2τ

Figure 3.8. Correlation functions of the radial velocity component normalised by u2τ at

r/R = 0.5 (left) and r/R = 0.9 (right) as a function of normalised time. The results arefor Reτ = 362. Symbols as in figure 3.2. Solid: Lagrangian, dotted: Eulerian O(1), dashed:Eulerian O(t/tc), dash-dotted: Eulerian O((t/tc)

2).

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(t − t0)uτR−1

Brru−

2τ

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(t − t0)uτR−1

Brru−

2τ

Figure 3.9. Same as figure 3.8 but for Reτ = 647. Correlation functions of the radialvelocity component normalised by u2

τ at r/R = 0.5 (left) and r/R = 0.9 (right) as a functionof normalised time. Solid: Lagrangian, dotted: Eulerian O(1), dashed: Eulerian O(t/tc),dash-dotted: Eulerian O((t/tc)

2).

3.4 Discussion 47

3.4 Discussion

If we apply the expansion to the correlation function, without averaging but just forone fluid particle, we get

v′µ(t0)v

′ν(t) = u′

µ(x0, t0)u′ν(xt

0, t)

+ u′µ(x0, t0)

∫ t

t0

∂u′ν(xt

0, t)

∂xt1i0

u′i(x

t10 , t1)dt1 + O((t/tc)

2). (3.10)

We can average this for many realizations which leads to

⟨v′

µ(t0)v′ν(t)

⟩=

⟨u′

µ(x0, t0)u′ν(xt

0, t)⟩

+

⟨

u′µ(x0, t0)

∫ t

t0

∂u′ν(xt

0, t)

∂xt1i0

u′i(x

t10 , t1)dt1

⟩

+ O((t/tc)2). (3.11)

A general expression for u′µ(x0, t0) is given by

u′µ(x0, t0) = u′

µ(xt0, t) −

∫ t

t0

du′µ(xt2

0 , t2)

dtdt2. (3.12)

If we substitute (3.12) in the second term in (3.11) we get

⟨v′

µ(t0)v′ν(t)

⟩=

⟨u′

µ(x0, t0)u′ν(xt

0, t)⟩

+

⟨

u′µ(xt

0, t)

∫ t

t0

∂u′ν(xt

0, t)

∂xt1i0

u′i(x

t10 , t1)dt1

⟩

−⟨∫ t

t0

du′µ(xt2

0 , t2)

dtdt2

∫ t

t0

∂u′ν(xt

0, t)

∂xt1i0

u′i(x

t10 , t1)dt1

⟩

+ O((t/tc)2). (3.13)

The second term on the r.h.s. can be simplified due to incompressibility, viz. theu′

i(xt10 , t1) term can be put in the spatial derivative. And in case ν = µ, the u′

µ(xt0, t)

term can be put in the derivative as well, allowing to take the spatial derivative outof the averaging brackets. One is then left with a spatial derivative of an averagedquantity, which for pipe flow is only unequal to zero for the radial direction. This termis always small compared to the first term on the r.h.s., which is exactly what onewould expect from an asymptotic theory. The third term on the r.h.s. can be brought

in a different form using the Navier-Stokes equation to replace thedu′

µ(xt20

,t2)

dt term.In the next chapter we will show that the pressure gradient and viscous contributionare small compared to this term and that this term is dominated by the convective


term which then leads to (only if ν = µ)

⟨v′

µ(t0)v′µ(t)

⟩'

⟨u′

µ(x0, t0)u′µ(xt

0, t)⟩

+1

2

∂

∂xt1i0

⟨∫ t

0

u′µ(xt

0, t)u′µ(xt

0, t)u′i(x

t10 , t1)dt1

⟩

−⟨∫ t

t0

∫ t

t0

∂u′µ(xt2

0 , t2)

∂xt2j0

u′j(x

t20 , t2)

∂u′µ(xt

0, t)

∂xt1i0

u′i(x

t10 , t1)dt1dt2

⟩

+ O((t/tc)2). (3.14)

The next chapter will focus for a great deal on the fourth order moment which arises(third term on the r.h.s.), which is for t1 = t2 = t exactly the convective part ofthe Navier-Stokes equation squared. It will turn out that this fourth order momentbecomes of the same order of magnitude as the first term in equation (3.14) wellwithin the inertial range. This limits the application of the expansion to the viscoussubrange. However, in order to determine the coefficients in the Langevin equation,Lagrangian correlation functions in the inertial range are necessary. Note that thisterm arises in the O((t/tc)

2) term of the expansion when statistics at two differenttime levels are studied. In a similar way one can show that for single-time statisticssuch a term arises in the O((t/tc)

3) term, but the effect is less pronounced.

3.5 Summary

In this chapter an expansion is presented to describe the statistics of Lagrangianevolving fluid particles. The main advantage of the expansion is that it uses onlyEulerian statistics, which are more easily measurable than Lagrangian velocities. Tocheck the expansion fluid particle-tracks have been calculated within the DNS andstatistical properties of these particles have been calculated. Both approaches, theLagrangian statistics and the expansion based on Eulerian statistics, should giveidentical results for statistical properties of interest for times that are much smallerthan the Lagrangian correlation time tc. There are however questions on the validityin connection with the sweeping hypothesis of Tennekes, which is investigated in thischapter.

From the results it can be concluded that the expansion is suitable to describesingle-time statistics, e.g. the drift velocity or standard deviation of fluid particles.For properties like these the expansion is useful to describe Lagrangian behaviour,since the leading term is rather simple to measure and higher order terms are alwayssmall compared to this term. For two-time properties, e.g. correlation- or structurefunctions, always a fourth order moment appears in the higher order terms which isonly small compared to the first term in the viscous subrange. As a result, manyhigher-order terms in the expansion become equally important in the inertial range,and the expansion looses its practical relevance. At our lowest Reynolds numberobtained with the DNS, these higher order terms are still of minor importance, butat the higher Reynolds number the influence starts to become evident. In the next

3.5 Summary 49

chapter we will try to derive scaling behaviour for this fourth order moment and wewill see that the term is related to Tennekes ”random-sweeping” hypothesis.

Chapter 4

Acceleration Statistics

4.1 Introduction

In many stochastic models the fluid particle acceleration is modelled using a δ− corre-lated process, which is called a Markov process for particle velocity. The assumptionthat the acceleration becomes uncorrelated if the Re−number is high enough is asimplification and resulted in attempts to quantify acceleration statistics using bothnumerical and experimental techniques. The Lagrangian acceleration is defined bythe material derivative of the velocity vector

a ≡ Du

Dt=

∂u

∂t+ (u · ∇)u. (4.1)

Using the Navier-Stokes equations this can be written as

a ≡ Du

Dt=

∂u

∂t︸︷︷︸

aL

+(u · ∇)u︸︷︷︸

aC

= −1

ρ∇p

︸︷︷︸

aI

+ ν∇2u︸︷︷︸

aS

. (4.2)

Using the notation introduced by Tsinober et al. [101], we can also write a =aL + aC = aI + aS , where aL ≡ ∂u/∂t is the local acceleration, aC ≡ (u · ∇)uis the convective acceleration, aI ≡ −∇(p/ρ) is the irrotational pressure gradientand aS ≡ ν∇2u is the viscous acceleration. Several physical issues in turbulencebased on this decomposition of the acceleration vector have been studied. The firsttheoretical approaches date back to Monin and Yaglom [58] and Batchelor [9]. Theydeveloped traditional theories for the scaling properties of acceleration and pressure.According to Monin and Yaglom [58] and Lin [53] the total acceleration a is largelydominated by the irrotational pressure gradient and the solenoidal viscous part issmall, especially at high Re−numbers. More recent theoretical work has been doneby Hill [35]. Only recently researchers have succeeded to measure fluid particle tra-jectories and in this way deduce acceleration information. Conventional, pixel based,particle tracking methods are limited to low Re−numbers. Ott and Mann [63] usedParticle Tracking (PT) to measure turbulent diffusion of particle pairs in turbulent

52 Acceleration Statistics

flow generated by two oscillating grids, with Reλ close to 100. Here Reλ = σλ/ν isthe Reynolds number based on the Taylor length scale λ and σ, which is the r.m.s. ofthe velocity fluctuations. Voth et al. [105] used a photodiode in combination with thePT-technique to increase the spatial resolution and was in this way able to measurein a Re−number range of 900 ≤ Reλ ≤ 2000. The turbulence was created betweentwo counter-rotating disks. The acceleration variance as a function of the Reλ showedgood agreement with Kolmogorov scaling. La Porta et al. [68] performed direct mea-surements of the Lagrangian acceleration of individual particles. The method wasthe same as that of Voth et al. [105] in the same experimental setup. The Reynoldsnumber range was 140 ≤ Reλ ≤ 970. They found that the acceleration is highlyintermittent, i.e. an instantaneous value of the acceleration can be many times largerthan the root-mean-square value. Tennekes’ hypothesis [95] assumes that the to-tal acceleration a is small compared to the local and convective accelerations andhence aL and aC are strongly negatively correlated. Due to the increase of computerpower, these findings where later confirmed by DNS results of incompressible, forced,isotropic turbulence (Tsinober et al. [101], Vedula and Yeung [103]).

How all these findings relate to inhomogeneous turbulence is yet unknown. Ourprimary interest in this chapter is to investigate the various contributions to theacceleration term obtained from DNS of inhomogeneous pipe flow and to comparethe results with those obtained from isotropic turbulence. The remainder of thechapter is organised as follows. First, we will show the DNS results at two differentRe−numbers for the various contributions to the total acceleration. Single-pointvariances and geometric statistics of vector alignment as well as PDF’s of the variousterms and pressure are shown. Since we are mainly interested in the Lagrangianacceleration, which is important for the derivation of stochastic models, the pressurestatistics are examined in depth as the pressure gradient will turn out to be the largestcontributor to the Lagrangian acceleration. After that, we will more closely examinethe properties of the total, local and convective acceleration also in respect with therandom sweeping hypothesis of Tennekes and scaling with Reynolds number. As wewill see, this hypothesis is exactly the reason why the expansion from the previouschapter does not converge. Finally, conclusions are summarised.

4.2 Single-point variances

In this section we study the components of the one-point acceleration variance scaledwith Kolmogorov variables. In isotropic, homogeneous turbulence statistical proper-ties of variables of interest can be obtained by averaging over time and the completecomputational domain. In inhomogeneous pipe flow all statistical properties dependon the radius. A problem that arises here is the definition of the Re−number. Inview of Reynolds number dependence we will study results at two different Reynoldsnumbers based on the bulk velocity and consider the same radial position for bothcases. One might argue that this is not correct and we should compare results basedon local Reynolds numbers, i.e. depending on the radial position. The problem ofdefining such a Reynolds number limits the possibility to collapse all results for var-ious components of the acceleration. Since there is a dependence of statistics on the

4.2 Single-point variances 53

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

r/R

<a2 L

>

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

r/R

<a2 C

>

Figure 4.1. Acceleration variances normalised by 〈ε〉3/2 ν−1/2 as a function of the radius.Solid: radial component, dotted: tangential component, dashed: axial component. Lineswithout and with the markers are at Reτ = 362 and Reτ = 647 respectively. Left: varianceof the local acceleration < a2

L >, right: variance of the convective acceleration < a2C >.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

r/R

<a2 I

>

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

r/R

<a2 S

>

Figure 4.2. Acceleration variance normalised by 〈ε〉3/2 ν−1/2 as a function of the radius.Solid: radial component, dotted: tangential component, dashed: axial component. Lineswithout and with the markers are at Reτ = 362 and Reτ = 647 respectively. Left: varianceof the pressure gradient < a2

I >, right: variance of the viscous acceleration < a2S >.


radial position, the local Reynolds number should depend on the radial position aswell. Hill [36] suggested not to use Reλ to scale the results, since it is based on σ (ther.m.s. of velocity fluctuations) and hence not universal, i.e. it contains informationof the large, anisotropic scales and therefore cannot reveal the true scaling behavioursince these large scales are not universal. Instead he defined a new Re−number, Ra,

as the ratio between 〈a〉1/2and 〈aS〉1/2

, i.e. total (or fluid-particle) and viscous ac-celeration. This scaling did not give satisfying results in our case. Therefore, we willpresent all results as a function of the radius (when appropriate).

Since there is a large mean flow the local and convective acceleration stronglydominate over the pressure gradient and viscous term. Hence the mean flow contri-butions are subtracted from the acceleration terms by moving with the local meanaxial velocity uz(r). Figures 4.1 and 4.2 show ensemble averaged results for

⟨a2

L

⟩,

⟨a2

C

⟩,⟨a2

I

⟩and

⟨a2

S

⟩. Since we are dealing with inhomogeneous flow the result is

plotted for every individual velocity component. For a more quantified comparisonTable 4.1 shows the ratios between the components at a radial position of r/R = 0.5for Reτ = 362 and Reτ = 647 respectively. Several aspects also observed for isotropicturbulence (Tsinober et al. [101], Vedula and Yeung [103]) can be observed. Ac-cording to Tennekes’ hypotheses the total acceleration a should be small comparedto its local and convective contributions (aL and aC), which on their turn shouldhave variances close to each other. The total acceleration should also be close to thecontribution from the pressure gradient aI . The values for

⟨a2⟩

are not shown inthe figure, but results from Table 4.1 indeed indicate that 〈a〉 ≈ 〈aI〉. Tsinober etal. [101] remarked that since aI , which is irrotational, dominates over aS , a should benearly irrotational as well. On the other hand aL is solenoidal (= divergence free). Inhomogeneous turbulence irrotational and solenoidal vectors are uncorrelated. Theyindeed find no correlation between a and aL, even at very small values for the Tay-lor Reynolds number Reλ. Our results clearly show that a and aL are no longercompletely uncorrelated, due to inhomogeneity. Since a = aL + aC and a is small,aC and aL must cancel each other, i.e. they should be negatively correlated. Thecorrelation coefficients ρ(aL,aC) shown in Table 4.1 show a slight decrease towards-1 with increasing Re-number for the radial and tangential component, but oppositebehaviour for the axial component. Whether or not this is a low Reynolds numbereffect is unclear at this moment. This opposite behaviour of the axial component isalso observed in figure 4.2 for aS . The numerical values of around -0.7 for ρ(aL,aC)are relatively far from the values found by Tsinober et al. [101]. They find a value ofapprox. -0.9 at their highest Reynolds number.

4.3 Geometrical statistical properties

In line with Tsinober et al. [101] we will discuss here alignment properties of a, aL

and aC and alternatively a, aI and aS relative to each other. The notation θ(V1,V2)is used for the angle between any two vectors V1 and V2. Due to the dependence ofstatistics on the radius we will restrict ourselves to one single radial position whenconvenient. The cancellation between local and convective acceleration indicates thatthese two vectors should be antiparallel, i.e. the angle should be close to 180 degrees.

4.3 Geometrical statistical properties 55

Table 4.1. Ratios of variances of acceleration components and correlation coefficients ρ.The results are for a radial position of r/R = 0.5 and at the Re−numbers of Reτ = 362 andReτ = 647.

Reτ = 362 Reτ = 647r φ z r φ z

⟨a2⟩/⟨a2

L

⟩0.9845 0.8585 0.8380 0.7736 0.7165 0.9588

⟨a2⟩/⟨a2

C

⟩0.6011 0.5274 0.4510 0.4924 0.4539 0.5012

⟨a2

L

⟩/⟨a2

C

⟩0.6106 0.6143 0.5381 0.6365 0.6334 0.5227

ρ(a, aL) 0.1747 0.1245 -0.0111 0.1152 0.0814 0.0234ρ(a, aC) 0.6388 0.6287 0.6797 0.1152 0.0814 0.0234ρ(aL, aC) -0.6460 -0.6934 -0.7410 -0.7170 -0.7410 -0.7064⟨a2⟩/⟨a2

I

⟩1.0125 1.0365 1.1127 1.0208 1.0363 1.0592

⟨a2⟩/⟨a2

S

⟩22.0145 20.5903 14.8064 25.7896 24.8437 26.2856

⟨a2

I

⟩/⟨a2

S

⟩21.7420 19.8657 13.3063 25.2650 23.9732 24.8159

ρ(a, aI) 0.9772 0.9754 0.9658 0.9805 0.9797 0.9808ρ(a, aS) 0.1356 0.1900 0.3249 0.1501 0.1876 0.2409

This was already confirmed by Tsinober et al. [101] for isotropic turbulence. Infigure 4.3 Probability Density Functions (PDF’s) for θ(aL,aC) (a) and also θ(a,aC)and θ(a,aL) (b) are shown for the two Re−numbers at r/R = 0.5. The same isrepeated in figure 4.4 for the PDF’s for θ(aI ,aS) (a) and also θ(a,aI) and θ(a,aS)(b). There is indeed a large peak of the PDF of θ(aL,aC) close to 180o. Althoughnot as clear as in Tsinober’s simulations there is a tendency for the angle to get moreantiparallel with increasing Re−number. Tsinober et al. [101] simulated almost anorder of magnitude in the Taylor Re−number ranging from Reλ = 38 to Reλ = 243.In our simulations there is merely a factor 1.9 in Re−number range. The trend withRe−number is also not very clear in figure 4.5 where the mean values of the cosinesof the angles between a, aL and aC and a, aI and aS as a function of the radius aregiven. The anti-alignment between aL and aC does slightly increase to approx. -0.6,but is still far from Tsinober’s values of around -0.8 at their highest Reλ−numberof 243. Figure 4.5 also shows that all angles exhibit a constant, or nearly constant,behaviour over a large range of the radial coordinate. Only in the near-wall regionthere is deviation from constant behaviour. Here, cos(a,aI), cos(a,aC), cos(a,as)and cos(aC ,aI) all go to zero. Near the wall all components of the velocity tend tozero. However, as the convective terms scale with u2, they tend to zero faster thanthe local acceleration. Hence, close to the wall the total acceleration is dominatedby local acceleration and cos(a,aL) → 1. On the other hand a = aI + aS and aS

is unequal to zero at the wall, since ∂2u/∂r2 6= 0. This implies that the pressuregradient is unequal to zero as well and cos(aI ,aS) → −1.

Figure 4.3(b) shows that a is positively aligned with aL and aC . There is hardlyany Re−number dependence for both θ(a,aC) and θ(a,aL). However, this is probablydue to the slight increase in Re−number since Tsinober et al. [101] found that the


alignment of a and aC decreases significantly at higher Re−number. In agreementwith their results there is the lack of Re−number dependence for the alignment of a

with aL. They find a nearly constant mean angle between the vectors a and aL overthe whole Re−number range of approx. arccos(0.1). Our results give a constant, butmuch higher value of approx. arccos(0.25). This is easily seen in figure 4.5. Whatis surprising, and up till now not clear, is how a can be positively aligned with bothaL and aC since θ(aL,aC) is close to 180. The suggested conditional PDF [102]P (θ(a,aC)|0 < θ(a,aL) < α), where α is some threshold value, did not clarify thisbehaviour.

In figure 4.4 similar information is shown as in figure 4.3, but for the vectors a,aI and aS . The outcome is comparable with results for isotropic turbulence. ThePDF of θ(aI ,aS) is relatively flat, meaning that there is no preferential alignmentbetween these two vectors, and has no Re−number dependence. Since 〈a〉 ≈ 〈aI〉 itis expected that a is almost aligned with aI and less with aS . This is illustrated infigure 4.4(b) and in agreement with Tsinober et al. [101].

0 20 40 60 80 100 120 140 160 1800

1

2

3

(a)

0 50 100 150

100

0 20 40 60 80 100 120 140 160 1800

1

2

3

(b)

θ(aL, aC)

θ(a, aC) and θ(a, aL)

PD

FP

DF

Figure 4.3. (a) PDF’s of θ(aL,aC) in degrees at Reτ = 362 and Reτ = 647 (solid anddashed respectively.). The smaller figure is the same but with a logarithmic scale. Figuresare comparable with Tsinober et al. [101]. (b) PDF’s of θ(a, aC) (lines including markers)and θ(a, aL) (lines without markers) at Reτ = 362 (solid) and Reτ = 647 (dashed).

4.4 Pressure statistics

The standardised, i.e. scaled with its variance, one-point pressure PDF is shown infigure 4.6. The plot is in a semi-logarithmic scaling to observe the behaviour in the

4.4 Pressure statistics 57

0 20 40 60 80 100 120 140 160 1800

0.5

1

(a)

0 20 40 60 80 100 120 140 160 1800

1

2

3

(b)

0 50 100 150

100

θ(aI , aS)

θ(a, aI) and θ(a, aS)

PD

FP

DF

Figure 4.4. (a) PDF’s of θ(aI ,aS) in degrees at Reτ = 362 and Reτ = 647 (solid anddashed respectively.). (b) PDF’s of θ(a, aI) (lines including markers) and θ(a, aS) (lineswithout markers) at Reτ = 362 (solid) and Reτ = 647 (dashed). The smaller figure is thesame but with a logarithmic scale.

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

cos(aC

,aL)

cos(a,aL)

cos(a,aC

)

cos(a,aI)

cos(a,aS)

cos(aI,a

S)

r/R

Figure 4.5. Mean values of the cosines of the angles between a, aL and aC and a, aI andaS as a function of the radius. The solid lines are for Reτ = 362 and the dashed lines arethe results for Reτ = 647.


tails. The solid line represents the result at the lowest Re−number and the dottedline is the result at the higher Re−number. Both results are at a position r/R = 0.5.Figure 4.9 essentially shows the same results, but now only for the higher Re−numberat three radial positions. The pressure PDF is negatively skewed (Cadot et al. [18]).For isotropic turbulence the shape of the PDF for positive fluctuations is close tothe Gaussian distribution. Vedula and Yeung [103] suggest that this part could alsobe described by exponential decay. That would mean in this semi-logarithmic plotthat this part follows a straight line. They find the exponential decay to be closeto the Gaussian profile. In our case, however, the shape of the PDF for positivefluctuations clearly deviates from Gaussian behaviour, but still follows a straight(exponential) line. Also Cadot et al. [18], who performed experiments in a closed,stirred cylindrical box, found a perfect Gaussian fit for positive pressure. However,this was only observed for sufficiently high Re−numbers. At lower Re−numbers theyalso find deviations from Gaussian behaviour. In regions close to the wall (r/R > 0.9),the PDF starts to deviate from the PDF of more inner regions as can be observed infigure 4.9.

The shape of the PDF for negative fluctuations shows no Re−number dependence.Vedula and Yeung [103] find universal behaviour for this part of the PDF. Further-more, they find exponential tails (i.e. straight lines) up to p′/σp′ ≈ −10 (startingfrom p′/σp′ ≈ −2) for all Re−numbers. At around p′/σp′ ≈ −13 the tail changesshape by stretching more outwards. Our Re−number range is too small to observewhether or not this phenomenon also takes place for pipe flow, but an exponentialdecay is not observed for the region −10 < p′/σp′ < −2. If a ”stretched exponential”

fit is used for this region at Reτ = 647 using P (p′/σp′) = e−β(p′/σp′)α

as suggestedby Vedula and Yeung [103], the result in figure 4.7 shows that there is an increasinglydiscrepancy for more negative values of the pressure. The value of α in this case is≈ 0.94 whereas Vedula and Yeung find a value of ≈ 0.8. The fit indicates that thereis some departure from a exponential tail. If we take a smaller region for the fittingprocedure, i.e. −10 < p′/σp′ < −5 the result changes and the value for α decreases to≈ 0.83, which is more in line with Vedula and Yeung. This is indicated with the dot-ted line in figure 4.7. So the left part of the tail more or less has the same slope as inisotropic turbulence. Cao et al. [20] also studied the PDF of the pressure in isotropicturbulence. They re-plotted figure 4.7 by magnifying the negative pressure region byplotting ln(−ln(P (p′/σp′))) = ln(β)+αln(−p′/σp′) as a function of ln(−p′/σp′). Thisenables the use of a linear fit rather than a nonlinear fit. In figure 4.8 the result isshown for the two Re−numbers. The results for the two Re−numbers collapse, sothere seems to be no clear dependence on the Re−number. We obtain a value ofα = 0.8783 for ln(−p′/σp′) from 0.51 to 1.89, α = 0.8155 for ln(−p′/σp′) from 1.20to 1.89 and α = 0.7802 for ln(−p′/σp′) from 1.40 to 2.78, indicating a decreasingslope for low-pressure regions. If we use the region −10 < p′/σp′ < −5 as beforeagain, which corresponds to the region 1.62 < ln(−p′/σp′) < 2.31 an value α = 0.81is obtained which is close to the earlier found value α = 0.83.

The pressure is obtained by solving the Poisson equation. Results from previousnumerical simulations have demonstrated that even using purely Gaussian velocityfields, the pressure is still negatively skewed, so that associating the pressure PDF


−15 −10 −5 0 5 10 1510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

p′/σp′

log10P

DF

Figure 4.6. Base-10 logarithm of the standardised pressure fluctuation PDF. Results areshown for r/R = 0.5. The solid line is for Reτ = 362 and the dashed line is for Reτ = 647.The smooth dashed line represents a Gaussian distribution.

with intermittency effects must be done with some care, which was also pointed outby Nelkin [60]. The pressure head, or total pressure, is defined as P = p + 1

2u2. Notethat this is the pressure as calculated by the DNS code, as explained in Chapter 2.The PDF of the pressure head is shown in figure 4.10 for the two Re−number cases.Both PDF’s show a distribution which is close to Gaussian, something that was alsoobserved by Cao et al. [20] for their isotropic case.

The standardised PDF of the pressure gradient is shown in figure 4.11. Here, onlyresults for Reτ = 647 are included. To distinguish between the various components,we use black triangles for ∂p/∂r, squares for ∂p/∂φ and diamonds for ∂p/∂z. The fig-ure shown is at a radial position of r/R = 0.5. There are small differences between thethree separate components. These differences become bigger in the region r/R > 0.9(not shown here). There seems to be no or very little Re−number dependence (there-fore the results for Reτ = 362 are not included). This is in contradiction with Vedulaand Yeung [103]. As the Reynolds number increases, their PDF stretches out towardslarger magnitudes of ∇p. This is already observed at relatively low Reynolds num-bers. The PDF for the tangential component of the pressure gradient, i.e. 1

r ∂p/∂φ,

fits reasonably well to a function of the form P (x) = Ce−x2/((1+|αβ/σ|γ)σ2). This isshown in figure 4.12. The other two directions ∂p/∂r and ∂p/∂z show a less accu-rate result for this fit (not shown). La Porta et al. [68] performed measurements ofacceleration statistics at relatively high Re−numbers between two counter-rotatingdisks. The turbulence generated in this way was not isotropic. They used the samefunction P (x) = Ce−x2/((1+|αβ/σ|γ)σ2) to fit it to the total Lagrangian acceleration


−15 −10 −5 0 5 10 1510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

p′/σp′

log10P

DF

Figure 4.7. Same as figure 4.6, however results are only for Reτ = 647. The markers

indicate the region used for fitting with a stretched exponential P (p′/σp′) = e−β(p′/σp′)α

.The solid line is the result for this fit, which deviates for larger negative values. For the dottedline a smaller region than indicated by the markers is used for the fit (−10 < p′/σp′ < −5).

1 1.5 2 2.5 31

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

α=1

ln(p′/σp′)

ln(−

ln(P

(p′ /

σp′))

)

Figure 4.8. ln(−ln(P (p′/σp′))) = ln(β) + αln(−p′/σp′) as a function of ln(−p′/σp′). (�):Reτ = 362, (+): Reτ = 647. There seems to be no clear Re−number dependence. α = 1corresponds to an exponential tail.


−15 −10 −5 0 5 10 1510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

p′/σp′

log10P

DF

Figure 4.9. Base-10 logarithm of the standardised pressure fluctuation PDF, but now onlyresults for Reτ = 647 at various radial positions are shown. (H): r/R = 0.1, solid: r/R = 0.5,(�): r/R = 0.95.

−15 −10 −5 0 5 10 1510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

P ′/σP ′

log10P

DF

Figure 4.10. Base-10 logarithm of the standardised pressure head fluctuation PDF. Resultsare shown for r/R = 0.5. The solid line is for Reτ = 362 and the circles (◦) are for Reτ = 647.For comparison also the PDF of the pressure p is included (dashed line) for Reτ = 647. Thedistribution of the pressure head PDF is much more Gaussian.


−30 −20 −10 0 10 20 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

∇p′/σ∇p′

log10P

DF

Figure 4.11. The three components of the pressure gradient for Reτ = 647. (H): radialcomponent, (�): tangential component, (�): axial component. The results are for r/R = 0.5.

a. This fit-result is included in figure 4.12. They found no Re−number dependenceon the shape of the PDF of a. Since a is almost identical to aI it is remarkable thatVedula and Young find a strong dependency on the Re−number of the PDF of aI .La Porta et al. [68] find values β = 0.539, γ = 1.588 and σ = 0.508, in our case weobtain the values β = 0.612, γ = 1.373 and σ = 0.614. So, they find a much wider, i.e.more intermittent, behaviour for the Lagrangian acceleration compared to our results.A different way of studying the broadness, or intermittency, of the acceleration is

looking at the skewness and flatness values given by⟨a3⟩/⟨a2⟩3/2

and⟨a4⟩/⟨a2⟩2

.For a Gaussian signal the skewness should be zero and the flatness should be 3. Infigure 4.13 the skewness is plotted for the total acceleration a for both Re−numbersas a function of the radius. There is a very large region 0.1 < r/R < 0.8 where theskewness is relatively constant. Near the wall large negative values for the skewnessare observed for the radial and axial components. This same behaviour is observedfor other components of the acceleration. Hence, they are not displayed here. Themost striking result is the zero skewness for the tangential component, in line with thesymmetric shape of the PDF of the tangential component of the pressure gradient,since the total acceleration is dominated by the pressure gradient. The radial andaxial components have positive skewness, in contradiction with findings in isotropicturbulence. For all results the influence of the Re−number is small. In figure 4.14 theflatness is shown. Although there is still quite some statistical noise it is clear thatthere is a more or less constant platform in the region 0.0 < r/R < 0.8 with a valueof around F (a) ≈ 15, which is far from the Gaussian value of 3. The statistical noiseis too large to be able to distinguish any Re−number dependence.

4.5 Tennekes’ hypothesis 63

−30 −20 −10 0 10 20 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

∇p′/σ∇p′

log10P

DF

Figure 4.12. Same as figure 4.11 but only for the tangential component. Results are forr/R = 0.5. (H): Reτ = 362, (�): Reτ = 647. The solid and dashed line are fits for Reτ = 362

and Reτ = 647 respectively using the function P (x) = Ce−x2/((1+|αβ/σ|γ)σ2). The dottedline is the fit-result from La Porta et al. [68] who find no Re−number dependence.

The results for other components of the acceleration and the pressure are sum-marised in table 4.2 for the radial position r/R = 0.5. The skewness for the tangentialcomponent is very small for all acceleration components, indicating a symmetric dis-tribution. The flatness however, is not equal to the Gaussian value of 3 meaning thatthe tails of the distribution tend to stretch out.

4.5 Tennekes’ hypothesis

In this section we will focus on the predictions of Brouwers [17]. He presented anasymptotic analysis of viscous and inertial subrange behaviour of Eulerian accelerationcorrelations for general forms of turbulence. First, the general ideas of Brouwers [17]are pointed out. We will only consider the Lagrangian acceleration in terms of localand convective parts

aµ ≡ Duµ

Dt=

∂uµ

∂t+ ujσµj . (4.3)

We use a slightly different notation here compared to section 4.1. In equation (4.3) aµ

is one of the three components of the Lagrangian acceleration and uµ is the velocity.For cartesian coordinates σµx = ∂uµ/∂x, σµy = ∂uµ/∂y and σµz = ∂uµ/∂z. In cylin-drical coordinates one obtains σµr = ∂uµ/∂r, σµφ = 1

r ∂uµ/∂φ and σµz = ∂uµ/∂z.In case µ = r or µ = φ the extra curvature terms of respectively −u2

φ/r and uruφ/rshould be added to equation (4.3). This can be done by adding them to σµr and


Table 4.2. Skewness and flatness of acceleration components and the pressure at r/R = 0.5.

Reτ = 362 Reτ = 647r φ z r φ z

Sa 0.86 -0.014 0.80 0.79 0.05 0.70SaL

-0.03 -0.004 -2.02 -0.17 -0.03 -1.17SaC

0.63 -0.046 2.49 0.57 0.09 1.89SaI

1.13 -0.014 0.74 0.97 0.05 0.68SaS

0.18 -0.003 0.16 0.32 0.01 0.01Fa 13.77 10.78 16.07 16.93 16.12 17.84FaL

13.54 15.26 22.97 15.60 15.77 16.75FaC

16.70 16.18 23.95 17.85 18.51 21.32FaI

14.41 11.08 16.57 17.37 16.30 18.41FaS

9.69 10.74 10.97 11.29 11.79 11.26

Sp -0.40 -0.55SP -0.01 -0.03Fp 5.24 -5.05FP -2.86 -3.10

0 0.2 0.4 0.6 0.8 1−4

−3

−2

−1

0

1

2

3

4

r/R

Skew

nes

s(a)

Figure 4.13. Skewness of the Lagrangian acceleration a as a function of the dimensionlessradius. Solid line: radial component, dotted line: tangential component, dashed line: axialcomponent. Lines without markers are for Reτ = 362, the lines including the square markersare for Reτ = 647.


0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

50

r/R

Fla

tnes

s(a)

Figure 4.14. Flatness of the Lagrangian acceleration a as a function of the dimensionlessradius. Symbols are the same as in figure 4.13.

σµφ respectively. We would like to evaluate the separate terms in equation (4.3) ina moving coordinate system, i.e. moving with the local mean velocity. In this waythe large contribution of the mean flow is eliminated. For pipe flow, the only meanflow stems from the axial velocity uz(r), since ur(r) = uφ(r) = 0. In mathematicalform the moving system for pipe flow is described by r∗ = r, φ∗ = φ, z∗ = z − uz(r)tand t∗ = t, where the ′∗′ denotes variables in the new, moving system. Therefore,evaluation of equation (4.3) in moving coordinates leads to (where we use Einsteinnotation and dropped the ′∗′)

aµ ≡ Duµ

Dt=

∂uµ

∂t+ ujσµj − uz

∂uµ

∂z. (4.4)

This equation can be averaged, the result can be subtracted from equation (4.4) andwe end up with an expression describing the fluctuating part of the acceleration inthe moving coordinate system.

a′µ =

∂u′µ

∂t+ u′

jσ′µj − u′

jσ′µj +

∂

∂xj

(u′

juµ

). (4.5)

This can be re-grouped to

∂u′µ

∂t= −u′

jσ′µj + a′

µ + u′jσ

′µj −

∂

∂xj

(u′

juµ

)

= −u′jσ

′µj + a′

µ + γµ, (4.6)


where we have introduced a new variable γµ = u′jσ

′µj − ∂

∂xj

(u′

juµ

). For pipe flow, the

second term of γµ only has a contribution for the axial component of the acceleration.To assess the magnitude of the various terms in the before mentioned equation, oftenmean square values are used. Brouwers [17] theoretically derived scaling propertiesfor these contributions and here we will verify his findings using the DNS results. Themean square of the fluctuating part of the convective acceleration, i.e. the first termon the r.h.s. of equation (4.6), can be written as

⟨(u′

jσ′µj

)2⟩

=⟨u′

iu′jσ

′µiσ

′µj

⟩=⟨⟨

u′iu

′jσ

′µiσ

′µj

⟩⟩+⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩+

+⟨u′

iσ′µj

⟩ ⟨u′

jσ′µi

⟩+

∂

∂xi

⟨u′

iu′µ

⟩ ∂

∂xj

⟨u′

ju′µ

⟩. (4.7)

The 〈〈[·]〉〉 denotes correlation or cumulant, which can be splitted into moments andsubmoments, see e.g. Stratonovich [87] and van Kampen [42]. The cumulant of thecombination of two stochastic processes a and b is defined as 〈〈ab〉〉 = 〈ab〉 − 〈a〉〈b〉.Similarly, a third order cumulant can be written as 〈〈abc〉〉 = 〈abc〉 − 〈a〉〈b〉〈c〉 −〈ab〉〈c〉− 〈ac〉〈b〉− 〈bc〉〈a〉. Following Brouwers [17], moments of two or three randomvariables with zero mean are equal to their correlation or cumulant, whereas momentsof four variables are not. Fluctuations associated with small viscous scales and largeenergetic scales become increasingly uncorrelated with increasing Reynolds number.As Brouwers [17] points out, to assess the magnitude of each of the correlationsand second-order moments on the r.h.s. of equation (4.7) one should take for eachof the velocities and derivatives the characteristic values for either small or largescales, whatever leads to the largest value. Following this approach, the first termon the r.h.s. of equation (4.7) is completely dominated by small scales. Accordingto Kolmogorov scaling the small scales scale with σRe−1/4, where σ is the root meansquare fluctuating velocity and Re is the Reynolds number based on σ, a typical lengthscale of the large scales L0 and the kinematic viscosity ν, Re = σL0/ν. The spatialderivatives of fluctuating velocities have a characteristic value of σL−1

0 Re1/2 at thesmall scales. These scaling arguments are derived as follows. The average dissipationrate 〈ε〉 should balance the total production rate of the turbulence which is injected atthe large scales and hence scale with σ3/L0. Using the definitions of the Kolmogorov

length and velocity scales, η ∝(ν3/ 〈ε〉

)1/4and uk ∝ (ν 〈ε〉)1/4

respectively, and thedefinition of the Reynolds number as discussed earlier, Re = σL0/ν, the followingrelation is formulated for the velocity fluctuations at the small scales

uk ∝ (ν 〈ε〉)14 =

(σL0

Re〈ε〉) 1

4

∝(

σL0

Re

σ3

L0

) 14

= σRe−14 . (4.8)

In a similar way one can derive for the spatial derivatives of the velocity fluctuationsat the small scales

∂uk

∂η∝ (ν 〈ε〉)

14

(ν3/ 〈ε〉)1/4= ν− 1

2 〈ε〉12 ∝

(σL0

Re

)− 12(

σ3

L0

) 12

= σL−10 Re

12 . (4.9)


The magnitude of the 〈〈[·]〉〉-term on the r.h.s. of equation (4.7) is then given by

⟨⟨u′

iu′jσ

′µiσ

′µj

⟩⟩∼ σ4

L20

Re1/2. (4.10)

This term, which has been obtained by substituting relations (4.8) and (4.9) into theleft hand side of equation (4.10) is bigger then the one obtained if only large scaleswere taken into account. In that case the term has a magnitude of σ4/L2

0. Notethat these scaling properties only indicate an upper limit, i.e. their real values canpossibly be smaller. In a similar way we can estimate the magnitudes of the other(sub)moments by examining the large and small scales individually. The results are

⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩∼ σ4

L20

Re1

⟨u′

iσ′µj

⟩ ⟨u′

jσ′µi

⟩∼ σ4

L20

(4.11)

∂

∂xi

⟨u′

iu′µ

⟩ ∂

∂xj

⟨u′

ju′µ

⟩∼ σ4

L20

.

From these scaling properties it can be concluded that in the limit of large Reynoldsnumber the variance of the convective acceleration becomes equal to the second termon the r.h.s. of equation (4.7). This however, with a relative error of O(Re−1/2). Sowe get

⟨u′

iu′jσ

′µiσ

′µj

⟩=

⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩. (4.12)

In a similar way one can show that 〈γµ〉 ∼ σ4/L20 and

⟨a′

µ

⟩∼ σ4Re1/2/L2

0, sothat with increasing Reynolds number, γµ can be neglected compared to a′

µ, whilethe convective and local acceleration dominate over a′

µ but are complementary, i.e.cancel each other

∂u′µ

∂t= −u′

jσ′µj . (4.13)

Finally, invoking the properties of isotropy and inhomogeneity of small-scale turbu-lence one obtains

⟨(∂u′

µ

∂t

)2⟩

=〈ε〉15ν

(

2⟨

u′i2⟩

−⟨

u′µ2⟩)

. (4.14)

This equation is valid for general forms of turbulence. For isotropic flow the right handside of this equation is equal to 3 〈ε〉 / (15ν) and the well-known result of Tennekesis retrieved. We will now investigate the above findings using the DNS databaseof the pipe flow, starting with showing the result for all separate contributions inequation (4.7). On the l.h.s. and r.h.s of figure 4.15 the results for respectivelyReτ = 362 and Reτ = 647 are shown. From top to bottom the results are for theradial, tangential and axial directions. For the typical scale σ4/L2

0 we take the friction


0 0.2 0.4 0.6 0.8 1−1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

(a), (b) and (c)

(d), (e) and (f)

0 0.2 0.4 0.6 0.8 1−1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

(a), (b) and (c)

(d), (e) and (f)

0 0.2 0.4 0.6 0.8 1−2000

0

2000

4000

6000

8000

10000

12000

14000

(a), (b) and (c)

(d), (e) and (f)

0 0.2 0.4 0.6 0.8 1−2000

0

2000

4000

6000

8000

10000

12000

14000

(a), (b) and (c)

(d), (e) and (f)

0 0.2 0.4 0.6 0.8 1−2

−1

0

1

2

3

4x 10

4

(a), (b) and (c)

(d), (e) and (f)

r/R0 0.2 0.4 0.6 0.8 1

−2

−1

0

1

2

3

4x 10

4

(a), (b) and (c)

(d), (e) and (f)

r/R

Figure 4.15. Contributions of equation (4.7). The l.h.s. and r.h.s are the results forrespectively Reτ = 362 and Reτ = 647. From top to bottom the results are for the ra-

dial, tangential and axial directions. Solid:⟨(

u′jσ

′µj

)2⟩

(a), dashed:⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩(b),

dotted:⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩+⟨u′

iσ′µj

⟩ ⟨u′

jσ′µi

⟩+ ∂

∂xi

⟨u′

iu′µ

⟩∂

∂xj

⟨u′

ju′µ

⟩(c). The small terms

(closer to zero) are: solid:⟨u′

iσ′µj

⟩ ⟨u′

jσ′µi

⟩(d), dashed: ∂

∂xi

⟨u′

iu′µ

⟩∂

∂xj

⟨u′

ju′µ

⟩(e), dotted:

⟨⟨u′

iu′jσ

′µiσ

′µj

⟩⟩(f). All results have been normalised by u4

τ/D2.


0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

r/R

αµ

Figure 4.16. Scaling exponent αµ in the equation⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩∼ σ4

L20

Reαµ for µ = r

(solid), µ = φ (dashed) and µ = z (dotted). Theory predicts αµ = 1.

velocity and the diameter u4τ/D2 to make the results dimensionless. For Reτ = 362

and Reτ = 647 the results are plotted on the same scale for comparison. Even atthese relatively low Reynolds numbers relation (4.12) holds, indicated by the (upper)solid and dashed lines in the figure. This is true for all three velocity components,although for the axial component there are some differences in the near wall region.The contributions from

⟨u′

iσ′µj

⟩ ⟨u′

jσ′µi

⟩(solid) and ∂

∂xi

⟨u′

iu′µ

⟩∂

∂xj

⟨u′

ju′µ

⟩(dashed)

are very small.

Equation (4.11) predicts the scaling behaviour for⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩. If we assume

⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩∼ σ4

L20

Reαµ and determine αµ as a function of the radius using the

two available Reynolds numbers, the result looks like figure 4.16. Over a very largeregion it is indeed found that αµ ≈ 1 for all three directions. For a small regionnear the pipe axis the

⟨⟨u′

iu′jσ

′µiσ

′µj

⟩⟩term shows scaling behaviour according to

equation (4.10), but less convincing (not shown here). The other scaling dependenciesin equation (4.11) are not confirmed by the DNS results. These terms are alreadyvery small and do not show any scaling.

In figure 4.17 the l.h.s. (solid) and r.h.s. (dashed) of equation (4.14) are shownas functions of the radius for both Reτ = 362 (without markers) and Reτ = 647(including markers). The results shown here are for the radial components only, sincethe tangential and axial directions show similar results. Clearly, the balance betweenlocal acceleration and dissipation rate does not hold. Over a large region only afraction of the r.h.s. is obtained. This equation is derived from three assumptions.The first assumption contains the scaling rules for large and small-scale turbulence


0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

1400

1600

1800

2000

r/R

Figure 4.17. Balance of equation (4.14). Lines with and without markers are at

Reτ = 647 and Reτ = 362 respectively. The solid line is

⟨(∂u′

µ

∂t

)2⟩

and the dashed line is

〈ε〉15ν

⟨

u′i2⟩

(2 − δµi).

and the decorrelation between large and small scales which leads to equation (4.7).This assumption holds, even at these moderate Reynolds numbers as we saw earlier.

This leaves the possibility that either the second assumption that∂u′

µ

∂t and −u′jσ

′µj

dominate over a′µ + γµ or that local isotropy, the third assumption, do not hold.

The second assumption is validated in figure 4.18 where variances of all terms inequation (4.5) are plotted as functions of the radius (for the radial component only).Here, the variance of γµ is negligible but that of a′

µ is of the same order as that of∂u′

µ

∂t , although there is some improvement with increasing Reynolds number. Thiswas also more extensively studied in section 4.2. So, the second assumption, thevalidity of equation (4.13), is already introducing some deviations at these relativelylow Reynolds numbers. In the next chapter we will focus on the local isotropy of thepipe flow.

4.6 Summary

The various acceleration components in the Navier-Stokes equations have been anal-ysed for inhomogeneous pipe flow. Although there are many similarities with isotropicturbulence, some differences can be observed as well.

A question which requires further investigation is whether discrepancies from

4.6 Summary 71

0 0.2 0.4 0.6 0.8 1−200

0

200

400

600

800

1000

1200

1400

r/R

Figure 4.18. The reason why the balance of equation (4.14) is poorly represented. Lineswith and without markers are at Reτ = 647 and Reτ = 362 respectively. The solid line is⟨γµ

2⟩, the dashed line is

⟨(u′

jσ′µj

)2⟩

and the dotted line is⟨aµ

2⟩.

isotropic results by Tsinober et al. [101] are caused by the inhomogeneity of theflow or by the small value of the Re−number. To investigate this, simulations athigher Re−number should be performed.The theoretically derived scaling behaviour for

⟨u′

iu′j

⟩ ⟨σ′

µiσ′µj

⟩is observed, even at

these relatively low Reynolds numbers. However, the Lagrangian acceleration a′µ is

still of the same order of magnitude as the local and convective accelerations∂u′

µ

∂t and−u′

jσ′µj respectively.

Chapter 5

The Small Scale Structure of

Pipe Flow

5.1 Introduction

In the previous chapter various acceleration contributions have been investigated onReynolds number scaling. An important conclusion was that the total Lagrangianacceleration at the rather low Reynolds number is still relatively large compared tothe local and convective acceleration. This is one reason why the variance of thelocal acceleration is not adequately described by equation (4.14). Another possiblecontribution to the discrepancy in equation (4.14) is the validity of local isotropy.Therefore, in this chapter we will investigate whether the flow has the tendency tobecome more isotropic at the small scales with increasing Reynolds number. This isalso of direct relevance to understanding and modelling turbulent flow.

5.2 Local Isotropy

At sufficiently high Reynolds number the smallest fluctuations appear to be approxi-mately statistically isotropic. In case of anisotropic flow, like in our case, it is called’locally isotropic’ turbulence. Or as Nelkin [60] puts it:

’isotropy of the small scales in the presence of large scale anisotropy is called lo-cally isotropic turbulence’.

This local isotropy could also lead to the more restrictive property of universalityof the small scales. Hill [36] rephrases it as:

’universality of the small scales of turbulence is the hypothesis that every statisticof the small scales, when appropriately scaled, should become independent of the typeof flow as the Reynolds number increases’.

74 The Small Scale Structure of Pipe Flow

The scale region which obeys universality and local isotropy follows from Kol-mogorov’s scaling assumption [45] and is called the inertial sub-range. If universalityand local isotropy exist, then the small scales, part of the inertial sub-range, can berepresented by their universal statistical properties, irrespective of the flow geome-try. The properties of the large, anisotropic scales are defined by the external lengthscale L0, indicating that these scales vary from one geometry to another. These arethe scales which are interesting for engineering applications and will need modellingor measurements for each separate problem we are interested in. If we summarisethe main ideas of Kolmogorov’s work from 1941 using Sreenivasan and Antonia’sformulation [81] it can be split into two parts.

• ’When the fluid viscosity ν is small, the average energy dissipation rate 〈ε〉 isindependent of ν’

• ’Small-scale turbulence at sufficiently high Reynolds numbers is statistically in-dependent of the large scale, and is homogenous, isotropic and steady. This isknown as ”locally isotropic”’

These two hypotheses led to the famous self-similarity hypotheses which are oftenreferred to as K41 (Kolmogorov 1941) and state:

• ’The statistical properties of the dissipation scales are determined universally by〈ε〉 and ν’

• ’The statistical properties of scales in the inertial range, if the Reynolds numberis high enough for one to exist, are determined by 〈ε〉 only’

G.I. Taylor [93] introduced the idealization of homogeneous isotropic turbulence.This idealization has also been by far the most studied flow using direct numericalsimulations. Real flows, like jets, wakes and wall bounded flows are neither homoge-neous nor isotropic. Moreover, simulations of isotropic flows are either decaying ordepend on a rather non-physical type of forcing to prevent the flow from decaying.The study of local isotropy at real flows, using numerical simulation, is restrictedby the Reynolds number. For these flows, the Reynolds numbers attained so far arenot high enough to obtain a clear inertial sub-range, something that nowadays isonly starting to become possible for isotropic simulations. This has restricted theresearch to experiments, which however face other problems. In most experimentsthe quantity that is measured is the velocity along the direction of the mean flow(see e.g. Shen and Warhaft [85]). It is then assumed that the mean flow, u, sweepsthe turbulence by the point of observation without distortion. This is called thefrozen turbulence approximation of G.I. Taylor. This hypothesis is used to transformu(t) at a fixed point in space to the spatial distribution of the velocity u(x) usingu(x, t) = u(x0, t − (x − x0)/u). The measurement of transverse velocity incrementsis much more complicated since it requires arrays of measurement probes (see e.g.Nelkin [60] or Saddoughi and Veeravalli [74]).

Now that local isotropy has been labelled as an important property, we have toformulate a measurable quantity to describe it. A very suitable way to do this is bymeans of correlation- and structure functions. The low order two-point correlations

5.2 Local Isotropy 75

already describe many of the physical properties of turbulence. The higher ordervariants will be discussed in section 5.4. Let u = (u, v, w), then the second orderlongitudinal and transverse correlation function are given by [58] [60]

BLL(r, t) = 〈u(x, y, z, t)u(x + r, y, z, t)〉 (5.1)

BNN (r, t) = 〈v(x, y, z, t)v(x + r, y, z, t)〉 . (5.2)

In case of stationary flow the brackets 〈·〉 denote equally time- or ensemble averages.For isotropic flow, the cross correlation BLN (r, t) vanishes. In a similar way the secondorder structure functions DLL(r, t) and DNN (r, t) are given by

DLL(r, t) =⟨

(u(x + r, y, z, t) − u(x, y, z, t))2⟩

(5.3)

DNN (r, t) =⟨

(v(x + r, y, z, t) − v(x, y, z, t))2⟩

. (5.4)

When higher order structure- or correlation functions are meant, the number of sub-scripts in the equations is increased. A third order longitudinal structure function

for example is defined as DLLL(r, t) =⟨

(u(x + r, y, z, t) − u(x, y, z, t))3⟩

. Sometimes

we will also use the notation Dp(r, t) = 〈(u(x + r, y, z, t) − u(x, y, z, t))p〉. We will

restrict ourselves now to spatial correlation and structure functions so that they arefunctions of the separation r only. For convenience we will now omit x, y, z andt from the equations. In case of isotropic, incompressible turbulence there exists arelation between longitudinal and transverse structure functions (see [60]):

DNN (r) = DLL(r) +r

2

∂DLL(r)

∂r. (5.5)

Equation (5.5) is a measure for local isotropy. If this relation is investigated for thepipe flow results, it becomes clear that it does not hold at these moderate Reynoldsnumbers. On the left hand side of figure 5.1 the results for the two Reynolds numbersReτ = 362 and Reτ = 647 are shown if r is taken in the axial direction of the pipeand the axial velocity component uz is investigated, i.e. the longitudinal structurefunction. This is done for a radial position of r/R = 0.5. With increasing Reynoldsnumber the validity of equation (5.5) improves.

In a similar way as for the second order structure function, there exists a theo-retically exact relation, in fact the only exact relation, for the third order structurefunction, which originates from the second similarity hypothesis of K41 (see [58])

−DLLL(r) =4

5〈ε〉 r. (5.6)

This relation is known as the 4/5 law. It is derived under the assumption of isotropyat all scales. Since the equation only concerns structure functions, it can be assumedthat the result is correct for the inertial range scales where the turbulence is locallyisotropic even if the large scales are not. From the right hand side of figure 5.1, wherethe result is shown at the two Reynolds numbers at the radial position r/R = 0.5, wecan conclude that there is not yet a clear inertial range.


101

102

10−2

10−1

100

r/η10

010

110

210

310

−4

10−3

10−2

10−1

100

101

r/η

Figure 5.1. Left: DNN (r) (solid) and DLL(r) + (r/2)∂DLL(r)/∂r (dashed) normalised byu2

τ as functions of the separation distance r normalised by the Kolmogorov length scale η.The results are for r/R = 0.5. Lines without and with the markers are at Reτ = 362 andReτ = 647 respectively.Right: DLLL(r) in axial direction as a function of the separation distance r normalisedby the Kolmogorov length scale η. The results are for r/R = 0.5. (◦): Reτ = 362, (+):Reτ = 647. The solid and dashed lines are Kolmogorov’s 41 prediction, DLLL(r) = 4/5 〈ε〉 r,for Reτ = 362 and Reτ = 647, respectively.

5.3 Dissipation rate as a discriminator for local iso-

tropy

Although a clear inertial range is not present in our current direct numerical simu-lation results, we will attempt to investigate to which extent the pipe flow is locallyisotropic and if there is a trend towards isotropy with increasing Reynolds number.The second similarity hypothesis of K41 implies that at sufficiently high Reynoldsnumber small-scale turbulence is homogeneous and isotropic and solely depends onthe average dissipation rate 〈ε〉. In cartesian coordinates 〈ε〉 is given by

〈ε〉 =ν

2

⟨(∂ui

∂xj+

∂uj

∂xi

)2⟩

= ν

⟨(∂ui

∂xj+

∂uj

∂xi

)∂uj

∂xi

⟩

. (5.7)

In Hinze [37] a relation for the dissipation rate is derived for homogeneous, isotropicturbulence. In this case the expression for the dissipation rate simplifies (due to sym-

metries) to 〈ε〉 = 15ν⟨

(∂u1/∂x1)2⟩

, so the dissipation rate can be fully expressed in

longitudinal derivatives. Hence, in isotropic experiments, e.g. measurements in tur-bulence behind a grid, the average dissipation rate can be obtained from the measuredtime record using Taylor’s frozen turbulence hypothesis. Obviously this relation canbe used to investigate how local isotropy is violated or satisfied in the pipe flow case.It is also possible to assign different velocity derivatives to portions of the dissipation

5.4 Intermittency 77

rate. One then obtains

⟨∂ui

∂xk

∂uj

∂xl

⟩

=

115

εν , if i=j =k= l.

215

εν , if k= l and i=j and i 6= k.

± 130

εν , if (j = l and i=k) or (i= l and j =k and i 6= k).

0, other combinations.

(5.8)

In cylindrical coordinates the same result is obtained albeit with some extra termsdue to curvature. This results in fifteen different correlations unequal to zero, thatcontribute to the total dissipation rate. The first two contributions in equation (5.8),i.e. the correlations for i = j = k = l (three in total) and k = l ∧ i = j ∧ i 6= k (six intotal), add up to the total dissipation term 〈ε〉. The third term has contributions of± 1

30εν for the individual components, but the sum of all six components should equal

zero. In the pipe flow case the average dissipation rate 〈ε〉 is a function of the radius.An appropriate way to check for local isotropy is looking at the normalised quantity⟨

∂ui

∂xk

∂uj

∂xl

⟩

/(C εν ), which should equal unity and where C is the value according to

relation (5.8). The normalised quantity should be unity for all contributions in caseof local isotropy.

Figure 5.2 shows the normalised quantity⟨

∂ui

∂xk

∂uj

∂xl

⟩

/(C εν ) for i = j = k = l

(left) and k = l ∧ i = j ∧ i 6= k (right). The lines without and with markers are atReτ = 362 and Reτ = 647 respectively. For most correlations there is the tendency tobecome more isotropic, i.e. get closer to unity, especially near the pipe axis. Similarbehaviour is observed for the correlations with (j = l ∧ i = k) ∨ (i = l ∧ j = k ∧ i 6=k), which are six terms in total and can be observed in figure 5.3. The sum of thesesix terms should be close to zero, which is nicely obeyed in a large region of the pipeas is illustrated in figure 5.4. The region increases with increasing Reynolds number.The figure shows the correlations with (j = l ∧ i = k) ∨ (i = l ∧ j = k ∧ i 6= k),where the squares are the sum of the six terms.

5.4 Intermittency

Kolmogorov’s hypotheses from 1941 predict that structure functions of order p arerelated to the average dissipation rate and separation distance r by

Dp(r) = Cp [〈ε〉 r]p3 , (5.9)

in case r is within the inertial range. In this equation Dp is the longitudinal structurefunction of order p

Dp(r) = 〈(u(x + r, y, z, t) − u(x, y, z, t))p〉 , (5.10)


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

r/R0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

r/R

Figure 5.2. The normalised quantities⟨

∂ui

∂xk

∂uj

∂xl

⟩

/(C εν) for i = j = k = l (left). Solid:

radial, dashed: tangential, dash-dotted: axial. For k = l ∧ i = j ∧ i 6= k (right) a selection

of results is made. Solid:

⟨(∂ur

∂φ− uφ

)2⟩

/(C εν), dashed:

⟨(∂uφ

∂z

)2⟩

/(C εν), dash-dotted:

⟨(∂uz

∂r

)2⟩

/(C εν). In both figures the lines without and with the markers are at Reτ = 362

and Reτ = 647 respectively.

as shown in equation (5.3) and the Cp are universal constants. The structure functionsscale with r as Dp(r) ∼ rζp , with ζp = p/3. It was first objected by Landau in1944 [51], shortly after the presentation of the K41 hypotheses, that this scaling couldnot be correct due to intermittency. A broad variety of models has been introducedsince that time to capture the correct scaling behaviour. A few examples are the log-normal model, stemming from Kolmogorov’s refined similarity hypothesis [46], thep-model, and more recently fractal models. An overview of these models can be foundin [34] or [60]. We have to remark here that in the log-normal model ζp has a parabolicform and that this prediction does not agree with experiments for higher values of p.

It is known that for low values of p all models more or less coincide with availableexperimental data, and that differences between the various models only becomeevident at higher values of p. At these high values of p the statistical accuracy forboth experiments and numerical results starts to become problematic, since the tails ofthe probability density function are multiplied by higher values and therefore becomeincreasingly important. It are these tails that are difficult to measure or calculate withhigh accuracy. The numerical attempts so far are restricted to isotropic, homogeneousturbulence (see e.g. Vincent and Meneguzzi [104]) and channel flow (see e.g. Toschiet al. [100] and Amati et al. [1]). Results for pipe flow are not available until now.For the low to modest Reynolds numbers obtained with numerical simulation, thereis the problem of lack of a clear inertial range. This was also remarked in the previoussection. It complicates the extraction of the scaling exponents p from the calculatedstructure functions. In figure 5.5 Dp(r) is shown for p = 1 . . . 8. The separation r isin the axial direction. A clear scaling region where Dp(r) ∼ rζp , is not present.


0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

r/R

Figure 5.3. Typical results for the normalised quantities⟨

∂ui

∂xk

∂uj

∂xl

⟩

/(± 130

εν) for (j = l ∧ i =

k) ∨ (i = l ∧ j = k ∧ i 6= k). Solid:⟨

∂ur

∂r∂uz

∂z

⟩/(C ε

ν), dashed:

⟨1r

(∂ur

∂φ− uφ

)∂uφ

∂r

⟩

/(C εν),

dash-dotted:⟨

1r

∂uφ

∂z∂uz

∂φ

⟩

/(C εν). Lines without and with the markers are at Reτ = 362 and

Reτ = 647 respectively.

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/R0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r/R

Figure 5.4. Same as figure 5.3 but now unnormalised and for all six components at Reτ =362 (left) and Reτ = 647 (right). (�): sum of the six terms.


100

101

102

103

10−4

10−2

100

102

104

← p=1← p=2

← p=3

← p=4

← p=5

← p=6

← p=7

← p=8

r/η

Dp(r

/η)

Figure 5.5. Longitudinal structure functions Dp(r) for p = 1 . . . 8 as a function of axial sep-aration r normalised by the Kolmogorov length scale η. The slope increases with increasingvalue of p. The results are for Reτ = 647 and for the radial position r/R = 0.5.

Several attempts have been made to extend the region to extract scaling behaviour.Benzi et al. [11] found that plotting the absolute value of the structure function Dp(r)as a function of Dq(r), with p 6= q, resulted in very clear scaling behaviour, includingthe dissipative range. This property is called extended self similarity (ESS), since thescaling region extends from the inertial range to the dissipative range. Self-scalingmeans that the structure function of order p obeys the relation

Dp(r) ∼ Dq(r)ζp/ζq . (5.11)

Compared to Kolmogorov’s 1941 prediction of Dp(r) ∝ rp/3 (since D3 ∝ r and 〈ε〉 isindependent of r) which becomes valid at high Reynolds numbers, the ESS appears tobe valid at moderate Reynolds numbers as well. The ESS turns out to be detectabledown to scales of the order of 5η, whereas K41 scaling fades away if r is of the orderof 30η [1]. In figure 5.6 Dp(r) is shown as a function of D3(r) for p = 1 . . . 8. Nowscaling exponents can easily be extracted from the slopes of the dashed lines. Theselines are least-square fits of Dp(r) = CpDq

ζ , with ζ = ζp/ζq. This can be writtenas log(Dp(r)) = log(Cp) + ζlog(Dq), which is a linear least-square problem. Higherorder structure functions, p > 8, suffer from too much statistical noise and are notdisplayed here. The fit directly gives the scaling exponents ζ as a function of p. This isshown in figure 5.7. The squares and black triangles are the results for Reτ = 362 andReτ = 647 respectively at r/R = 0.5. The solid line is Kolmogorov’s 1941 predictionζ = p/3. She and Leveque [83] proposed a hierarchy structure model yielding thescaling exponents for the velocity increment ζ = p/9 + 2(1 − (2/3)p/3), denoted bythe dashed line. The log-normal model corresponds to the dash-dotted line. For this


10−3

10−2

10−1

100

101

10−4

10−2

100

102

104

← p=1← p=2

← p=3

← p=4

← p=5

← p=6

← p=7

← p=8

D3(r/η)

Dp(r

/η)

Figure 5.6. Longitudinal structure functions Dp(r) for p = 1 . . . 8 as a function of D3(r)normalised by the Kolmogorov length scale η at the radial position r/R = 0.5. The slopeincreases with increasing value of p. The results are for Reτ = 647. The dashed lines arefits of Dp(r) = CpDq

ζ , with ζ = ζp/ζq.

regime of powers p, the results for the She-Leveque and log-normal model are closetogether and in good agreement with the numerical results. The result for Reτ = 647corresponds better with the models compared to the Reτ = 362 result. The modelsstart to deviate for values p > 10, see e.g. Chen and Cao [21], but these values arenot attainable using the current DNS results since the higher moments have not yetstatistically converged.

In 1962 Kolmogorov [46] refined his hypotheses from 1941 by averaging the localdissipation rate over a sphere with radius r

εr(r, t) =

∫

Vr

ε(x)dV, (5.12)

such that equation (5.9) transforms to

Dp(r) = C∗p [〈εr〉 r]

p3 , (5.13)

where the C∗p are no longer universal constants. The locally averaged dissipation rate

scales according to 〈εr〉p ∝ rτp and combined with equation (5.13) this results inscaling exponents for Dp equal to ζp = p/3 + τp. The refined hypotheses were neededto account for the failure of the K41 theory to predict the different scaling behaviour,i.c. deviations from ζ = p/3. Kolmogorov postulated that the logarithm of εr(r, t)has a log-normal distribution with increasing variance if r increases. The refined


1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

p

ζ p

Figure 5.7. Scaling exponent ζp as a function of order p. The solid line is K41-predictionζp = p/3, the dashed line is the She-Leveque model ζp = p/9 + 2(1 − (2/3)p/3), the dash-dotted line is the log-normal model ζp = p/3 − µp(p − 3)/18, with µ = 0.2. (�): CurrentDNS at Reτ = 362, (H): Current DNS at Reτ = 647. The radial position equals r/R = 0.5.

hypotheses are still subject to investigation, see e.g. [39], [70], [88], [97] and [119].With the current simulation it is possible to calculate the dissipation rate accordingto its definition and hence integrate it over a sphere with radius r. In this way thescaling exponent τp can directly be obtained without the use of the ESS method.

Figure 5.8 shows 〈εr〉p as a function of r for various values of p. There is a largeregion where scaling behaviour is observed, allowing a least square fit to obtain thescaling exponent. For larger values of r the dependence of the dissipation rate on theradial coordinate starts to influence the results. The final result for ζp = p/3 + τp

is shown in figure 5.9 for both Reynolds numbers for a position halfway the radiusof the pipe. Again, there is good agreement between the models and the numericalresults as seen earlier with the ESS theory. In this case the result for Reτ = 362 alsois in better agreement with the models.

Until now we have only studied the scaling exponents at position r/R = 0.5.Antonia [4] et al. used DNS data from channel flow, with the Reynolds number basedon half the channel height, friction velocity and dynamic viscosity equal to 180. Theseare one of the few results reported for the scaling exponents in inhomogeneous flowuntil now (see e.g [12] or [1] for other examples). They studied the variation of ζp

with the distance from the wall for p = 4, p = 6 and p = 8. For channel flow ζp

increases with the distance from the wall, but also shows a local maximum at aroundy+ ≈ 5 and a local minimum at around y+ ≈ 20. Our pipe flow results at Reτ = 362shown in figure 5.10, which correspond with the channel flow results of Antonia et al.,are in good agreement. Only in the very vicinity of the wall, i.e. for very small values


100

101

100

101

102

p=1 →

p=2 →

p=3 →

p=4 →

p=5 →

p=6 →

p=7 →

p=8 →

r/η

〈εr〉p

Figure 5.8. 〈εr〉p as a function of r for p = 1 . . . 8. Symbols are the same as in figure 5.5.

The results are for Reτ = 647 at the radial position r/R = 0.5. The dotted lines are fits of〈εr〉

p = C∗prτp .

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

p

ζ p

Figure 5.9. Same as figure 5.7, but now the numerical scaling exponents are obtained byintegrating the dissipation rate over a volume with radius r. (�): Current DNS at Reτ = 362,(H): Current DNS at Reτ = 647.


0 50 100 150 200 250 300 3501.1

1.2

1.3

p=4

0 50 100 150 200 250 300 350

1.4

1.6

1.8

p=6

0 50 100 150 200 250 300 3501.5

2

2.5

p=8

y+

ζ pζ p

ζ p

Figure 5.10. Variation of the scaling exponent ζp for p = 4 (top), p = 6 (middle) and p = 8(bottom) with the wall coordinate y+. (�): Channel flow by Antonia [4], (H): Current DNSat Reτ = 362, (�): Current DNS at Reτ = 647. The lines through the symbols are addedto clarify the trend.

of y+, small differences can be observed. For large values of y+, the pipe flow resultsshow more scatter, especially for the larger values of p, due to the worse statisticalaveraging near the pipe axis. Included in the figures are the results for Reτ = 647.Clearly, the scaling exponents are now approximately constant in a relatively largeregion, i.e. 125 ≤ y+ ≤ 325. The results also coincide with the results at Reτ = 362,indicating there is no Reynolds number dependence.

5.5 Summary

In this chapter local isotropy of the flow has been investigated. At the low to mediumReynolds numbers attainable with the current Direct Numerical Simulations, no clearinertial subrange is present. If the various contributions of the dissipation rate areused as a measure for local isotropy, the flow tends to become more isotropic withincreasing Reynolds number. This, however, is not sufficient for K41 theory to be validin the limit of large Reynolds numbers. For higher order statistics, we have shown thatthis theory does not hold, due to the increasing influence of intermittency. The scalingbehaviour of these higher order statistics is in agreement with channel flow resultsat the same (low) Reynolds number. It also corresponds with various intermittencymodels, where we should note that differences between these models are only visibleat very high orders, not accessible with the current numerical approach. The scalingexponents do not depend on the Reynolds number, but only depend on y+.

Chapter 6

Lagrangian Stochastic Model

for Pipe Flow

6.1 Introduction

To investigate the dispersion in turbulent flows, Taylor introduced a stochastic modelfor the position of a fluid particle x(t). Later on it was found that this model isidentical to the stochastic equation proposed in 1908 by Langevin to model the velocityof particles undergoing Brownian motion. That is why nowadays stochastic models aresometimes also called Langevin models. An overview of the history and developmentsof Lagrangian stochastic models can be found in Sawford’s review article [79].

Apart from a model for the position of a fluid particle as proposed by Taylor,there are stochastic models for the velocity of a fluid particle. If we restrict ourselvesto homogeneous, isotropic turbulence, the general Langevin equation for the velocityof a fluid particle is given by

dv′µ

dt= Aµjv

′j + Bµjwj(t), (6.1)

where Aµj is called the drift tensor or damping tensor, Bµj is the diffusion tensorand wj(t) is random white noise. For homogeneous, isotropic turbulence the only

sensible choice of coefficients is Aµj = −δµj/tc and Bµj =(2σ2

v/tc)1/2

δµj where tcis the Lagrangian correlation time (or integral time scale) and σ2

v is the variance ofthe velocity. For this type of flow tc = 2σ2

v/(C0ε), so that Bµj = (C0ε)1/2δµj . Here,

C0 is called Kolmogorov’s constant and ε is the dissipation rate. Several propertiescan be derived from the result for the Lagrangian autocorrelation obtained from thisLangevin model: ρ(τ) = 〈v′

i(t)v′i(t + τ)〉 = exp (−|τ |/tc). The first property is that it

contains only one time scale tc characteristic for the large, energy containing scales.Secondly, there is no Reynolds dependence. And finally the slope of ρ(t) is discontin-uous at t = 0, indicating that the velocity from the Lagrangian stochastic model iscontinuous, but not differentiable. These properties also reveal the shortcomings of

86 Lagrangian Stochastic Model for Pipe Flow

the Langevin model when finite Reynolds numbers are considered. This can be over-come by introducing a refined model as proposed by Sawford [75] and Pope[66]. Withthis model a second time scale is introduced, namely a time scale proportional withthe Kolmogorov time scale τη, which introduces a Reynolds number dependence sincethe ratio tc/τη increases with Reynolds number. Also the velocity autocorrelation isnow differentiable at τ = 0.

In the past, the velocity models have been compared with homogeneous andisotropic turbulence, usually with good results. This is due to the property thatfor this type of flow analytical relations can be derived for the damping coefficients(sometimes also referred to as the ’drift’-term) in the stochastic model. This was workdone by Thomson [96] where he showed how to ensure that the damping coefficients inthe stochastic model are consistent with prescribed Eulerian statistics. While Thom-son’s approach ensures consistency with the prescribed Eulerian statistics, it doesnot give a unique stochastic model in two or three dimensions when the flow is notisotropic and homogeneous. This is called the ’non-uniqueness’ problem and reducedmany investigations to finding the ’best performing’ model out of the many that areconsistent with the Eulerian statistics. A model that satisfies this condition is calleda model from the ’well-mixed’ class. As a consequence, models were tested on theiroutcome by comparing the results with experimental data. A more robust methodis to use Lagrangian statistics, which have been calculated with a DNS, to constructclosures for the stochastic models.

Only recently people have started to develop models for inhomogeneous and aniso-tropic flows using this approach. For example a general linear model for the velocitywas used by Pope [67] for homogeneous turbulent shear flow by using DNS data.Sawford and Yeung [78] compared two different Lagrangian models with DNS resultsof uniform shear flow. However, for a real anisotropic flow, like pipe flow, the perfor-mance of stochastic models is not really known. In the next section a linear stochasticmodel for the velocity is introduced. Lagrangian DNS data is used to fit the unknowndamping coefficients and Kolmogorov’s constant. In section 6.3 the model results arecompared with the Lagrangian DNS data. Correlation functions of the DNS are com-pared with those of the model as well as statistics of the position and velocity of fluidparticles. After that, the dispersion of a point source is modelled with the stochasticmodel and compared with two DNS results and experiments. Finally in section 6.4,the conclusions are summarised and recommendations for improvements are given.

6.2 Linear stochastic model

6.2.1 Model properties

In this subsection the derivation and some properties of the linear stochastic modelare discussed. If we start with the Navier-Stokes equations

∂uµ

∂t+ uj

∂

∂xjuµ +

1

ρ

∂p

∂xµ= ν

∂2uµ

∂x2j

, (6.2)

6.2 Linear stochastic model 87

decompose the velocity and pressure into its mean and fluctuating part uµ = u0µ(x)+

u′µ(x, t) and p = p0(x)+p′(x, t) and use this in equation (6.2) we can make a Reynolds

decomposition which leads to

∂u′µ

∂t+ u0

j

∂

∂xju′

µ + u′j

∂

∂xju0

µ + u′j

∂

∂xju′

µ +1

ρ

∂p0

∂xµ+

1

ρ

∂p′

∂xµ= ν

∂2u0µ

∂x2j

+ ν∂2u′

i

∂x2j

. (6.3)

Here we already used the property u0j

∂∂xj

u0µ = 0 which holds for unidirectional flows,

such as pipe and channel flow. Equation (6.3) can be averaged which results in

∂

∂xj

⟨u′

ju′µ

⟩+

1

ρ

∂p0

∂xµ= ν

∂2u0µ

∂x2j

. (6.4)

Now let us look at the Lagrangian frame. The Lagrangian velocity is given by

vµ(t) = uµ(x(t), t), (6.5)

and the fluctuating velocity by

v′µ(t) = uµ(x(t), t) − u0

µ(x(t)). (6.6)

The acceleration of a fluid particle is then equal to

dv′µ(t)

dt=

∂uµ

∂t+

∂uµ

∂xj

dxj

dt−

∂u0µ

∂xj

dxj

dt=

∂uµ

∂t+ uj

∂

∂xjuµ − uj

∂

∂xju0

µ. (6.7)

Using equation (6.2) this leads to

dv′µ(t)

dt= −1

ρ

∂p

∂xµ+ ν

∂2uµ

∂x2j

− uj∂

∂xju0

µ

= −1

ρ

∂p0

∂xµ+ ν

∂2u0µ

∂x2j

− 1

ρ

∂p′

∂xµ+ ν

∂2u′µ

∂x2j

− u′j

∂

∂xju0

µ

Using eq. (6.4)=

∂

∂xj

⟨u′

ju′µ

⟩− 1

ρ

∂p′

∂xµ+ ν

∂2u′µ

∂x2j

− u′j

∂

∂x′j

u0µ. (6.8)

The average acceleration of the Lagrangian velocity is obtained by averaging thisequation and results in the simple relation

⟨dv′

µ

dt

⟩

=∂

∂xj

⟨u′

ju′µ

⟩. (6.9)

For the fluctuating velocity of a fluid particle a simple Markov model, or Langevinequation, reads

dv′µ

dt= gµ(v′,x, t) +

√

C0εwµ(t)

dxµ

dt= vµ.

(6.10)


The C0ε term on the r.h.s. follows from K41 theory and C0 is assumed to be a universalconstant, in case the Reynolds number is large enough. wµ(t) is Gaussian white noiseand properties of the noise will be discussed later on. The form and identity of thefunctions gµ(v′,x, t) are not known for inhomogeneous flow. A reasonable assumptionfor a first approach would be to let the function vary linearly in vµ, i.e. gµ(v′,x, t) =

g0µ(x) + aµjv

′j . For this model

⟨dv′

µ

dt

⟩

= g0µ(x) = ∂

∂xj

⟨u′

ju′µ

⟩, which follows from

equation (6.9). This term represents the average drift and should be added to theLangevin equation. For pipe flow the ∂

∂xj

⟨u′

ju′µ

⟩term reduces to 1

rddr

⟨ru′

ru′µ

⟩since

averaged quantities do not depend on the tangential or axial direction in a stationaryflow.

If we assume a linear model in Cartesian coordinates then in cylindrical coordinatesthe complete first order Markov/Langevin model is given by

dv′r

dt= arrv

′r + arzv

′z +

1

r

d

dr

⟨ru′2

r

⟩+

1

rv′2

φ + (C0ε)1/2

wr(t)

dv′φ

dt= aφφv′

φ − 1

rv′

rv′φ + (C0ε)

1/2wφ(t)

dv′z

dt= azrv

′r + azzv

′z +

1

r

d

dr〈ru′

ru′z〉 + (C0ε)

1/2wz(t).

(6.11)

Note that due to the transformation extra curvature terms arise in the model, i.e. theterms 1

r v′2φ and − 1

r v′rv

′φ. Hence, a linear model in Cartesian coordinates automatically

means a nonlinear model in cylindrical coordinates. Furthermore, we made use of thefact that the tangential direction is fully decoupled from the other two directions.

This means that correlations like e.g.⟨

u′φu′

z

⟩

and⟨

u′φu′

r

⟩

are zero. The same holds

for the damping coefficients arφ, aφr, azφ and aφz. This simplifies the problem,since only the axial and radial component of the velocities are coupled. For thedissipationrate ε which appears in the model, the value from the DNS calculationsis used. The remaining task is to determine the Kolmogorov constant C0 and thedamping coefficients aµj . This is discussed in section 6.2.2 and 6.2.3 respectively.

6.2.2 Kolmogorov constant

The model discussed in the previous section contains the Kolmogorov constant C0.This constant can be estimated from the second order Lagrangian structure function

Dµµ(t) =⟨(

v′µ(t) − v′

µ(0))2⟩

which according to K41-theory should equal C0 〈ε〉 t in

the inertial subrange. Although a clear inertial subrange is not present in our currentDNS simulations, we can still select a time interval to determine an estimate for C0.During the DNS calculation passive fluid particles have been released at 10 differentradial positions, viz. r/R = 0.1 . . . 0.9 with steps of 0.1, and one additional positionat 0.95. This allows to determine the Kolmogorov constant C0 and the functions aµj

as a function of the radius.In the limit of very large Reynolds numbers the structure functions in the three

principal directions will be equal. At our moderate Reynolds numbers this only


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

r/R0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

3.5

4

r/R

Figure 6.1. Cµ0 for the three directions at Reτ = 362 (left) and Reτ = 647 (right). (�):

radial, (H): tangential, (�): axial.

holds for regions far enough from the wall. Sawford and Yeung [77] [78] for exampleinvestigated the performance of two dispersion models for homogenous shear flowand took the random term in the stochastic equation to be isotropic. It seems morenatural to define a Cµ

0 that differs for each direction µ. This was also proposed byPope [67] when he investigated a linear stochastic model for homogeneous shear flow.The results for an anisotropic choice for Cµ

0 are plotted in figure 6.1. The structurefunctions are fitted with a linear function in a time interval that is assumed to be inthe inertial range. As mentioned before, a clear inertial range is not present at thesemoderate Reynolds numbers, but changing the time interval does not significantlyalter the results for Cµ

0 . In a large part of the pipe C0 has a constant value ofapproximately 2− 2.5 for Reτ = 362 and around 3 for Reτ = 647. The results do notstrongly depend on the direction, meaning that Cµ

0 is only slightly anisotropic. Onlynear the wall some differences can be observed. The linear fit has been made for eachindividual record. The obtained variation for Cµ

0 is an indication of the error that ismade in the estimation of its values. The error bar calculated in this way is includedin the figures. Near the pipe axis the error is relatively large, since due to the reducedtangential distance the number of independent statistical samples is relatively low.

In the past, the value of C0 has been widely studied by means of Lagrangianvelocity measurements, direct numerical simulations and observation of dispersion oftracer particles in turbulent flow. Hanna [31] suggested C0 = 4.0±2.0 using neutrally-buoyant balloons to measure Lagrangian velocities. Sawford [75] found C0 = 7.0 bycomparing the ratio tc/tη (Lagrangian correlation time over Kolmogorov time) withDNS results of Yeung and Pope [117] of isotropic homogeneous turbulence. Du etal. [24] found C0 = 3.0 ± 0.5 by fitting a Markov model to dispersion measurementsusing C0 as the fitting parameter. Similar results with the same technique werefound by Anand and Pope [2] and Borgas and Sawford [14]. Fox and Yeung [30]

proposed a Reynolds-dependence for C0, C0 = 6.5 [1 + 8.1817/Rλ (1 + 110/Rλ)]−1

,


−1 0 1−1

−0.5

0

0.5

1

−1 0 1−1

−0.5

0

0.5

1

−1 0 1−1

−0.5

0

0.5

1

Figure 6.2. DNS result of the distributions of fluid particles initially starting from r/R = 0.1(left), r/R = 0.5 (middle) and r/R = 0.9 (right) at time tuτ/D = 0.048.

to compensate for low Reynolds number effects. For very large Reynolds numbers,the expression gives a value C0 = 6.5. Rλ is related to the square-root of the Reynoldsnumber based on the bulk velocity: Rλ ∝

√Reb. So our increase of Reb with a factor

2 amounts to an increase of Rλ with merely a factor 1.4. This explains why ourC0 at Reτ = 647 is close to the one determined at Reτ = 362. In our Reynoldsnumber regime Rλ ≈ 30− 40, which would give a value of C0 ≈ 2.8− 3.7 when usingFox’ relation. This is in relatively good agreement with the current results. Theselow values of C0 are also in line with results found by e.g. Wilson et al. [110] forcomparing numerical results with field data from the atmosphere where Rλ ≈ 1000.It is remarkable that they found such a low C0 at this rather high Rλ. At themoment the generally accepted idea is that C0 increases with the Reynolds numberat low Reynolds number and reaches an asymptotic value of approximately 6-7. Thevalues for C0 shown in figure 6.1 are used in the Langevin model.

The fluid particles start at 10 different radial positions. Once they are releasedthey will immediately drift to other radial positions where other statistical propertiesapply. One could thus argue that e.g. the obtained correlation time attributed toa specific radial position is not representative for the Lagrangian correlation time atthat position. However, for times smaller than the Lagrangian integral time scale tcthe radial displacement of the fluid particles is not very large, as can be observedin figure 6.2. Here the distribution of particles is plotted which initially startedfrom r/R = 0.1 (left), r/R = 0.5 (middle) and r/R = 0.9 (right) at time tuτ/D =0.048. This time is approximately half the correlation time of the axial velocitycomponent of the fluid particles (the correlation time for the radial and tangentialvelocity components is smaller). At this time the particles are still relatively closeto their initial radial starting point so that averaged statistical properties can beattributed to this radial position. This time is approximately the upper value of theinertial range and since the inertial range is used to obtain our values for C0 (and alsothe damping coefficients) it seems justified to attribute C0 (and aµj) to the initialstarting points of the fluid particles.


6.2.3 Damping coefficients

The next step in the derivation of the stochastic model is the determination of thedamping coefficients aµj . In equation (6.11) we assumed, based on the DNS results,that the tangential direction is completely decoupled from the radial and axial direc-tions. This means that out of the originally 9 coefficients aµj , only 5 are unequal tozero, i.e. the diagonal coefficients in the 3 directions and 2 cross-coefficients.

The coefficient for the tangential direction can be determined independently fromall others. If the Langevin equation for this component is multiplied with the startingvelocity of the fluid particle v′

φ(0) and the result is ensemble averaged over manyparticles we get

⟨dv′

φ(t)

dtv′

φ(0)

⟩

=⟨aφφv′

φ(t)v′φ(0)

⟩−⟨

1

rv′

r(t)v′φ(t)v′

φ(0)

⟩

+⟨

(C0ε)1/2

wφ(t)v′φ(0)

⟩

. (6.12)

On the l.h.s. the averaging brackets and the velocity at time t = 0 can be broughtinto the time derivative. Then the time derivative operation can be taken outside theaveraging, meaning that the l.h.s is equal to the time derivative of the Lagrangianautocorrelation function. On the r.h.s. the second and third terms are equal to zeroif t > 0, since the tangential velocity component is uncorrelated from the other twovelocity components and wφ(t) is only correlated with v′

φ(t). This leads to

d

dt

⟨v′

φ(t)v′φ(0)

⟩= aφφ

⟨v′

φ(t)v′φ(0)

⟩. (6.13)

The solution for the Lagrangian velocity autocorrelation is hence a single exponential

function:⟨

v′φ(t)v′

φ(0)⟩

=⟨

v′φ(0)2

⟩

eaφφt. Once the correlation function is known,

and in this case we take the result from the DNS calculation, the damping coefficientcan easily be determined.

In a similar way as for the tangential component the radial and axial componentscan be treated, but here a coupled system remains since the equations for v′

r and v′z

in equation (6.11) are coupled. If we denote B′µν(t) =

⟨v′

µ(t)v′ν(0)

⟩, we obtain a set

of four equations for the four unknown damping coefficients:

B′rr(t) B′

zr(t) 0 0

B′rz(t) B′

zz(t) 0 0

0 0 B′rr(t) B′

zr(t)

0 0 B′rz(t) B′

zz(t)

arr

arz

azr

azz

=

d

dtB′

rr(t)

d

dtB′

rz(t)

d

dtB′

zr(t)

d

dtB′

zz(t)

.

This set of equations is solved for arr, arz, azr and azz in the time interval which isassumed to be in the inertial range, i.e. tη � t � tc. At high Reynolds numbers this


0 0.2 0.4 0.6 0.8 1−60

−40

−20

0

20

40

60

80

r/R0 0.2 0.4 0.6 0.8 1

−60

−40

−20

0

20

40

60

80

r/R

Figure 6.3. aµj at Reτ = 362. (�): arr, (∗): arz, (◦): azr, (�): azz, (/): aφφ. The dashedlines are added for a clearer view of the trend.

will result in unique values for the damping coefficients. In this case, however, somevariation of the coefficients can be expected. This variation is an indication of theerror that is made in the estimation of its values. The statistical error made due tothe finite number of records that are generated is small compared to the first errormade by varying over time.

The set of equations is solved not only for each time within the inertial range butalso for all radial positions used as starting positions for the fluid particles, leadingto a radial dependence of the damping terms in the same way as for the Kolmogorovconstant C0. Figure 6.3 shows the result for the 5 damping coefficients as a functionof the radius (left: Reτ = 362, right: Reτ = 647). The first thing that can beremarked is that the dependence on the Reynolds number is very small. For bothReynolds numbers an error estimation of the damping coefficients has been included.The error of the estimates increases in the near wall region. Ensured is that theeigenvalues of the damping matrix aµj have negative real parts, since otherwise thecorrelation functions would keep increasing (see also Pope [67]). In a large part ofthe pipe the coefficients are more or less constant, only near the wall they stronglychange. The region where the coefficients are more or less constant increases withincreasing Reynolds number. The diagonal damping terms are all negative. The crosscoefficient azr is positive for all radial positions, increasing strongly in the wall region.At the pipe axis arr and aφφ should get closer for mathematical reasons, but this isnot completely captured by the fitting procedure. For arr there is a sudden decreasenear r/R = 0.1 which is probably not physical but due to poor sampling. Possibly,it would be better to work with the Cartesian coefficients axx and ayy. Remarkableis that the cross term arz is close to zero for both Reynolds numbers, over the wholeradius. If this coefficient is indeed zero, it would imply that the well-mixed problemis solved. This is the subject of the next subsection.


6.2.4 Well-mixed condition

In the previous section we saw that one of the cross-damping terms, i.e. arz, is (closeto) zero. With this knowledge the well-mixed problem would be solved, meaning thatthe non-zero damping coefficients can also be derived from Eulerian based statistics.We will give the derivation of these Eulerian-based coefficients here. For the whitenoise the following properties apply

〈wi(t)wj(τ)〉 = δijδ(t − τ)

〈v′i(t)wj(t)〉 =

1

2(C0ε)

1/2δij .

(6.14)

If the tangential component of equation (6.11) is multiplied with v′φ and ensemble

averaged this results in (note that now the velocity is at the same time level, incontradiction to what we did to derive equation (6.12) where the velocities were atdifferent times)

⟨d

dt

1

2v′2

φ

⟩

= aφφ

⟨v′2

φ

⟩− 1

r

⟨v′

rv′2φ

⟩+ (C0ε)

1/2 ⟨wφv′

φ

⟩. (6.15)

Using the definition of the material derivative, the first term on the l.h.s. can bereplaced by Eulerian velocities. Furthermore, correlations of v′

φ with other velocitycomponents are equal to zero and if also equation (6.14) is applied this leads to

⟨∂

∂t

1

2u′2

φ

⟩

+

⟨

u · ∇1

2u′2

φ

⟩

= aφφ

⟨u′2

φ

⟩+

1

2C0ε. (6.16)

For stationary flow, which we consider here, the first term on the l.h.s. of this equationis zero. And since the flow is incompressible the second term on the l.h.s. can besimplified. What remains is

∇ ·⟨

1

2uu′2

φ

⟩

= aφφ

⟨u′2

φ

⟩+

1

2C0ε. (6.17)

Since the tangential and axial directions are homogenous for pipe flow, the l.h.s. ofthis equation can be simplified leading to

1

r

⟨

r1

2u′

ru′2φ

⟩

= aφφ

⟨u′2

φ

⟩+

1

2C0ε. (6.18)

Now the l.h.s. vanishes because moments of the tangential velocity component withother velocity components are zero. So once the Kolmogorov constant C0 is known,the damping coefficient aφφ can be determined from Eulerian statistics only using

aφφ = −12C0ε⟨

u′2φ

⟩ . (6.19)


A similar derivation can be made for arr, assuming that arz = 0, by multiplying theradial component of equation (6.11) with v′

r. The final result is

arr = −12C0ε

〈u′2r 〉

+1

r

ddr

⟨12ru′3

r

⟩

〈u′2r 〉

. (6.20)

Note that⟨u′2

r

⟩=⟨

u′2φ

⟩

implies that arr 6= aφφ at r = 0. For the remaining two

coefficients, azr and azz, the axial component of equation (6.11) is multiplied by v′z

and averaged which gives

azr 〈u′ru

′z〉 + azz

⟨u′2

z

⟩= −1

2C0ε +

1

r

d

dr

⟨1

2ru′

ru′2z

⟩

, (6.21)

and the radial component of equation (6.11) is multiplied by v′z and averaged in

combination with the axial component of equation (6.11) being multiplied by v′r and

averaged. The result is summed and gives

azr

⟨u′2

r

⟩+ azz 〈uru

′z〉 = −arr 〈uru

′z〉 +

1

r

d

dr

⟨1

2ru′2

r u′z

⟩

. (6.22)

The coefficients azr and azz can then be obtained from equations (6.20) and (6.21).With equations (6.19)-(6.22) the remaining coefficients can be calculated assumingthat C0 is known. In this case we will use the values of C0 as obtained in the previoussection.

The damping coefficients are calculated using this well-mixed condition, referredto as awm

µj , and used to normalise the ones obtained from the Lagrangian data. Theresult, aµj/awm

µj is shown in figure 6.4 as a function of the radius for arr, azr, azz

and aφφ. In general the ratio is close to unity for azz and aφφ. For azr the ratio isunderpredicted over the whole radius and tends to zero for r = 0. In the near wallregion the ratio tends to zero for arr, azr and azz. Based on these results, takinginto account the various error contributions, it is difficult to conclude whether thewell-mixed problem is solved. The first order model is not good enough at this lowReynolds number.

6.2.5 Numerical technique

In this section the numerical techniques to solve the stochastic model will be brieflydiscussed. It is very inconvenient to solve the model in the presented form, dueto the 1/r terms in equation (6.11). Therefore, we will solve this set of equations inCartesian coordinates transforming each time step between a cylindrical and Cartesiancoordinate system. All equations in the model are made dimensionless in the samemanner as done in the DNS. Also the same boundary conditions have been appliedin the model, i.e. elastic collisions with the wall. To integrate equation (6.10) in timea 2-stage Runge Kutta scheme is used for both the velocity and position.

Special care has to be taken here to get the right statistical properties for thenoise, since we have to work with discrete values. In general Gaussian white noise isdefined as

w(t)w(t′) = Γδ(t − t′). (6.23)


0 0.2 0.4 0.6 0.8 10

1

2

0 0.2 0.4 0.6 0.8 10

1

2

0 0.2 0.4 0.6 0.8 10

1

2

0 0.2 0.4 0.6 0.8 10

1

2

r/R

arr

azr

azz

aφ

φ0 0.2 0.4 0.6 0.8 1

0

1

2

0 0.2 0.4 0.6 0.8 10

1

2

0 0.2 0.4 0.6 0.8 10

1

2

0 0.2 0.4 0.6 0.8 10

1

2

r/R

arr

azr

azz

aφ

φ

Figure 6.4. Ratio aµj/awmµj at Reτ = 362 (left) and Reτ = 647 (right). (�): arr, (◦): azr,

(�): azz, (/): aφφ. The dashed lines are added for a clearer view of the trend.

This means thatT∫

0

w(t)w(t′)dt′ = Γ if 0 < t < T. (6.24)

So

1

T

T∫

0

T∫

0

w(t)w(t′)dtdt′ = Γ. (6.25)

These properties hold for continuous Gaussian white noise. Now consider discreteGaussian white noise w(tj) with tj = j∆t, j = 0, . . . , N and T = N∆t, where ∆t isthe time step and N the total number of time steps. Then we require the white noiseto satisfy

1

T

N∑

i=0

∗ N∑

j=0

∗

w(ti)w(tj)∆t2 = Γ, (6.26)

where the summation with the ′∗′ is defined asN∑

i=0

∗ai = 1

2 (a0+aN )+N−1∑

i=1

ai (trapezium

rule). The sum in equation (6.26) is equal to zero unless j = i. So equation (6.26)can be simplified to

1

T

N∑

i=0

∗

w(ti)2∆t2 = Γ, (6.27)

which is identical to

1

N

N∑

i=0

∗

w(ti)2 =Γ

∆t. (6.28)

This means that the variance of the discrete noise array depends on the time step andis given by σ2 = Γ

∆t .


For the spectrum of the noise we consider the discrete Fourier transform

w(tj) =1√N

N/2−1∑

k=−N/2

wke2πikj

N , (6.29)

where wk = 1√N

N−1∑

j=0

w(tj)e− 2πikj

N . Using the variance of the noise array it follows that

σ2 =1

N

N−1∑

j=0

w(tj)2Using Parseval’s identity (see [19])

=

=1

N

N/2−1∑

k=−N/2

wkwk∗ =

1

N

N/2−1∑

k=−N/2

|wk|2. (6.30)

For Gaussian white noise |wk| is independent of k so σ2 = |wk|2. This defines ourchoice for the magnitude of the noise term, i.e. |wk| = Γ/∆t. In practice this meansthat the noise terms for the stochastic model are generated in Fourier space. Byselecting a random phase in the range 0 ≤ φ ≤ 2π, an amplitude of Γ/∆t, andtransforming the result back to physical space, a Gaussian noise array can be obtainedwith far better properties than standard random number generators.

The values of the damping coefficients and the Kolmogorov constant are calculatedat only 10 radial positions. As a fluid particle will drift to other radial positions, thefunctions aµj and Cµ

0 have been extrapolated to obtain the values at the wall and atthe pipe axis. For intermediate points a simple linear interpolation scheme suffices.

Figure 6.5 shows a typical result of the first order stochastic model. On the toppart of this figure the radial velocity of a fluid particle is shown as calculated with thedirect numerical simulation. On the bottom part, the same is done for the stochasticmodel. Comparing these two results, it is clear that the stochastic model assumes alarge Reynolds number, viz. that the viscous part of the Navier-Stokes equations hasbeen neglected. The fluctuating part of the velocity contains much more small scalesthan the DNS results. The large scale behaviour, on the other hand, is in much betteragreement.

6.3 Results

6.3.1 Lagrangian statistics

As a first validation we compare results from the Langevin model with DNS resultsfor particles released at two radial positions: r/R = 0.5 and r/R = 0.9. The particlesare homogeneously distributed over the tangential and axial directions. These areexactly the same initial positions as in the direct numerical simulation, which enablesus to make a comparison of various statistical properties. The DNS results have beenensemble averaged as discussed in previous chapters. The statistical error still presentin the DNS results will be indicated with an error bar. For the stochastic model 20000

6.3 Results 97

0 1 2 3 4 5

−2

0

2

4

0 1 2 3 4 5

−2

0

2

4

tuτR−1

Figure 6.5. Time series of v′z(t) at Reτ = 362. Top: DNS, bottom: Markov model.

particles, which can be considered as independent samples, have been released whichensures convergence of the statistics. The statistical error in the stochastic model issmaller than in the DNS results and hence not shown in the figures. Further increaseof the number of particles did not show any differences.

As an initial condition for the three velocity components a Gaussian distributionis used with the standard deviations of the Eulerian velocity components at the cor-responding radial position calculated with the DNS. For the axial velocity componenta fraction of the radial velocity component was added to incorporate the correlationbetween the radial and axial velocity component.

In figure 6.6 the same correlation functions are shown which were used to extractthe damping coefficients. Due to the small-scale characteristics of the stochasticmodel, the correlation functions start with a slope, in contradiction with the DNSresults. Only for larger times the slopes of the two results should be the same. Ingeneral a reasonable agreement is achieved with this rather simple, first order model.

From an engineering point of view it would be interesting to have a model capableof predicting the spread or dispersion of fluid particles. A practical application is e.g.the spread of pollutants from an industrial chimney. In figure 6.7 the average radialposition 〈r〉 is shown along with the standard deviation 〈σr〉 as functions of time.For these properties a reasonable agreement between the stochastic model and DNSresults is obtained.

The comparison worsens for velocity statistics. In figure 6.8 this is illustrated bythe average radial drift velocity, the standard deviation of this radial drift velocity,the skewness and flatness. The agreement between the DNS and the stochastic modelis much worse compared to statistics of the position. For the drift velocity 〈v′

r〉 andits standard deviation

⟨σv′

r

⟩the trend is somewhat captured.

We can use the starting velocities of the fluid particles from the DNS as an inputfor the stochastic model as an alternative for the Gaussian random number generator.This will give a non-Gaussian initial velocity distribution for the fluid particles. In


0 0.2 0.4

0

0.5

1

µ=ν=r

0 0.2 0.4

0

0.5

1

µ=ν=φ

0 0.2 0.40

0.5

1

1.5

2

µ=ν=z

0 0.2 0.4−0.2

0

0.2

0.4

0.6

µ=r,ν=z

µ=z,ν=z

tuτR−1tuτR−1

0 0.2 0.4

0

0.2

0.4

µ=ν=r

0 0.2 0.4

0

0.5

1

µ=ν=φ

0 0.2 0.40

2

4

6

8

µ=ν=z

0 0.2 0.4

0

0.5

1

µ=r,ν=z

µ=z,ν=z

tuτR−1tuτR−1

Figure 6.6. Lagrangian correlation functions of velocity fluctuations:⟨v′

µ(0)v′ν(t)

⟩. The

markers are the results from the DNS and the solid lines are results from the first orderstochastic model. All presented results are for Reτ = 362 at the radial positions r/R = 0.5(left four figures) and r/R = 0.9 (right four figures). For r/R = 0.9 also results for thestochastic model are included in case a non-Gaussian initial velocity is used (dashed).

this way also the cross correlation between the radial and axial velocity componentsis automatically accounted for. In the DNS 256 particles where released for eachrecord. With 100 records this gives enough samples that can be used as an initialcondition for the stochastic model. An interesting question is whether the stochasticmodel will return to Gaussian behaviour after the particles have been released. Thecalculation for r/R = 0.9 is repeated with these velocities as initial condition. Atthis radial position the deviation from Gaussianity is larger and the effects are moreeasily visible. The results have been included in figure 6.6 for the correlation functionsand in figures 6.7 and 6.8 for the position and velocity statistics respectively usinga dashed line. The effects of a non-Gaussian initial condition is negligible on thecorrelation functions, average radial position and its standard deviation. Also themean drift velocity and its standard deviation are hardly affected. For the skewnessand flatness of the velocity the model returns to Gaussianity instants after the start.At tuτ/R ≈ 0.02 the influence of a non-Gaussian initial condition has vanished. Theskewness of the Lagrangian data from the DNS also has the tendency to return tozero, although on a different time scale.

6.3.2 Point source

Using the results for the damping coefficients aµj and Kolmogorov constant Cµ0 of

the previous sections, in principle it is now possible to start with any desired ini-tial condition for the particle simulation. For example a point source can easily besimulated by releasing all the particles in the same starting point. Since the noiseterm is different for each individual particle, each particle will follow a different path,

6.3 Results 99

0 0.1 0.2 0.3 0.4 0.50.5

0.52

0.54

0.56

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

tuτR−1

〈r〉R

−1

〈σr〉R

−1

0 0.1 0.2 0.3 0.4 0.50.82

0.84

0.86

0.88

0.9

0.92

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

tuτR−1

〈r〉R

−1

〈σr〉R

−1

Figure 6.7. Average radial position and its standard deviation for DNS and the first orderstochastic model for Reτ = 362 at the radial positions r/R = 0.5 (left) and r/R = 0.9(right). For r/R = 0.9 also results are included in case a non-Gaussian initial velocity isused (dashed). These are indistinguishable from the Gaussian results.

or in other words, due to the different noise term, each particle can be seen as anindependent realization.

Brethouwer [15] simulated the turbulent mixing of a passive scalar in pipe flowusing the numerical code developed by Eggels [27]. Brethouwer added an advection-diffusion equation to the existing code. The equation that is solved is

∂c

∂t+ ∇ · uc = κ∇2c, (6.31)

where c is the concentration of a scalar and κ is the molecular diffusivity. In case thescalar is e.g. a chemical species which is formed or destructed a source term is added tothe equation which represents the source or sink. Here we only consider passive scalarswhere this term is omitted. When this equation is written in dimensionless form κ isreplaced by 1/(ReSc), where Re is the Reynolds number based on the friction velocityand Sc = ν/κ is the Schmidt number which is the ratio of the kinematic viscosityand the molecular diffusivity. The range of length scales of the scalar field dependson the turbulent length scales. The large scales of the scalar field are mostly of thesame order as that of the turbulence. The smallest length scales of the scalar field areexpressed by the Batchelor length scale ηB = 1/

√

(Sc)η. So with increasing Schmidtnumber, i.e. smaller diffusivity, the length scales of the scalar field can be smallerthan that of the turbulence. We will restrict ourselves in this case to Sc = 1, so thatthe smallest scales of the scalar and turbulence field are of the same order.

Brethouwer uses a combination of a second-order centered scheme and a totalvariation diminishing scheme to discretise in space. This scheme preserves mono-tonicity so that negative concentrations cannot occur. For the flow field a domainlength of 5D is used. For the passive scalar three of these flow fields are used in a rowto get a total domain length of 15D, since otherwise the development of the scalarfield is inadequately captured. The boundary conditions for the scalar are no flux


0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0 0.1 0.2 0.3 0.4 0.50.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5−0.5

0

0.5

0 0.1 0.2 0.3 0.4 0.50

2

4

tuτR−1

0 0.1 0.2 0.3 0.4 0.5−0.4

−0.2

0

0 0.1 0.2 0.3 0.4 0.50.4

0.6

0.8

0 0.1 0.2 0.3 0.4 0.5−2

0

2

0 0.1 0.2 0.3 0.4 0.50

5

10

tuτR−1

Figure 6.8. Average radial drift velocity (top) and its standard deviation (second), skewness(third) and flatness (bottom) for DNS and the first order stochastic model. The results arefor Reτ = 362 for particles initially starting at the radial positions r/R = 0.5 (left) andr/R = 0.9 (right).

through the pipe wall, a Dirichlet boundary condition at the inflow and a Neumannboundary condition at the outflow. The time integration is fourth order accuratefor the advective terms and second-order accurate for the diffusive terms which aretreated implicitly only in the tangential direction. As an initial condition an ax-isymmetric Gaussian shaped concentration profile is used with a standard deviationσ = 8.5 · 10−3D in the radial direction. The Reynolds number is equal to 5300 basedon the bulk velocity and diameter, identical to our Reτ = 362-case. Since Sc = 1 thegrid resolution used for the scalar field can be the same as for the velocity field, inthis case Nr × Nφ × Nz = 96 × 128 × 256 for the velocity field (see Table 2.1) andNz = 3×256 for the scalar field due to the three pipe lengths used for the scalar field.Brethouwer collected nearly eighty independent concentration fields.

De Hoogh [38] extended the current pseudo-spectral DNS with a passive scalarequation. He investigated several spatial and temporal discretisation methods tosolve this equation. The numerical approach for the velocity field remained unchangedand is calculated as discussed in Chapter 2 using the pseudo-spectral method. Thispseudo-spectral method appeared not suitable for the advection-diffusion equation,since non-physical negative concentrations occurred. Hence, a finite volume methodwas selected, which does not suffer from this problem. For the advection term inthe equation the MUSCL-scheme developed by van Leer [52] is used: a third orderaccurate upwind scheme. The diffusion part is discretised with a second order centeredscheme which is implicit in the tangential direction. The integration in time is treatedin exactly the same manner as done for the velocity field which yields a second orderaccurate time splitting method. The use of a finite volume methods implies that adifferent grid is used than in the pseudo-spectral method. In practice this means thata new velocity field is interpolated from the field obtained from the pseudo-spectralgrid. De Hoogh performed a similar simulation as Brethouwer with the same initial

6.3 Results 101

conditions for the scalar field and identical boundary conditions. He restricted thecomputational domain for the passive scalar to the same length as used for the velocityfield, i.e. 5D. The total number of independent scalar fields was 222. For more detailson the numerical techniques for solving the advection-diffusion equation in the currentDNS the reader is referred to [38]. Figure 6.9 shows a snapshot of an isocontour ofthe concentration field after the solution has reached a statistically stationary state.

The scalar source at the beginning of the pipe continuously feeds concentrationto the pipe. To capture this within the stochastic model, each discrete time step20000 particles are added to the simulation at z = 0 with σ = 8.5 · 10−3D aroundr = 0 and uniformly distributed over the tangential direction. After the first releasedparticles have reached a desired distance the simulation is stopped, and the velocitiesand positions of all particles are stored at that moment in time. With a total of2000 time steps this means that the information of 40 million particles that describesthe development of the plume in time and space is stored. The bottom part offigure 6.9 shows the distribution of half a million of these particles at the end ofthe simulation. This plot also illustrates that the stochastic model is only capableof producing average statistical information and gives no information on the spatialdistribution of the scalar. To be able to compare statistics of the stochastic modelwith the Eulerian based convection-diffusion equation, the computational domain ofthe stochastic model is divided into cells. Within each cell a concentration can beattributed based on the number of particles within the cell. We will now compare somestatistical results that Brethouwer [15] investigated with results from the stochasticmodel, and results by De Hoogh [38].

Figure 6.10 shows the mean centerline concentration as a function of the axialdistance from the source normalised with the diameter of the pipe D. The stochasticmodel does not include molecular diffusion. This means that the stochastic modelassumes an infinite Schmidt number. The influence of this assumption is clearlyvisible in the result for the centerline concentration in the region z/D ≤ 2. Where theDNS results of De Hoogh and Brethouwer are in excellent agreement, the stochasticmodel predicts a centerline concentration which is too high in this region. Aanen [3]performed measurements under the same conditions as in the DNS calculations tomeasure the concentration using laser induced fluorescence (LIF). This is indicatedby the square markers. The Schmidt number used in these experiments is about 2000,so that the molecular diffusion is of negligible influence on the spreading of the plume.Similarly as for the stochastic model, the mean concentration of the experiments ispositioned above the DNS results for z/D < 2. From there on the experimental resultsof Aanen are in good agreement with the DNS results of De Hoogh and Brethouwer.Further away from the source turbulent transport dominates the process, and this isexactly the time scale on which we fitted our stochastic model. One would thus expectthat for larger distances from the source the stochastic model produces results morein agreement with the DNS and experimental results. The statistical noise is stillrelative large in this region, but the stochastic model still predicts a mean centerlineconcentration which is higher than in the DNS results. The error made for short timesdue to either the infinite Schmidt number or leaving out viscous damping is still ofconsiderable influence at later times. From the experiments it can be concluded that


Figure 6.9. Dispersion of the point source located at z = 0. Top: an isocontour ofthe concentration using the advection-diffusion equation. Bottom: distribution of 500000particles using the stochastic model.

the effect of a different Schmidt number is not of influence in the final regime, implyingthat the differences produced by the stochastic model are caused by neglecting theviscous damping.

The standard deviation of the spreading of the concentration in radial directionas a function of the normalised axial position is shown in figure 6.11. For the finitevolume method it is defined as

σ(z)2 =1

2

∞∫

0

r3c(r, z)dr

∞∫

0

rc(r, z)dr

. (6.32)

For the stochastic model, the distribution of particles is used to calculate a concen-tration from which these integrals can be calculated. The standard deviation is madedimensionless with the pipe diameter and is corrected for the finite source size, i.e.plotted is

√

(σ(z)2 − σ20)/D with σ(z) defined by equation (6.32). The DNS results

of Brethouwer and De Hoogh are again in excellent agreement.According to Warhaft [107] three different stages can be distinguished in the de-

velopment of the plume. These stages are valid for stationary homogeneous turbulentflow at high Reynolds number. For t � Dmol/v′v′, where Dmol is the molecular dif-

6.3 Results 103

10−2

10−1

100

101

10−3

10−2

10−1

100

101

z/D

c(z/D

)/c(

0)

Figure 6.10. Mean centerline concentration as a function of the normalised distance to thesource. Solid line: DNS results from De Hoogh [38], (4): DNS Brethouwer [15], (◦): Linearstochastic model, (�): Experiments of Aanen [3].

fusion coefficient and v′ the Lagrangian fluctuating velocity, the process is dominatedby molecular diffusion. In this stage the variance grows according σ(z)2 = 2Dmolt(σ(z) ∝

√t). For Dmol/v′v′ � t � tc, where tc is the Lagrangian correlation time,

the plume is in its second stage: turbulent convection. In this regime the standarddeviation grows linearly in time according to σ(z)2 = v′v′t2. For times much largerthan the Lagrangian correlation time turbulent diffusion is the decisive factor and thevariance of the spreading is given by σ(z)2 = 2v′v′tct.

Brethouwer estimated the Lagrangian correlation at the pipe axis to be approxi-mately 0.06D/uτ , and hence that for z � 1.1D the plume is at its first and secondstage and for z � 1.1D turbulent diffusion dominates. He correctly mentions that atthese low Reynolds numbers the first and second stage are indistinguishable. With anestimation for v′v′ using the r.m.s. velocity at the pipe axis he finds a good agreementbetween the models and the DNS result for z � 1.1D. The stochastic model devi-ates significantly from the DNS results in this stage. Since this model assumes thatthere is no molecular diffusion, the plume does not grow according σ(z)2 = 2Dmoltin the early stages. The standard deviation of the plume is underestimated up tillz/D ≈ 1, where the slope is different compared to both numerical results. Aanen [3]finds similar results with his experiments (filled black squares). As mentioned before,in this case the Schmidt number is not infinite but around 2000. In this regime thestochastic model and the experiments agree reasonably well.

Brethouwer found that the third stage, where turbulent diffusion dominates, is notwell described by σ(z)2 = 2v′v′tct. This relation was derived for stationary homoge-


10−2

10−1

100

101

10−3

10−2

10−1

z/D

√

(σ2−

σ2 0)D

−1

Figure 6.11. Standard deviation in the radial direction normalised with the diameter andcorrected for the source size as a function of the axial distance to the source. Solid line:DNS results from De Hoogh [38], (4): DNS Brethouwer [15], (◦): Linear stochastic model,(�): Experiments of Aanen [3].

neous turbulence, but for these times the plume has already entered inhomogeneousregions were the influence of the wall is present. In this region the stochastic modelperforms better than at small times, but still underpredicts the standard deviation.In this stage, starting from z/D ≈ 1, the slopes of all the results are equal. Thisin agreement with the result from figure 6.10 where the centerline concentration wasoverpredicted. A way to overcome this problem is by correcting for low Reynoldsnumber effects. The first order model discussed in this section assumes an infiniteReynolds number where viscous effects have been neglected. This means that thefirst order model assumes an infinitely small Kolmogorov time scale. At these lowReynolds numbers this is a crude assumption and a better approach would be to ac-count for a finite viscous time scale. To model the second (small) time scale the noiseterm in the r.h.s. of equation (6.10) is replaced by the random process ζ(t) whichobeys the equation (see e.g. Stratonovic [87] and also Sawford [75])

τη ζ + ζ = (C0ε)1/2w2(t), (6.33)

where τη is the Kolmogorov time-scale and w2(t) another white-noise term with unitvariance. This kind of modelling could give an improvement of the results.

Figure 6.12 shows the averaged concentration profiles scaled with the centerlinevalues as a function of the radial position for three axial positions. The radial positionhas been scaled by the diameter. Brethouwer scaled the radial position with thestandard deviation at the current axial position (σ(z) in equation (6.32)) and found

6.3 Results 105

0 0.05 0.1 0.15 0.2 0.250

0.5

1

0 0.05 0.1 0.15 0.2 0.250

0.5

1

0 0.05 0.1 0.15 0.2 0.250

0.5

1

0 0.05 0.1 0.15 0.2 0.250

0.5

1

r/D

c(r)

/c(

0)

c(r)

/c(

0)

c(r)

/c(

0)

c(r)

/c(

0)

Figure 6.12. Averaged concentration profiles as a function of the normalised radial position.The concentration profile has been normalised with the centerline value. The radial positionis scaled with D. First: z/D = 0.5, second: z/D = 1.0, third: z/D = 3.0, fourth: z/D = 5.0.Solid line: DNS results from De Hoogh [38], (◦): Linear stochastic model.

that the concentration profiles are nearly Gaussian up till z/D = 13. At this positionthe plume has reached the wall and deviation from Gaussianity occurs. The stochasticmodel also predicts Gaussian profiles. Since the standard deviation is underpredictedfor all axial position the radial position is scaled with the diameter only so that thisdifference can be observed. The axial positions for which the results are shown arez/D = 0.5, z/D = 1.0, z/D = 3.0 and z/D = 10.0. The solid lines are the resultsby De Hoogh, the circles are results from the stochastic model. The DNS results ofDe Hoogh have error bars included. These are calculated using approximately 200individual independent samples. The results of Brethouwer have not been includedsince for all axial positions both DNS calculations give results which fall inside theerror bars. The stochastic model underpredicts the standard deviation of the plumecompared to the DNS results by approximately 20 percent. Also the poorer samplingat larger axial distances from the source becomes detectible.

Finally, in figure 6.13 the same averaged concentration profiles are shown but nowfor z/D = 9.0 and z/D = 13.0. At these large distances from the source De Hooghhas no data available, since he modelled only 5 pipe diameters in axial direction. Thetriangles are data from Brethouwer, and the circles are results from the stochasticmodel. The radial position is now normalised by the standard deviation of the plumeat the axis op the pipe. The results are in good agreement.


0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

r/σ

c(r)

/c(

0)

c(r)

/c(

0)

Figure 6.13. Same as figure 6.12 only the radial position is now scaled with σ(z). Top:z/D = 9.0, bottom: z/D = 13.0. (N): DNS Brethouwer, (◦): Linear stochastic.

6.4 Summary

In this chapter a linear stochastic model for pipe flow is introduced. The modelassumes a Markov approximation for the velocity of a fluid particle. The unknowndamping coefficients in this model are obtained by fitting the stochastic model toLagrangian (cross)correlation functions which were obtained by the DNS. It appearedthat one damping coefficient, i.e. arz, is close to zero. This would mean that using thewell-mixed condition, where the damping coefficients can be calculated using Eulerianbased statistics, would be sufficient to solve the non-uniqueness problem. Using thisapproach however, leads to different values for the damping coefficients.

Due to the low Reynolds number the DNS results predict an anisotropic Kol-mogorov constant which is therefore used in the stochastic model. This constantwas obtained by using Kolmogorov’s hypotheses. The values found for Cµ

0 are onlyslightly anisotropic and vary from 2− 2.5 to around 3 for respectively Reτ = 362 andReτ = 647 in the inner region of the pipe. Closer to the wall the anisotropy increasesand smaller values for Cµ

0 are found. The results from the linear stochastic model arein reasonably good agreement with the Lagrangian DNS data. This agreement couldindicate that the Lagrangian velocity is well represented by a linear process.

As a second test case the model is compared with the dispersion of a passivescalar. The passive scalar is modelled in the DNS using a convection-diffusion equationwith a Schmidt number equal to unity and starts as a continuous point source atthe beginning of the pipe. A limited amount of experimental results are available.Statistics that have been compared are the centerline concentration as a function of thedistance from the source z, the spreading of the plume in radial direction as a functionof z and radial concentration profiles at various values of z. In general a reasonableagreement between the stochastic model, DNS and experiments is obtained. Thedeviation in the model can be attributed to the infinite Schmidt number assumed bythe model and the Markov assumption for the velocity. A better agreement is possibleif the model is extended to incorporate finite Reynolds number effects. This requiresfurther research.

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Summary

The objective of this thesis is to make a contribution in the understanding of thestochastic nature of inhomogeneous turbulence. More specifically, Eulerian and La-grangian statistics are studied following a numerical approach based on direct numeri-cal simulations (DNS) of developed pipe flow. For this type of flow many experimentalresults are available to validate the results. Moreover, the flow is the simplest inho-mogeneous flow, which is realizable in experiments.

The DNS that has been developed is based on a pseudo-spectral method. Withthis numerical code large databases are generated for Reynolds numbers as large aspossible using today’s super computers. Two Reynolds numbers are considered, viz.Reτ = 362 and Reτ = 647, where Reτ is based on the wall shear velocity and pipediameter.

Attention is focussed on statistical parameters which are of importance to La-grangian based stochastic models of turbulence, i.e. models which describe the ran-dom motion of a passively marked fluid particle or a contaminant that moves withthe flow as if it is part of it. Statistical properties of a marked particle are diffi-cult to assess experimentally. Therefore many efforts have been made to establishthese properties through relationships of Lagrangian based statistics with Eulerianbased statistics. These connections play an important role in the establishment andcalibration of Lagrangian based models.

Theoretical relationships exist between the evolution in time of the velocity of amarked fluid particle and the Eulerian velocity in a frame that moves with the meanfluid velocity. These velocity relationships can be used to construct single-time ve-locity statistics with their Eulerian counterparts, a connection which leads to resultssimilar to those known from the so-called well-mixed criterion. The relationship is con-firmed for pipe flow by the present DNS results. However, for velocity auto-correlationfunctions the connection is different. In this case the slopes of the Lagrangian andEulerian correlation functions are found to deviate significantly, which is line withrecent theoretical predictions in this field.

Another parameter of interest is Lagrangian acceleration. Theoretical analysisand hypotheses involving homogeneous, isotropic, stationary turbulence indicate thatin the limit of large Reynolds number, temporal and convective accelerations nearlycancel each other and dominate over the total or Lagrangian acceleration. The nu-merical results tend to confirm these findings for the case of inhomogeneous pipe flowand are in line with recent theoretical work.

116 Summary

Stochastic models rely on the similarity hypotheses of Kolmogorov. For the Kol-mogorov hypotheses to hold, the break up of large eddies into smaller eddies shouldbe accompanied by local isotropy of the flow although the flow itself can be stronglyanisotropic. To investigate the extent of local isotropy in anisotropic pipe flow thevarious components of the dissipation rate have been compared. The results indicatethat with increasing Reynolds number local isotropy increases, which is in agreementwith Kolmogorov’s predictions. For higher order statistics, these predictions fail dueto the increasing importance of intermittency. The calculated scaling exponent whichpredicts the deviation from Kolmogorov’s hypotheses is in close agreement with inter-mittency models. The expectation is that deviation from Kolmogorov’s hypothesesis of minor importance for statistical properties that are of interest for engineeringpurposes when using a stochastic model.

A promising model for the statistical description of turbulence concerns the Lang-evin equation for fluid particle velocity. It is based on the property that, accordingto Kolmogorov’s scaling laws, in the limit of infinite Reynolds number particle accel-erations become δ-correlated, causing the particle velocity to become Markovian. Forinhomogeneous flow the equation can only partially be specified by Eulerian statis-tics, which is known as the non-uniqueness problem. The present DNS results provideLagrangian information by which a complete specification of the equation becomespossible. The resulting stochastic model is used to model the dispersion of a passivescalar from a point source in pipe flow. Dispersion statistics thus obtained are inreasonable agreement with experiments and DNS results. Further improvement canbe expected if the model is extended to incorporate finite Reynolds number effects.

Samenvatting

Het doel van dit proefschrift is een bijdrage leveren aan het begrip van het stochas-tische karakter van inhomogene turbulentie. In het bijzonder zijn Euleriaanse en La-grangiaanse statistieken bestudeerd met behulp van een directe numerieke simulatie(DNS) van turbulente pijpstroming. Voor dit type stroming zijn veel experimenteleresultaten bekend die gebruikt kunnen worden om de numerieke methode te vali-deren. Daarnaast is het een van de meest simpele inhomogene stromingen die nogrealiseerbaar zijn in een experiment.

De numerieke methode is gebaseerd op een pseudo-spectrale techniek. Met behulpvan de ontwikkelde code zijn databases aangelegd voor zo groot mogelijke Reynolds-getallen die haalbaar zijn met de hedendaagse supercomputers. De simulaties zijnuitgevoerd bij twee Reynoldsgetallen, namelijk Reτ = 362 en Reτ = 647. HetReynoldsgetal is hierbij bepaald door gebruik te maken van de wandfrictiesnelheiden de diameter van de pijp.

De nadruk in dit onderzoek is gelegd op statistische parameters die belangrijkzijn voor de ontwikkeling van Lagrangiaanse stochastische modellen van turbulen-tie. Dit zijn modellen die de fluctuerende snelheid beschrijven van passief gemar-keerde vloeistofdeeltjes of een verontreinigende stof. De statistische eigenschappen vandergelijke Lagrangiaanse variabelen zijn moeilijk meetbaar in de praktijk. Daaromwordt geprobeerd om deze Lagrangiaanse statistiek uit te drukken in of te verbindenmet Euleriaanse statistiek. Die connectie speelt een belangrijke rol in de kalibratie enontwikkeling van Lagrangiaanse stochastische modellen.

Er bestaan theoretische koppelingen tussen de evolutie in de tijd van passief gemar-keerde vloeistofdeeltjes en de Euleriaanse snelheid gemeten in een stelsel dat meebe-weegt met de gemiddelde lokale vloeistofsnelheid. Deze connectie kan gebruikt wordenom Lagrangiaanse statistiek geevalueerd op een tijdstip uit te drukken in Euleriaansestatistiek: een connectie die tot vergelijkbare resultaten leidt als het ”well-mixed” cri-terium. Uit de huidige DNS resultaten blijkt dat dit verband ook geldt voor pijpstro-ming. Voor Lagrangiaanse autocorrelatiefuncties is de connectie anders. In overeen-stemming met recente theoretische voorspellingen op dit gebied wijken de hellingenvan Euleriaanse snelheidscorrelaties in de tijd fundamenteel af van de Lagrangiaansevariant.

Een andere interessante parameter is de Lagrangiaanse versnelling. Theoretischeanalyses en hypothesen die toepasbaar zijn op homogene, isotrope, stationaire tur-bulentie voorspellen dat de lokale en convectieve acceleraties elkaar vrijwel opheffen

118 Samenvatting

en dat beide domineren over de Lagrangiaanse versnelling in de limiet van oneindigReynoldsgetal. De numerieke resultaten van de inhomogene pijpstroming neigen ditte bevestigen en dit is in overeenstemming met een recente veralgemenisering van detheorie.

Stochastische modellen leunen sterk op de gelijkvormigheidshypothesen van Kol-mogorov. Deze hypothesen gaan op hun beurt weer uit van de vooronderstelling datin een anisotrope stroming door het opbreken van grote wervels naar kleinere wervelsde stroming steeds isotroper wordt. Om de mate van lokale isotropie te onderzoekenzijn de verschillende bijdragen tot de dissipatie bestudeerd. Het resultaat is dat mettoenemend Reynoldsgetal de stroming lokaal isotroper wordt, in overeenstemmingmet de hypothesen van Kolmogorov. Voor statistiek van hogere orde zullen deze hy-pothesen falen vanwege de toenemende invloed van het intermitterend fluctueren vande dissipatie. De berekende schalingsexponent die deze afwijking van de Kolmogorovhypothesen beschrijft, stemt goed overeen met intermitterende modellen. Er wordtechter verwacht dat de afwijking van de Kolmogorov hypothesen maar een kleine in-vloed heeft op statistische grootheden die van belang zijn voor praktische toepassingenwanneer deze met een stochastisch model berekend worden.

Een veelbelovend model voor de statistische beschrijving van turbulentie is deLangevinvergelijking voor de snelheid van een vloeistofdeeltje. Deze stochastischevergelijking maakt gebruik van de eigenschap dat de acceleratie van een vloeistofdeeltjeδ-gecorreleerd wordt wanneer het Reynoldsgetal oneindig wordt, wat volgt uit de Kol-mogorov hypothesen. De snelheid van het deeltje is dus Markoviaans. Het probleemis dat voor inhomogene stromingen de vergelijking maar ten dele gespecificeerd kanworden uit Euleriaanse statistiek, wat bekend staat als het ”non-uniqueness” prob-leem. Met behulp van de Lagrangiaanse statistiek uit de DNS is echter de volledigespecificatie van de vergelijking mogelijk. Het stochastische model dat op deze maniergeconstrueerd is, is gebruikt om de dispersie van een passieve scalar uit een punt-bron in pijpstroming te modelleren. De zo verkregen dispersiestatistiek stemt re-delijk overeen met DNS resultaten en experimenten en verdere verbeteringen kunnenverwacht worden wanneer het model rekening houdt met de lage Reynoldsgetallen.

Dankwoord

Terugkijkend op de afgelopen vier jaren wil ik een aantal personen bedanken voor hunsteun. Zij hebben zeker bijgedragen aan de succesvolle afronding van mijn promotie.

In de eerste plaats gaat mijn dank uit naar degene die mij gedurende de afgelopenjaren dagelijks heeft begeleid: Hans Kuerten. Ik heb onze samenwerking altijdals ontzettend prettig ervaren. Je hebt me voortreffelijk geholpen met fysische,wiskundige en programmeer problemen, de deur stond altijd open voor vragen endiscussies, en op mijn e-mailtjes op de meest onmogelijke tijden kreeg ik steevastantwoord op nog onmogelijkere tijden. Hans, heel erg bedankt voor de vier plezierigeen leerzame jaren.

Verder wil ik natuurlijk Bert Brouwers bedanken, de grote mot(ivat)or achter hetonderzoek. Op momenten dat het onderzoek wat minder soepel verliep, kwam Bertmet nieuwe inzichten en ideeen die weer een verfrissende werking hadden en nieuwemotivatie brachten. Bert, hartelijk dank voor het vertrouwen in mij. Tevens wilik Philip de Goey en Willem van de Water bedanken voor het bestuderen van hetconcept en het geven van nuttige suggesties.

Voor het gebruik van supercomputer-faciliteiten bij dit onderzoek is subsidie ver-leend door de Stichting Nationale Computer Faciliteiten (NCF), met financiele steunvan de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), waarvoordank. Als speciale vermelding hierbij wil ik graag Wim Rijks van SARA bedankenvoor het optimaliseren van een stuk van onze numerieke code.

Bij promoveren hoort ook ontspannen. Daar is ruimschoots in voorzien. Deelnameen organisatie van twee studiereizen naar Zwitserland hebben geleid tot een ontzettendleuke tijd met veel lol. Maar ook tijdens werkuren zijn er veel leuke momenten geweestmet promovendi binnen de sectie Procestechnologie. Speciale dank gaat uit naarmijn kamergenoot, Joost, voor de dagelijkse lach en discussies over film en sport, ennaar ex-kamergenoot Ruben, voor onze speciale competities, vele squash wedstrijdjes,fietsweekenden en je besmettelijke hardloopvirus.

Tot slot verdienen mijn vrienden en familie mijn uitgesproken waardering. In hetbijzonder bedank ik mijn ouders, mijn broer Patrick en mijn vriendin Natascha voorhun enorme steun en liefde.

Maurice

Beuningen, april 2004.

120 Dankwoord

Curriculum Vitae

Maurice Veenman was born in Nijmegen on October 27, 1973. From 1986 till 1992he followed his pre-university education (VWO) at the Canisius College Mater-Deiin Nijmegen. In 1992 he started his Mechanical Engineering studies at the DelftUniversity of Technology, where he specialised in thermal power engineering. Duringhis studies he had a trainee-ship at Sulzer Turbo AG in Zurich, Switzerland. His finalproject, which he finished in 1999, concerned the implementation and validation of acombustion model for the combustion of biogas in a gas turbine.

After obtaining his master’s degree he worked for six months as a researcher at theenergy producing company ENEL in Pisa, Italy. Here, he investigated the feasibilityof biogas-over-coal combustion in a small boiler.

In 2000 Maurice started his Ph.D. research in the group of prof.dr.ir. J.J.H.Brouwers at the Faculty of Mechanical Engineering at the Eindhoven University ofTechnology. During this period he developed numerical codes with which he was ableto predict the behavior of fluids in turbulent pipe and channel flow and homogeneousturbulence. In this way he was able to study important statistics related to stochasticmodels. He presented parts of his research at international conferences in Arlington(USA, 2001), Barga (Italy, 2002) and Sendai (Japan, 2003). Besides the research,Maurice guided six students during their trainee- and graduation-period and assistedwith the course ’physical phenomena’.

statistical analysis of turbulent pipe flow : a numerical ... · erally accepted phenomena of a...

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