Drag Reduction by Polymer Additives

in a Turbulent Pipe Flow:Laboratory and Numerical Experiments

Drag Reduction by Polymer Additives

in a Turbulent Pipe Flow:Laboratory and Numerical Experiments

proefschrift

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magni�cus Prof. ir. K.F. Wakker,in het openbaar te verdedigen ten overstaan van een commissie,

door het College van Dekanen aangewezen,op dinsdag 9 januari 1996 te 16.00 uur

door

Jacob Marinus Jan den Toonder

wiskundig ingenieur,geboren te Terneuzen

Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. F.T.M. Nieuwstadt

Toegevoegd promotor:Dr. ir. G.D.C. Kuiken

Samenstelling promotiecommissie:

Rector Magni�cus, voorzitterProf. dr. ir. F.T.M. Nieuwstadt, Technische Universiteit Delft, promotorDr. ir. G.D.C. Kuiken, Technische Universiteit Delft, toegevoegd promotorProf. dr. W.G.M. Agterof, Technische Universiteit EindhovenProf. dr. ir. M.E.H. van Dongen, Technische Universiteit EindhovenProf. dr. ir. A.J. Hermans, Technische Universiteit DelftDr. ir. G. Ooms, Koninklijke/Shell Exploratie en Produktie LaboratoriumProf. dr. ir. A.A. van Steenhoven, Technische Universiteit Eindhoven

This research has been supported by the Foundation forFundamental Research on Matter (FOM).

cip-data koninklijke bibliotheek, den haag

Toonder, Jacob Marinus Jan den

Drag reduction by polymer additives in a turbulent pipe

ow: laboratory and numerical experiments / JacobMarinus Jan den Toonder. - [S.l. : s.n.]Thesis Technische Universiteit Delft. - With ref. - Withsummary in Dutch.ISBN 90-9008990-XNUGI 841Subject headings: drag reduction / turbulence / rheology

Copyright c1995 by J.M.J. den ToonderAll rights reserved.

Printed by Ponsen & Looijen.

Aan Arianne

Preface

This thesis is the �nal result of four years of research at the Laboratory for Aero- andHydrodynamics of the Delft University of Technology. It is about turbulence and rhe-ology, and about computations and measurements. My interest in uid dynamics, andin turbulence in particular, was aroused when I did my �nal work as an undergraduateat the Maritime Research Institute Netherlands (MARIN). However, when I started inAugust 1991 at \the Lab", I knew nothing about rheology, and, being educated as amathematician, even less about doing measurements. In the course of the past four yearsI have bene�ted greatly from the knowledge and help of others in picking up on thesesubjects. Also in other ways, the support of many people has been essential for thesuccessful completion of this thesis.

In this preface, I thank all who have contributed in one way or another to this thesis.In particular, I would like to mention the following people.

I thank Frans Nieuwstadt and Gerard Kuiken for giving me the opportunity to carryout the research described in this thesis. Both Frans and Gerard have always shown largeinterest in my work, and I gratefully acknowledge their support and guidance.

I thank all of my colleagues at the Lab for their interest and support. However, twoof them deserve a special mention. In the �rst place, without the tremendous e�ortsof Aswin Draad the experimental setup as it is described in this thesis would not haveexisted, and \Part I" would probably have been a great deal shorter. Second, I havegreatly bene�ted from the sharp insight of Martien Hulsen; our frequent discussions werealways quite stimulating.

I also thank my former students Gabriel Tahitu and Edo Koevoet, who did a �ne jobin exploring the (im)possibilities of the LDA system. Together with them, I gained theexperience necessary to perform the measurements presented in this thesis.

The numerical simulations I carried out are founded upon the numerical code writtenby Jack Eggels and Mathieu Pourqui�e, for which I am very grateful.

Finally, I warmly thank my wife Arianne, and my two little girls, Sija and Lotte, forcontinuously reminding me that there is life beyond uid dynamics, although this lifetoo is mostly turbulent.

vii

Contents

Preface vii

Summary x

Samenvatting xii

Nomenclature xiv

1 General introduction 1

1.1 Background : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1

1.2 Turbulence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2

1.3 Rheology : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3

1.4 Aim and framework of this thesis : : : : : : : : : : : : : : : : : : : : : : : 3

2 Basic equations and literature review 5

2.1 Basic equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5

2.2 Review of polymeric drag reduction research : : : : : : : : : : : : : : : : : 7

I Laboratory experiments 15

3 Introduction 17

4 Description of the experimental setup 19

4.1 The pipe ow loop : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19

4.2 The LDA setup : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21

4.3 The polymer solution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 22

4.4 The experimental conditions : : : : : : : : : : : : : : : : : : : : : : : : : : 24

5 Results of the laboratory experiments 30

5.1 Gross ow measurements : : : : : : : : : : : : : : : : : : : : : : : : : : : 30

5.2 Turbulence statistics of the Newtonian uid : : : : : : : : : : : : : : : : : 31

5.3 Turbulence statistics of the polymer solution : : : : : : : : : : : : : : : : 37

5.4 Power spectra : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 48

6 Conclusions of Part I 54

viii

CONTENTS ix

II Numerical experiments 57

7 Introduction 59

7.1 Extensional viscosity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 607.2 Anisotropy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 627.3 Elasticity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63

8 Constitutive models 64

8.1 Extensional viscosity model : : : : : : : : : : : : : : : : : : : : : : : : : : 648.2 Viscous anisotropic model : : : : : : : : : : : : : : : : : : : : : : : : : : : 668.3 Viscoelastic anisotropic model : : : : : : : : : : : : : : : : : : : : : : : : : 71

9 Numerical procedure 74

10 Results of the numerical experiments 77

10.1 The role of extensional viscosity : : : : : : : : : : : : : : : : : : : : : : : : 7710.2 The role of stress anisotropy : : : : : : : : : : : : : : : : : : : : : : : : : : 8710.3 The role of elasticity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 104

11 Conclusions of Part II 112

12 Closing discussion and main conclusions 115

A LDA measurement procedures 118

A.1 Measurement errors : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 118A.2 Finite measurement volume correction : : : : : : : : : : : : : : : : : : : : 120

B Measurement of power spectra in turbulence 122

B.1 Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 122B.2 Practice : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 124B.3 The method used in this thesis : : : : : : : : : : : : : : : : : : : : : : : : 128

C A criterion for identifying strong ow regions in turbulence 129

C.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 129C.2 Review of existing criteria : : : : : : : : : : : : : : : : : : : : : : : : : : : 130C.3 A strong ow criterion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 132C.4 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 142

D The relation between the EV constitutive model models, the relaxation

time of the polymer and the onset hypothesis 144

Bibliography 145

Curriculum vitae 153

Summary

Drag Reduction by Polymer Additives in a Turbulent Pipe Flow:

Laboratory and Numerical Experiments

Jaap M.J. den Toonder

The addition of a minute amount of polymers to a turbulent pipe- or channel ow canresult in a large reduction of the frictional drag. Although this e�ect has been known foralmost half a century, the uid dynamics community still has not been able to clearlyidentify the physical mechanism that causes this drag reduction. Apart from the obviouspractical applications, it is interesting from a fundamental uid dynamics point of viewas well, since the study of polymeric drag reduction may give insight in turbulence itself.

A large number of measurements of drag-reduced ows in channels and pipes, andsome simple theoretical analyses have been published in the literature. The measure-ments have shown, that the polymers do not simply suppress the turbulence. On thecontrary, the stream-wise turbulence intensity is increased in most experiments, while theturbulence intensity normal to the wall is decreased. This hints at an essential changein the turbulence structure, instead of a suppression, by the polymers. In the theoreticalpapers, suggestions for three possible mechanisms have been proposed. These are anincrease in e�ective viscosity in extensional ow regions, an anisotropic e�ect caused byextended polymers, and the elasticity introduced in the uid behaviour by the polymers,respectively.

The purpose of the present thesis is to gain a better understanding of the physicalmechanism of polymeric drag reduction. For this we employ two techniques: laboratoryand numerical experiments. As the laboratory experiments we present Laser DopplerAnemometry (LDA) measurements of the turbulent pipe ow of water and dilute polymersolutions. For the numerical experiments we use Direct Numerical Simulations (DNS)of turbulent pipe ow using various uid models for the polymer solution. The resultsof the measurements serve as a data base against which the results of the numericalsimulations are interpreted and validated. The three uid models used in the simulationsstress one speci�c aspect of the polymer solution, each of them focusing on one of theproposed mechanisms mentioned above. Comparison of the results of the simulationswith the measurements then gives insight into the role of the various aspects describedby the models in the process of drag reduction. In this manner, for the �rst time severalhypotheses on polymeric drag reduction are tested systematically.

The LDA measurements were carried out in a pipe with an inner diameter of approx-imately 4 cm, with the help of a two-component LDA system. Both the stream-wise andthe radial velocity components were observed. Special measures were taken to preventoptical refractions by the curved pipe wall. The uids used were tap water and a 20 wppmpolyacrylamide-water solution. The Reynolds number (based on the pipe diameter andthe friction velocity) ranged from Re� � 625 to � 1390. From the measurements, pro�les

x

SUMMARY xi

of turbulence statistics were computed up to the fourth moment, as well as turbulentpower spectra. The results for the water and the polymer solution are compared at equalwall shear stress, normalised on inner variables.

When analysing the water data for the various Reynolds numbers, a clear Reynoldsnumber e�ect can be observed for the mean velocity pro�le in the logarithmic layer.The \universal" logarithmic law of the wall only appears to be valid for Re� > 1400,approximately. The addition of the polymers to the ow causes the following changes.The bu�er layer is thickened, leading to an o�set of the logarithmic region. The slopeof the logarithmic mean velocity pro�le is increased slightly for the lowest Reynoldsnumbers. The peak of the stream-wise turbulence intensity is increased and shifted awayfrom the wall. The radial turbulence intensity is decreased throughout the pipe. Thehigher order moments are barely inuenced, however there are some small consistentchanges. The turbulence shear stress is suppressed and shows a so-called \Reynoldsstress de�cit", i.e. there is a signi�cant contribution of the polymer to the mean shearstress. The measured spectra show that the stream-wise turbulent kinetic energy is re-distributed from small to large scales. The radial turbulent kinetic energy is dampedover all scales. In short, the turbulence structure becomes more anisotropic.

In DNS, no turbulence modelling is applied and all turbulent motions are resolved.The numerical code uses a �nite volume spatial discretisation and a mixed explicit/implicittime integration.

The �rst polymer model considered is a generalised Newtonian constitutive model,in which an increase in e�ective viscosity in extensional ow regions is applied. Theextensional ow regions are characterised with a new \strong ow criterion". Di�erentcomputations undertaken with this model show that ow �elds other than uniaxial ex-tension might be involved in drag reduction. Although the computation with the modelshows a small drag reduction if it is assumed that the e�ective viscosity increases inregions with uniaxial as well as regions with biaxial extension, many changes in theturbulence structure are not in accordance with the measurements.

Viscous anisotropic e�ects are modelled with the second polymer model. The inu-ence of polymer orientation is included under the assumption that the polymers are beingstretched signi�cantly by the ow. This model causes the relation between the deforma-tion and the stress in the ow to be anisotropic. The DNS with this model results in asigni�cant drag reduction, while almost all changes in the turbulence statistics and thepower spectra are qualitatively consistent with the LDA measurements.

The third uid model consist of an extension of the second model with an elasticcomponent, and can be interpreted as an anisotropic Maxwell model. The numericalsimulation with this model shows considerable less drag reduction in comparison withthe original viscous anisotropic model. Moreover, the behaviour of the mean velocitypro�le di�ers essentially from the measurements.

The main conclusion of our study is that the key property for drag reduction bypolymer additives is the purely viscous anisotropic stress introduced by the extendedpolymers, while elasticity has an adverse e�ect on the drag reduction.

Samenvatting

Weerstandsvermindering door Toevoeging van Polymeren aan

een Turbulente Buisstroming: Laboratorium en Numerieke

Experimenten

Jaap M.J. den Toonder

Door het toevoegen van een zeer kleine hoeveelheid polymeren aan een turbulente buis-of kanaalstroming kan de wrijvingsweerstand sterk afnemen. Dit fenomeen werd al bijnaeen halve eeuw geleden ontdekt, maar niettemin is men er nog steeds niet in geslaagdhet fysische mechanisme dat de weerstandsvermindering veroorzaakt te identi�ceren.Afgezien van de voor de hand liggende praktische toepassingen, is dit e�ect ook vanuit eenfundamenteel oogpunt interessant, omdat de bestudering van weerstandsverminderingdoor polymeren nieuw inzicht kan geven in turbulentie zelf.

Er zijn in de literatuur veel metingen in kanalen en buizen gepubliceerd, alsmede enigetheoretische beschouwingen. De metingen tonen aan, dat de turbulentie niet simpelwegonderdrukt wordt door de polymeren. Integendeel, de stromingsgewijze turbulentie-intensiteit neemt juist toe, terwijl de turbulentie-intensiteit loodrecht op de wand afneemt.De metingen suggereren eerder een wezenlijke verandering van de turbulentie-structuur,dan een onderdrukking. In de theoretische publikaties zijn een aantal speculatievevoorstellen gedaan voor het mogelijke mechanisme van de weerstandsvermindering. Hier-bij kunnen we drie verschillende principes onderscheiden. De eerste richting legt denadruk op een verhoging van de e�ectieve viscositeit in stromingsgebieden met sterkeelongatie, de tweede op anisotrope e�ecten die ge�ntroduceerd worden door gestrektepolymeren, en de laatste op elasticiteit van de polymeren.

Het doel van dit proefschrift is een beter inzicht te verwerven in het fysische mecha-nisme van weerstandsvermindering door polymeren. Hiertoe worden laser Doppler snel-heidsmetingen (LDA) in turbulente buisstromingen van water en van verdunde polymeer-oplossingen gepresenteerd, alsmede directe numerieke simulaties (DNS) van turbulentebuisstromingen gebruik makend van diverse vloeistofmodellen voor de polymeeroplossing.De meetuitkomsten worden gebruikt ter validatie van de numerieke resultaten. De driegebruikte vloeistofmodellen beschrijven elk een speci�eke eigenschap van de polymeerop-lossing, die correspondeert met een van de hierboven genoemde mogelijke mechanismen.Het vergelijken van de uitkomsten van de metingen met die van de berekeningen geeftinzicht in de rol die de afzonderlijke eigenschappen in het weerstandsverminderings-processpelen. Op deze wijze worden voor het eerst verscheidene hypothesen over de oorzaakvan de weerstandsvermindering systematisch onderzocht.

De LDA metingen zijn uitgevoerd in een buis met een binnendiameter van ongeveer 4cm, met een twee-componenten LDA systeem waarmee de axiale en de radiele snelheids-componenten gemeten zijn. Er zijn speciale maatregelen getro�en om de e�ecten vanoptische breking door de gekromde buiswand te vermijden. De gebruikte vloeisto�en zijn

xii

SAMENVATTING xiii

water en een 20 wppm polyacrylamide-water oplossing. Het Reynoldsgetal (gebaseerdop de buisdiameter en de wrijvingssnelheid) varieert van Re� � 625 tot � 1390. Uit demeetgegevens zijn pro�elen van de turbulentiestatistiek tot en met het vierde moment, enturbulente-energiespectra berekend. De resultaten voor water en de polymeeroplossingzijn vergeleken bij gelijke wandschuifspanning, en genormaliseerd met wandvariabelen.

Een vergelijking van de verschillende meetresultaten met water, toont aan dat er eenduidelijke Reynoldsgetal-afhankelijkheid optreedt voor het gemiddelde-snelheidspro�elin de logaritmische laag. De \universele" logaritmische wet lijkt slechts te gelden voorongeveer Re� > 1400. Het toevoegen van polymeren heeft de volgende veranderingentot gevolg. De bu�erlaag wordt dikker, hetgeen correspondeert met een verschuiving vande logaritmische laag. De helling van het logaritmische snelheidspro�el wordt iets steilervoor de kleinere Reynoldsgetallen. Het maximum van de axiale turbulentie-intensiteitwordt groter en verschuift van de wand af, terwijl de radiele turbulentie-intensiteit in degehele buis wordt verlaagd. Er is weinig invloed op de hogere-orde momenten, maar erzijn wel degelijk kleine consistente veranderingen waar te nemen. De turbulente schuif-spanning wordt onderdrukt en er treedt een zogenaamd \Reynoldsspaning tekort" op,hetgeen betekent dat de polymeren een substantiele bijdrage tot de totale schuifspanningleveren. De spectra tonen aan dat de axiale turbulente energie wordt herverdeeld vankleine naar grootschalige structuren. De radiele energie wordt in alle schalen onderdrukt.In het algemeen wordt de turbulentiestructuur meer anisotroop.

In DNS wordt geen turbulentiemodellering toegepast en alle turbulente snelheids-

uctuaties worden opgelost. Er wordt gebruik gemaakt van eindige-volume ruimtelijkediscretisatie, en een gemengd expliciete/impliciete methode voor de tijdsintegratie.

Het eerste vloeistofmodel betreft een gegeneraliseerd-Newtons constitutiemodel, waar-mee de toename van de e�ectieve viscositeit in gebieden met sterke elongatie wordt ge-modelleerd. De elongatiegebieden worden gekarakteriseerd met een nieuwe parameter.Simulaties met verschillende versies van het model laten zien, dat voor weerstandsvermin-dering andere stromingsdeformaties dan uniaxiale elongatie een rol spelen. Een simulatiewaarbij de e�ectieve viscositeit wordt verhoogd zowel in gebieden met uniaxiale als metbiaxiale elongatie geeft weliswaar enige weerstandsvermindering, maar veel veranderingenin de turbulentiestatistiek komen grotendeels niet overeen met de metingen.

In het tweede vloeistofmodel wordt de invloed van de orientatie van de polymerenbeschreven, waarbij wordt aangenomen dat de polymeren gestrekt worden door de stro-ming. De relatie tussen de deformatie en de spanning wordt daardoor anisotroop.De simulaties met het model laten een signi�cante weerstandsvermindering zien, ter-wijl bijna alle veranderingen in de turbulentiestatistiek en de energiespectra kwalitatiefovereenkomen met de metingen.

Het puur viskeuze tweede model kan worden uitgebreid met een elastische component.Hierdoor wordt een anisotroop Maxwell model verkregen. De simulatie met dit derdevloeistofmodel levert aanzienlijk minder weerstandsvermindering op ten opzichte van hetoriginele viskeuze model. Voorts verschilt het resultaat voor het snelheidspro�el wezenlijkvan de metingen.

De hoofdconclusie is, dat de puur viskeuze anisotrope spanning die wordt veroorzaaktdoor gestrekte polymeren, de voornaamste eigenschap is voor weerstandsvermindering.Elasticiteit heeft daarentegen een verminderend e�ect op de weerstandsreductie.

Nomenclature

symbol description de�ned in Sec.

Italic symbolsID ; IID; IIID tensor invariants of D

��8.1

A constant in log law 2.1a aspect ratio 8.2B constant in log law 2.1CP polymer concentration 4.4D pipe diameter 2.1D��

rate-of-strain tensor 2.1D(i) ith eigenvalue of D

��C.3

d width LDA measurement volume 4.4DRP drag reduction at constant �P 2.2DRQ drag reduction at constant Q 2.2E power spectrum B.1E���

permutation tensor C.2E1; E2; E3 power spectrum estimators B.2E1B block power spectrum estimator B.2Euiui power spectrum of the ui component 5.4F stress parameter in the VEA model 8.3F detectibility function A.2Fi atness of i-component 4.4f Darcy-Weisbach friction factor 2.2

frequency 5.4fc Nyquist frequency B.2fdr mean LDA data rate 4.4fip re-sampling frequency B.3fn discrete frequency B.2G elastic modulus 8.3I� integral A.2kz stream-wise wavenumber 10.2L length computational domain 9L��

velocity gradient tensor 2.1M number of blocks B.2N number of (LDA) samples 4.4N1; N2 1st, 2nd normal stress di�erence 8.2Np number of particles per unit volume 8.2Ns number of slots B.2n refractive index 4.1n�

unit vector, director 8.2

xiv

NOMENCLATURE xv

P mean pressure 2.1Pzz turbulent energy production 5.2P� enstrophy production 10.1; C.3P�;rel relative enstrophy production C.3p pressure 2.1Q ow rate 2.2R auto-correlation function B.1R elongation parameter 8.1; C.3R3 auto-correlation function estimator B.2R1;R2;R3;R4 strong ow criteria C.2RM strong ow criterion C.3RN strain state measure C.3r radial co-ordinate 2.1r�

position vector 8.2rc position of measurement volume centre A.2Re Reynolds number 2.2Reo drag reduction onset Reynolds number 2.2Re� friction velocity Reynolds number 4.4Si skewness of i-component 4.4T measurement time 4.4T integral time scale A.1T1 time interval B.1TB duration of block B.2t time 5.1tj discrete time B.1tcoinc LDA coincidence window 4.4U mean velocity A.1U velocity Fourier transform B.1U�

mean velocity vector 2.1Ub bulk mean velocity 2.2u velocity B.1u�

velocity vector 2.1u� friction velocity 2.1v velocity component A.1w(t) window function B.2xi Cartesian co-ordinate 7.1y distance to the wall 2.1z stream-wise co-ordinate 2.1

xvi NOMENCLATURE

Greek symbols_ shear rate 7.1�P mean pressure drop 2.2�B shift constant in log pro�le 2.2�t time step 9�� time slot width B.2� strain/extension 8.3_� true strain rate 7.1_�B biaxial strain rate 7.1�`quantity' relative error in `quantity' A.1� shear viscosity 7.1�B biaxial extensional viscosity 7.1�E uniaxial extensional viscosity 7.1� circumferential co-ordinate 2.1� TIF material constant 8.2� polymer relaxation time 7.1; D� Newtonian dynamic viscosity 2.1�0; �2; �3 material constants anisotropic model 8.2~�0; ~�1; ~�2; ~�3 TIF material constants 8.2�P EV model polymer dynamic viscosity 8.1� Newtonian kinematic viscosity 2.1��(i) ith eigenvector of D

��C.2

� uid density 2.1� stress/force 8.3

standard deviation B.2� time lag B.1���

stress tensor 2.1�m maximum time lag B.2�P mean polymeric shear stress 5.3�T mean turbulent shear stress 4.4���TIF TIF stress tensor 8.2�V mean viscous shear stress 5.2�w total wall mean shear stress 2.1���P mean polymeric stress tensor 2.1

distribution function 8.2uiui power spectrum of the ui component 5.4

��

vorticity tensor 8.2

��D strain rotation tensor C.2

��R relative vorticity tensor C.2

!�

vorticity vector C.2

NOMENCLATURE xvii

Miscellaneous(: : : )0 uctuating quantity 2.1(: : : )+ non-dimensionalised with inner variables 2.1(: : : )� non-dimensionalised with u� , D and � 8.1(: : : )gauss Gaussian process A.1(: : : )m measured A.2(: : : )N Newtonian 2.1(: : : )P polymeric 2.1(: : : )t true A.2h(: : : )i ensemble average 2.1(: : : ) averaging operator 9, B.1D(: : : )=Dt material derivative 2.1

AbbreviationsBSA burst spectrum analyzer 4.2det determinant 8.1DNS direct numerical simulation 1.2DFT discrete Fourier transform B.2EV elongational viscosity 8.1FFT fast Fourier transform B.2JPDF joint probability density function 5.3LDA laser Doppler anemometry 1.4PDF probability density function C.3PIV particle image velocimetry 1.2PM photo multiplier 4.2ppm parts per million 1.1rms root-mean-square 2.2TIF transversely isotropic uid 8.2tr trace 8.1var variance B.2VA viscous anisotropic 8.2VEA viscoelastic anisotropic 8.3

xviii NOMENCLATURE

Chapter 1

General introduction

1.1 Background

By dissolving a minute amount of long-chain polymer molecules in water or in organicsolvents, the frictional drag of turbulent ow through pipes and channels can be reduceddramatically. In pipe ows, for example, the drag can be reduced by up to 80 % byadding just a few parts per million (ppm) of polymer. The discovery of this phenomenonof turbulent drag reduction by polymer additives is generally ascribed to Toms (1949).He discovered it by chance in the summer of 1946, when he was actually investigatingthe mechanical degradation of polymer molecules using a simple pipe ow apparatus.Toms observed \the really astounding thing [...] that a polymer solution clearly o�eredless resistance to ow, under constant pressure, than the solvent itself" (see Toms 1977).Already one year earlier, Mysels had discovered that the addition of an aluminium soapto gasoline lowered the resistance of the uid to turbulent ow in a pipe. However, sincethis work was done under wartime restrictions, publication was delayed by several years(Mysels 1949). In hindsight, it has turned out that the corresponding e�ect in �bresuspensions had been recognised already 15 years earlier by a small circle of engineersworking with paper pulps (see Brautlecht & Sethi 1933).

The drag reduction e�ect is extremely interesting from a practical point of view.Liquids are mostly transported through pipes, and a drag reduction by adding a smallamount of polymers can o�er large economic advantages and a larger e�ectiveness of thistransportation. The most spectacular success in polymer applications for drag reductionhas been the use of oil-soluble polymers in the trans-Alaska pipeline system, where asa result the ow rate has been increased by 32,000 m3/d. The polymer is in this caseinjected downstream of pumping stations; polymer concentrations are of the order of10 ppm. In addition to a drag reduction, the polymer also causes a reduction in heattransfer, which is advantageous in maintaining low oil viscosity (Hoyt 1990). A similarapplication is the addition of polymers to oil being pumped from o�shore platformsto shore facilities (Beaty et al. 1984). Also, in sewerage pipes and storm-water drainspolymers have been used to increase the ow rates so that the peak loads do not resultin overowing; if only relatively infrequent use is required, this can be much cheaperthan constructing new pipes (Sellin 1988). Another application is the increase in therange and coherence of water jets from �re�ghting hoses, but this idea has not beenwidely exploited (Fabula 1971). A military application which has been patented is thereduction of the drag acting on a torpedo by ejecting a sea-water-polymer solution fromthe torpedo nose (Fabula et al. 1980). Finally, we mention a possible medical application:the addition of low concentrations of polymers might be capable of improving blood ow

1

2 CHAPTER 1. GENERAL INTRODUCTION

through stenotic vessels without altering ow through normal vessels, as is suggested bya study by Unthank et al. (1992).

In addition to these practical considerations, the phenomenon of drag reduction bypolymer additives is very interesting from a fundamental uid dynamics point of view aswell. The fact that such small changes in the uid can so drastically alter the turbulent

ow characteristics strongly hints at the existence of a key mechanism of turbulencemomentum transport with which the polymer interferes. That means that a study ofpolymeric drag reduction could help in gaining more knowledge about the turbulenceitself.

Hence, polymeric drag reduction is interesting in many ways and this is reected inthe virtual explosion of research and development work in many countries on the subjectduring the last three decades. A vast amount of publications has appeared, the majorityof which concern measurements. Some of the papers are theoretical reections, and alsoa respectable number of survey articles have been published. Despite this wealth ofinformation it cannot be said that the phenomenon is well understood. The physicalmechanism responsible for the drag reduction remains largely unclear. This is causedby the fact that not only it is necessary to consider the turbulence processes that arepresent in the ow, but also the inuence of the rheological properties of the uid. Hereone touches on two essentially incomplete areas within uid dynamics, viz. turbulenceand rheology.

1.2 Turbulence

Turbulence is a uctuating and chaotic state of uid motion which exists when in a ownon-linear inertial e�ects dominate over viscous e�ects. Apart from a rather qualitativeinsight into the dynamics a complete theory of turbulence is still lacking because weare not able to analyse in detail the non-linear equations that govern the turbulent ow.Turbulence is mostly studied from a statistical point of view and accurately predicting thedetailed evolution of a turbulent ow is in principle impossible. This makes, for example,weather-forecasting very di�cult, as the ow in the earth's atmosphere is turbulent.

Some decades ago, it was discovered by using visualisation techniques that large-scaleorganised structures exist in turbulence. These are generally called \coherent structures".Hence, a turbulent ow may not be so random as it may seem at �rst sight. It is assumedthat the coherent structures, evolving in time and in space, play an important role in thedynamics of turbulence. However, it is quite di�cult to determine the properties of thecoherent structures with traditional measurement techniques. In the past decade therehave been some developments in turbulence research that hold some promise for thefuture. One of these is Particle Image Velocimetry (PIV), which is an optical techniquethat opens the possibility to study the coherent structures quantitatively (see Wester-weel 1993). This technique is presently under swift development. Another importanttechnique which is relatively new is Direct Numerical Simulation (DNS). It is basedon solving the governing equations that describe the spatial and temporal evolutions of

ows numerically, without any approximation for the turbulence. The major advantageof DNS over laboratory experiments is, that it gives information which is very di�cultto obtain in the laboratory, such as higher order statistics in the near wall region, andthe evolution of vorticity structures. During the last decade DNS has developed into a

1.3. RHEOLOGY 3

well-established method to study turbulent ows (see Eggels 1994). However, due to thelarge computational e�orts required to perform a DNS, it has thus far been restricted tosimple geometries, low Reynolds numbers and Newtonian uids.

1.3 Rheology

Rheology is the branch of science that is occupied with the study of non-Newtonian uids.These are uids that cannot be described with the constitutive relation of Newton, inwhich the stress is linearly proportional to the deformation rate, and which for instancecan be applied to water or air. Examples of non-Newtonian uids are polymer solutions,molten polymers, blood, paints and liquid crystals. Non-Newtonian uids can showquite strange and unexpected ow behaviour, completely di�erent from that of wateror air, as described in Bird et al. (1987a). Drag reduction is one of these strange andcounter-intuitive e�ects.

The aim of rheology is to formulate constitutive equations that are able to describethe relation between the stresses and the deformation rates in the ow of non-Newtonian

uids. A large number of constitutive models for polymer solutions has been proposedin the literature. However, these models are based on measurements of the rheologicalproperties of non-dilute solutions in conventional rheometers that subject the uid tosteady ows, or on the behaviour of micro-structural polymer models in steady, homo-geneous ows. As a result, these constitutive models are only valid as approximations inthe limits of weak and/or slow deformations, and thus do not provide a useful basis fortheoretical prediction of dilute solution behaviour for more general ows, let alone tur-bulent ows (see Leal 1990). We must therefore conclude, that a satisfying constitutiveequation for dilute polymer solutions is still lacking.

1.4 Aim and framework of this thesis

The aim of the present thesis is to shed more light on the mechanism of drag reductionby polymer additives. We use two di�erent techniques to reach our goal, the �rst beinglaboratory experiments using Laser Doppler Anemometry (LDA) and the second numer-ical experiments using Direct Numerical Simulation (DNS). Both techniques are appliedto turbulent ow in a pipe.

The plan of the remainder of this thesis is as follows. First, in Ch. 2 we mention thebasic equations that apply to our problem, and we give a concise review of the literaturethat has appeared on polymeric drag reduction. Then we come to the main body ofthis thesis, that is divided into two parts. The �rst part consists of four chapters, anddeals with our laboratory experiments. After a short introduction in Ch. 3 a descriptionof our experimental setup follows in Ch. 4. This chapter contains descriptions of thepipe ow loop, the LDA setup and the polymer solution used, as well as a treatmentof the experimental conditions of the measurements. In Ch. 5 we present the results ofthe measurements, which consist of gross ow results, turbulence statistics and powerspectra. The conclusions from the laboratory experiments are listed in Ch. 6. Part twoof the thesis is concerned with our numerical experiments. This part, consisting of �vechapters, starts with an introduction in Ch. 7, in which we give the rationale behind ournumerical simulations. In the numerical simulations, the e�ect of the polymer additive

4 CHAPTER 1. GENERAL INTRODUCTION

is approximated by several constitutive models. The ideas by which these models areinspired are discussed in Ch. 7 as well, and a derivation of the constitutive equations isgiven in Ch. 8. The numerical procedures we used are briey discussed in Ch. 9. In Ch.10 we report the results of the numerical experiments and compare these with the resultsof the LDA measurements. The last chapter of part two, Ch. 11 contains the conclusionsdrawn from the numerical results. We end this thesis with some closing remarks and asummary of the main conclusions in the �nal chapter, Ch. 12.

Chapter 2

Basic equations and literature review

2.1 Basic equations

We consider a uid that consists of a Newtonian solvent to which a minute amount ofpolymers is added. The basic equations that describe the incompressible ow of such a

uid are given by:

r�� u�= 0; (2.1)

�Du�

Dt= �r

�p+r

�� ���: (2.2)

In these equations u�is the velocity vector, � is the density, p is the pressure and �

��is

the extra-stress tensor. D =Dt denotes the material derivative. The �rst equation is thecontinuity equation and the second the linear-momentum equation.

To close the problem, a relation must be given that expresses the stress in terms ofthe deformation history. To this end we split the stress tensor �

��into two parts, viz. a

part due to the Newtonian solvent and a non-Newtonian part caused by the polymers:

���= ���N + �

��P : (2.3)

For ���N the following well-known constitutive equation is valid:

���N = 2�D

��; (2.4)

where � is the dynamic viscosity, D��= (L

��+ L

��T )=2 is the rate-of-strain tensor in which

L��= (r

�u�)T is the velocity gradient tensor. For the non-Newtonian stress �

��P however,

a di�erent equation that takes the special properties of the uid into account, must besupplied. In Ch. 8 various non-Newtonian constitutive relations are derived. In thepresent chapter, we will remain with the general expression.

Substituting Eq. (2.3) into Eq. (2.2), we obtain the following equation of motion:

�Du�

Dt= �r

�p+ �r

�2u�+r

�� ���P : (2.5)

Turbulence can be described statistically by decomposing all quantities into a meanand a uctuating, i.e. turbulent part. The proper average to use in this procedure is theensemble average, which we will denote by h: : : i. As a consequence of the acceptance ofthe so-called ergodic theorem, in the case of spatial homogeneity a spatial average, andin the case of stationarity, a time average may be used. The decomposition amounts to:

u�= hu

�i+ u

�0 = U

�+ u�0; p = hpi+ p0 = P + p0; �

��P = h�

��P i+ �

��

0

P = ���P + �

��

0

P :(2.6)

5

6 CHAPTER 2. BASIC EQUATIONS AND LITERATURE REVIEW

Substituting Eq. (2.6) into Eqs. (2.1) and (2.5) and averaging the result, we obtain:

r�� U�= 0; (2.7)

�

�@U�@t

+ U�� r�U�+r

�� hu�0u�0i�= �r

�P + �r

�2U�+r

�����P : (2.8)

The last expression, which describes the evolution of the mean velocity �eld, is calledthe \Reynolds equation". Apart from �

��P , the unknown term �hu

�0u�0i appears; this term

is called the \Reynolds stress tensor". To close the problem for the mean velocity, anadditive relation for this extra unknown is needed. This is the \closure problem", whichis a central problem in the theory of turbulence. Possible ways to deal with this problemmay be found in Wilcox (1993) and Lesieur (1990). We will not concern ourselves withthis problem, because we do not use the Reynolds equation (2.8), but the full equation ofmotion (2.5) as a starting point in our simulations. However, to sketch the backgroundof the statistical quantities that we compute from our measurements and simulations, wewill pursue the analysis of Eq. (2.8) a little further.

Because we will study turbulent pipe ow, it is convenient to rewrite Eq. (2.8) interms of cylindrical co-ordinates. The axial, radial and circumferential directions aredenoted by z, r and �, respectively. We will assume a fully developed and stationarypipe ow, which means that the mean-ow conditions are independent of time and theaxial co-ordinate z, as well as being axisymmetric. The diameter of the pipe is D. Theequation of motion Eq. (2.8) then reduces to:

�r2

dP

dz= �hu0zu0ri � �

dUzdr

��P;rz: (2.9)

The right-hand side of this equation must be independent of z, due to the homogeneity inz-direction. Therefore, P is a linear function of z. Furthermore, it can be found from theradial equation of motion that P is not a function of r. The expression on the right-handside is the total mean shear stress, consisting of a turbulent, a viscous and a polymericpart.

Substituting r = D=2 in Eq. (2.9), we obtain:

�D4

dP

dz= �hu0zu0rijr=D=2 � �

dUzdr

jr=D=2 ��P;rzjr=D=2 � �w: (2.10)

�w is the total mean shear stress at the wall. We now de�ne the so-called friction velocityu� with u

2� = �w=�. Then Eq. (2.9) becomes, with Eq. (2.10):

�u2�2r

D= �hu0zu0ri � �

dUzdr

��P;rz: (2.11)

This equation states, that the total mean shear stress varies linearly across the pipediameter.

For the turbulent pipe ow of Newtonian uids, an elementary dimensional analysiscan give insight into the form of the mean velocity pro�le in the near-wall region (seeTennekes & Lumley 1972). The convenient scaling parameters are u� , � and �. The

2.2. REVIEW OF POLYMERIC DRAG REDUCTION RESEARCH 7

relation between the mean axial velocity and the distance to the wall, y = (D=2) � r,then equals:

U+z = f(y+); (2.12)

where U+z = Uz=u� and y+ = yu�=�, with the kinematic viscosity � = �=�.

1

A more detailed analysis then gives (e.g. Tennekes & Lumley 1972):

U+z = y+ if 0 < y+ < 5; (2.13)

U+z = A ln y+ +B if y+ > 30: (2.14)

The region y+ < 5 is called the viscous sublayer, y+ > 30 the logarithmic layer. Theinner region where 5 < y+ < 30 is the so-called bu�er layer. The value of the constants Aand B still is a matter of some minor dispute, since there is considerable scatter in thesevalues as determined from experiments. For fully developed pipe ow at high Reynoldsnumbers, the average of all experiments indicate that A = 2:5 and B = 5:0. However,for low Reynolds numbers the additive constant B takes a value of 5.5, as mentionedin Kim et al. (1987). In many experiments however Eq. (2.14) with the values A = 2:5and B = 5:0 is taken as a starting point to determine the non-dimensionalising u� ,instead of determining u� independently (e.g. Wei & Willmarth 1989). This might leadto systematic errors in the turbulence statistics plotted in wall variables and should beavoided, whenever possible.

For polymer solutions, Eqs. (2.13) and (2.14) may be di�erent, because of the possibleintroduction of an additional scaling parameter for the polymer. Due to the lack of insightin the drag-reducing process, it is yet unknown how the mean velocity should be scaled.

2.2 Review of polymeric drag reduction research

In this section we will give a concise review of existing literature in the �eld of dragreduction by additives. As the amount of literature published is enormous, the presentreview will be incomplete. For a more detailed overview on the subject, the reader isreferred to the numerous surveys already published (e.g. Lumley 1969, Virk 1975, Berman1978, Sellin et al. 1982, Lumley & Kubo 1985, Giesekus & Hibberd 1989, Hoyt 1990,Morgan & McCormick 1990, Tiederman 1990, Matthys 1991). Also, the recent book byGyr & Bewersdor� (1995) can be consulted.

Observations

The phenomenon of drag reduction has been studied experimentally mainly in channelsand pipes. In that case the reduction in drag manifests itself as a change in the relationbetween the mean pressure drop �P over the channel/pipe and the ow rate Q: the owrate is increased if the pressure drop is kept constant, or the pressure drop goes downif the ow rate is kept constant. This implies several ways of de�ning drag reduction,

1a superscript + will always denote non-dimensionalisation with the inner variables u� and �.

8 CHAPTER 2. BASIC EQUATIONS AND LITERATURE REVIEW

depending on the type of measurement. In this thesis, we use the following two de�nitions:

DRQ =

�1� �PP

�PN

�� 100% at equal Q; (2.15)

DRP =

�1� QN

QP

�� 100% at equal �P: (2.16)

In these equations the subscript P stand for \polymer solution" and the subscript N for\Newtonian solvent". In the literature several other, equivalent, de�nitions can be found.Instead of the pressure drop, one often uses the friction factor f , which is de�ned as:

f = 4�w

�U2b =2; (2.17)

where �w represents the total mean shear stress at the wall [see Eq. (2.10)], and Ub isthe bulk mean velocity. The friction factor f has the so-called Darcy-Weisbach form.2

Instead of the ow rate Q, one often uses the Reynolds number, which is Re = UbD=�for a pipe with diameter D, and � denotes the kinematic viscosity of the solvent.3

Measurements of the relation between the pressure drop and the ow rate (which canbe represented as friction factor versus Reynolds number plots) are called \gross-owmeasurements". From published gross-ow measurements the following picture emerges.At low Reynolds numbers, for which the ow is laminar, the addition of a small amountof polymers does not inuence the friction behaviour because a small amount of polymeradditives does not inuence the shear viscosity of the uid. That means that in a pipe

ow, the Newtonian solvent as well as the polymer solution follow the theoretical frictionlaw of Poiseuille:

f =64

Re: (2.18)

After transition to turbulence, at a certain Reynolds number, a Newtonian solvent followsthe empirical Blasius friction law for a smooth pipe:

f = 0:316Re�1

4 : (2.19)

However, for a dilute polymer solution the ow will follow Eq. (2.19) only up to a certainso-called \drag reduction onset Reynolds number" Reo. For higher Reynolds numbers,the corresponding friction factor will be lower for the polymer solution, indicating a re-duction in drag. The drag reduction behaviour is inuenced by many parameters: thepolymer concentration, the type of solvent, the type of polymer (exibility, molecularweight, chemical composition) and the diameter of the pipe. Also, it has been experi-mentally found that a so-called maximum drag reduction asymptote exists (Virk et al.1967): for each Reynolds number a lower bound for the friction factor that can be reached

2The Darcy-Weisbach friction factor is equivalent to four times the Fanning friction factor, which isthe other form of friction factor that is frequently used.

3For dilute polymer solutions showing drag reduction, it follows from standard rheological measure-ments (using simple shear ows) that the shear viscosity is no di�erent from that of the solvent. In thatcase one can use the kinematic viscosity of the solvent in the de�nition of the Reynolds number.

2.2. REVIEW OF POLYMERIC DRAG REDUCTION RESEARCH 9

exists, irrespective of the solvent/additive system that is used. The empirical relationfor this asymptote, which is also known as \Virk's asymptote", is:

2pf= 19:0 log10

�1

2Re

pf

�� 32:4: (2.20)

L

T

M

P

P’

Figure 2.1: The friction factor as a function of the Reynolds number in a pipe ow: possible dragreduction trajectories. L: Eq. (2.18), T: Eq. (2.19), M: Eq. (2.20), P, P': drag-reduction lines, see text.

In Fig. 2.1 we sketch the possible friction behaviour of drag-reducing polymer solutionsin a pipe. The plot contains the laminar friction law Eq. (2.18) (L), the turbulent frictionlaw Eq. (2.19) (T), and Virk's asymptote Eq. (2.20) (M). The lines denoted by P andP' show the possible behaviour of two speci�c polymer solutions. When increasing Re,the following trajectories are possible: L!T (i.e. Newtonian), L!T!P!M, L!M,L!M!P'. The exact positions of the lines P and P' depend on parameters alreadylisted in the previous paragraph; for speci�c information on this dependence, the readeris referred to Virk (1975), Morgan & McCormick (1990) or Sasaki (1991a, 1991b, 1992).We will mention here only the most important results.

In the �rst place, linear, high molecular weight polymers are most e�ective in reducingdrag. Furthermore, the experiments by Virk & Wagger (1990) show that the initialpolymer conformation has large inuence on the drag reduction. By varying the salinityof the solvent, Virk & Wagger (1990) varied the initial conformation of the polymers(i.e. the conformation in rest) from randomly-coiled to extended. For an initially coiledconformation, they found the most common trajectory, i.e. L!T!P!M; however, whenthe polymers were extended initially, the trajectory was L!M!P'. That means, thatin the latter case there was no drag reduction onset number, but the ow became drag-reduced as soon as it became turbulent. Sasaki (1991a) found that the drag reductionability decreases when the polymers become more exible. Also, Sasaki (1991b) showedthat increasing the elasticity of a drag-reducing uid (by adding a micro-gel to a polymersolution) has an adverse e�ect, rather than an enhancing inuence on drag reduction.

10 CHAPTER 2. BASIC EQUATIONS AND LITERATURE REVIEW

To be able to study polymeric drag reduction and its physical mechanism in moredetail, measurement of the turbulent velocity statistics are indispensable. Before 1970,measurements of velocities in ows were made using hot-wire anemometers or Pitot-tubes. However, it has been shown (Lumley 1973, Berman 1978) that the interactionbetween these devices and the ow causes considerable errors when measuring in non-Newtonian uids. This is mainly due to the large extensional viscosities than can occurin these uids (see Sec. 7.1). The development of Laser Doppler Anemometry (LDA) hasmade it possible to measure the ow without directly interfering with it. As a result,LDA has been used widely to study drag-reduced ow.

One of the �rst studies of a drag-reduced ow using LDA was made by Rudd (1972).He developed a novel type of LDA, which he used for measuring the ow of dilutepolymer solutions through a square pipe. While Rudd (1972) was able to measure onlyone component, Logan (1972) made use of two-dimensional LDA equipment; a squarepipe was selected for ease of making measurements near the wall. Both these LDAmeasurements su�er from the secondary ows induced by the corners of the square pipe,which makes the ow rather complex.

In the following years many LDA studies of drag-reduced ows were made. One-dimensional measurements (e.g. Chung & Graebel 1972, Mizushina & Usui 1977, Mc-Comb & Rabie 1982, Pinho & Whitelaw 1990) as well as two-dimensional measurements(e.g. Willmarth et al. 1987, Luchik & Tiederman 1988, Walker & Tiederman 1990, Harder& Tiederman 1991, Wei & Willmarth 1992) were performed. Most of the studies concernchannel ows. Of the references cited here, all the 1-D measurements concern pipes, andall 2-D measurements were done in channels.

Besides LDA, other techniques have been used to study drag-reduced ows, in partic-ular visualisation techniques. For example, Achia & Thompson (1977) used a hologram-interferometric technique in a partly circular pipe with a at wall. Tiederman et al.(1985) injected dye through the wall of a channel. Van Dam & Wegdam (1994) used adynamic light-scattering technique, viz. homodyne photon correlation spectroscopy, toobtain information on the turbulence statistics in polymer solutions.

From the measurements, the following general picture emerges of the inuence of poly-mer additives on the mean velocity pro�le in channels and pipes. In the viscous sublayer,the relation Eq. (2.13) is not changed by the addition of polymers in small quantities.The bu�er layer is thickened, which means that the logarithmic region accordingly hasan o�set. Hence, the pro�les fall above Eq. (2.14) if y+ > 30 for dilute polymer solutions,which is consistent with a reduction in drag. Most experiments reported that this shiftof the logarithmic pro�le is parallel when plotted in semi-logarithmic co-ordinates, i.e.for drag-reduced ows follows:

U+z = A ln y+ +B +�B: (2.21)

However, since the de�nition of the logarithmic zone can be somewhat subjective, thismay not be entirely correct. Although a systematic study by Virk (1975) indeed suggeststhat the shift is parallel, careful inspection of the recent measurements by Pinho &Whitelaw (1990), Harder & Tiederman (1991) and Wei & Willmarth (1992) shows thatthe slope A is actually increased for drag-reduced ow.

2.2. REVIEW OF POLYMERIC DRAG REDUCTION RESEARCH 11

LDA measurements of the root-mean-square of the stream-wise velocity uctuations,rms(uz) (the \stream-wise turbulence intensity") have been relatively common. All ofthe data show that the peak level for rms(uz) occurs at larger values of y

+ than inthe Newtonian case and that the peak is broader. Maximum values of rms(uz)

+ =rms(uz)=u� vary somewhat from study to study. Some of the variations are due tochanges in u� . However, the majority of the experiments indicate an increase in thepeak value for dilute polymer solutions (e.g. Rudd 1972, Logan 1972, Willmarth et al.1987, Luchik & Tiederman 1988, Pinho &Whitelaw 1990, Harder & Tiederman 1991, Wei& Willmarth 1992). For measurements in square ducts and circular pipes, the change inrms(uz)

+ is still subjected to discussion: peak values were found to be similar to (Berner& Scrivener 1980), smaller than (Bewersdor� 1984, Mizushina & Usui 1977) or largerthan (McComb & Rabie 1982, Usui et al. 1988, Pinho & Whitelaw 1990) the Newtonianvalue. McComb (1990) suggests that the \smaller than" data are due to experimentalerrors. Gampert & Rensch (1995) suggest that the\smaller than" data occur only if thepolymer solution cannot be considered dilute anymore, i.e. if the polymer concentrationis so large that polymer networks can form in the uid. Measurements of the root-mean-square of the velocity uctuations normal to the wall are less common than measurementsof the stream-wise root-mean-square velocity. They consistently show a shift of the peakposition away from the wall, and a decreased level, especially in the bu�er region of dilutepolymer channel and pipe ows. The same has been observed for the circumferentialroot-mean-square velocity in a pipe by Pinho & Whitelaw (1990).

Measurements of higher-order turbulence statistics are extremely di�cult, because ofthe large sensitivity to noise and the need to take a very large amount of velocity samplesto get accurate results. Reports of higher-order statistical moments of the turbulent

uctuations in drag-reduced ow are very rare. The only data known to us are those byWei & Willmarth (1992), who present skewness pro�les of the stream-wise and normalvelocity uctuations. The polymer additive has considerable inuence on the stream-wiseskewness for y+ > 40: it is lowered in that region. The normal skewness seems to bestrongly inuenced only in the near-wall region.

The turbulent shear stress has been shown to be suppressed by polymer additives.This implies a decreased correlation between the stream-wise and the normal velocity

uctuations. In addition, in most cases a so-called Reynolds stress de�cit occurs. Thatmeans that the turbulent shear stress and the viscous shear stress do not add up togive the linear variation of the total shear stress over the pipe- or channel cross section,i.e. the term ��P;rz in Eq. (2.11) has a considerable contribution to the total shearstress and cannot be neglected. However, this phenomenon has not been found in allexperiments, for example Harder & Tiederman (1991) found no Reynolds stress de�citwhile nevertheless a signi�cant drag reduction was obtained.

The changes in turbulent shear stress and the root-mean-squares of the velocity uc-tuations have also been illustrated by showing the joint probability density function(JPDF) of the stream-wise and the normal velocity uctuations. The general pictureone obtains, is a attening of the JPDF, while its principle axes are more aligned withthe axial and radial axes for drag-reduced ow. This corresponds to higher rms(uz),lower rms(uy) and a lower correlation for uzuy (see Luchik & Tiederman 1988, Harder& Tiederman 1991, Gyr & Bewersdor� 1990, Gampert & Yong 1990).

12 CHAPTER 2. BASIC EQUATIONS AND LITERATURE REVIEW

Harder & Tiederman (1991) and Wei & Willmarth (1992) show that the turbulent en-ergy production is decreased upon the addition of polymers. This quantity enters directlyas a production term in the equation for the normal axial Reynolds stress component. Ina Newtonian uid ow this production term is balanced primarily by destruction fromviscous dissipation and by redistribution to the other normal stress components throughthe pressure-strain correlation. The production term is important because there is noequivalent term (in fully developed channel ow) in the equations for the other two nor-mal stress components. In particular, the major source term for the normal Reynoldsstress component in the normal-to-the-wall direction is the pressure-strain correlation,which extracts turbulent energy from the stream-wise uctuations. It seems thereforerather surprising that the total turbulent energy production is decreased by the polymers,in view of the fact that the turbulent energy in the stream-wise velocity component isincreased. This might imply that the polymers interfere with the pressure-strain correla-tion. However, it is also possible that the polymers essentially change the energy budgetby introducing non-Newtonian terms, like in the shear stress balance.

Power spectra indicate that for the stream-wise component, turbulent energy is redis-tributed from small scales to large scales. The energy in the normal velocity uctuationsis suppressed over all scales (e.g. Wei & Willmarth 1992).

Injection experiments have been performed to study the dependence of the drag-reducing phenomenon on the localisation of the polymer additives in the ow. Theseexperiments make use of injecting the polymer at a certain location in the ow, andthen measuring the development of various quantities as the polymer spreads out acrossthe pipe/channel cross section going downstream. McComb & Rabie (1982) concludefrom their injection experiments in a pipe that the polymer molecules interact with theturbulence in an annulus which is near but not quite at the wall, viz. 15 < y+ < 100.The lower of these bounds was later modi�ed by Tiederman et al. (1985), who studiedthe injection of polymer solutions through slots in the wall into ow through a planechannel. They concluded that the e�ective zone is 10 < y+ < 100, although they werenot able to verify the upper limit, however it was clear that the viscous sublayer is apassive participant in the drag-reduction process.

Thus far, we have considered so-called \homogeneous" drag reduction, i.e. the poly-mer is homogeneously mixed in the ow. By injecting highly concentrated polymer solu-tions in a pipe or channel ow, a di�erent kind of drag reduction might occur. Namely,it is sometimes observed that the polymer forms a stable thread in the ow, while at thesame time signi�cant drag reduction is measured. This so-called \heterogeneous" dragreduction is in contradiction with the idea that polymers must be in the bu�er layer to bee�ective, which has been found in the injection experiments mentioned earlier. However,although the experiments of Vleggaar & Tels (1973) and Hoyt & Sellin (1991) seem tosuggest that indeed a di�erent mechanism is at work in heterogeneous drag reduction,the results of Smith & Tiederman (1991) and Bewersdor� et al. (1993) show that atleast a signi�cant part of the drag reduction is due to a dissolving process, and henceoriginates by the same mechanism as homogeneous drag reduction.

Finally, we mention that the addition of polymers to a Newtonian solvent not onlyresults in a reduction in drag, but also in a reduction in heat transfer (e.g. Matthys 1991),as well as in a reduction in noise (e.g. Bainbridge 1995). The research in these areas has

2.2. REVIEW OF POLYMERIC DRAG REDUCTION RESEARCH 13

not been developed much up to now.

Theory

Since the discovery of the drag reduction e�ect, several theories for the phenomenon havebeen proposed. All of these are semi-empirical or highly speculative, and all have beensubjected to criticism.

The �rst explanation of polymeric drag reduction was given by Oldroyd (1949), whoproposed the persistence of anomalous behaviour in a thin laminar layer at the pipe wallwhen the main stream has become turbulent. The principle is that an external constraintis imposed by the wall on the ways in which long-chain molecules can rotate near thewall; therefore an abnormally mobile laminar sublayer, of a thickness comparable withmolecular dimensions can exist at the wall, which causes an \e�ective slip". It is clearhowever, that such a slip does not show up in the experiments discussed in the previoussection.

Another kind of wall e�ect was proposed by Elperin et al. (1967). They suggestedthat an adsorbed layer of polymer molecules could exist at the pipe wall during ow andthis could lower the viscosity, create a slip, damp turbulence and prevent any initiationof vortices at the wall. However, from later experiments it has become clear that theadsorption of the additives on surfaces could in fact be an experimental artifact, but itcannot be the reason for the drag-reducing e�ect.

Later studies have put more emphasis on the interaction which takes place betweenthe polymer and the turbulence. For instance, the existence of a drag reduction onsetReynolds number Reo is a reection of this interaction. Virk et al. (1967) postulated thatonset occurs if the ratio between the turbulence an the polymer length scales reaches acritical value. Lumley (1973) found this mechanism unlikely, since the smallest turbulencelength scale is several orders of magnitude larger than the size of the polymer molecules,so that this parameter must be dynamically immaterial to the ow. As an alternative,he proposed that onset occurs when the ratio of turbulence and polymer time scales isof order one. This latter hypothesis was experimentally veri�ed by Berman & George(1974) (see also App. D).

After onset, some interaction between the polymer molecules and the turbulent owresults to give drag reduction. Lumley (1973, 1977) proposed that the increase of e�ectiveviscosity in regions with strong deformations (in which the polymers expand), may bethe key to drag reduction. These regions would primarily exist in the bu�er layer, wherethe strongest turbulence activity is present. By applying a rather crude scaling analysis,Lumley (1973) indeed obtained a reduction in drag by increasing the e�ective viscosityoutside the viscous sublayer. A similar analysis was carried out by de Gennes (1990), butwith the di�erence that his starting point was not an increased viscosity, but an elastice�ect. Also the analysis of de Gennes (1990) results in a drag reduction. The type ofanalysis used in these studies is not uncommon in turbulence investigations. However,the problem is that it ignores the fact that the primary e�ect of the polymers is localdepending on the ow deformation, so that it is unable to explain the detailed dynamicsoccurring in wall turbulence.

Another approach was made by Landahl (1973), who used a stability analysis toinvestigate the inuence of various uid models on a conceptually simple turbulent ow.

14 CHAPTER 2. BASIC EQUATIONS AND LITERATURE REVIEW

His study seems to point to the importance of an anisotropic relation between the stressand the deformation, introduced by extended polymers. However, his turbulence model isvery simpli�ed: for example, it is a two-dimensional model while turbulence is essentiallythree-dimensional.

A new tool used in the investigation of polymeric drag reduction is Direct NumericalSimulation (DNS). This technique has the advantage that no approximations whatsoeverare needed for the turbulence, so the attention can be fully concentrated on the inuenceof the polymer additive itself. Massah et al. (1993) computed the behaviour of a singlepolymer molecule in a turbulent channel ow; in this study, the interaction between thepolymer and the turbulence goes only one way, viz. from the turbulence to the polymer.Hence, only a �rst step is taken. However, Massah's conclusion is that it is plausible toassociate drag reduction with a preshearing of the polymers in the viscous sublayer, aswell as with large temporal changes of polymer con�gurations in the bu�er layer. Handleret al. (1993) obtained drag reduction in a simulated channel ow by randomising phasesof some of the Fourier modes. Much of the turbulence statistics they obtain are inagreement with experiments on polymeric drag reduction. However, there are essentialdi�erences that lead Handler et al. (1993) to conclude that drag reduction due to phaserandomisation may be due to a di�erent mechanism, namely the destruction of coherenceof the turbulence producing structures near the wall.

Closing this brief review, we conclude that the explaining theories for drag reductionmay be divided into three major categories. In the �rst place, an explanation in whichthe increase in extensional viscosity for the polymer solution is the main ingredient.Second, a theory that stresses the importance of anisotropic e�ects introduced by theextended polymer molecules. And �nally, a proposed explanation in which elastic e�ectsare responsible for drag reduction. We will elaborate on these three proposed mechanismsfurther in Ch. 7.

Part I

Laboratory experiments

15

Chapter 3

Introduction

Part I of this thesis deals with the laboratory experiments that we performed. Theseexperiments consisted of detailed two-dimensional LDA measurements of turbulent pipe

ow of water and a dilute polymer-water solution. The main purpose of the measurementswas to obtain a set of data against which the results of the numerical experiments, to bedescribed in Part II, could be tested.

One may ask, why these measurements have been necessary, considering the largebody of previously reported experiments on drag reduction, as reviewed in Ch. 2. Thereasons are the following. In the �rst place, we are interested in pipe ow, and mostof the existing observational studies deal with channel ows. This is probably causedby the fact, that in pipe ow LDA measurements are more di�cult to perform than inchannel ow, since for the former the refraction of the laser beams by the curved pipe wallmay cause major problems when measuring close to the wall. Second, experiments fordrag-reduced ows in pipes that have been reported so-far, show inconsistencies betweenthem, in particular for the behaviour of the stream-wise turbulence intensity. Also, thenature of the shift of the logarithmic velocity pro�le is not entirely clear: is it parallel ornot? Third, very few measurements exist of turbulence statistics very close to the wall,let alone higher-order statistics; these data however can be useful for an explanation ofthe drag reduction mechanism. Another important motivation for our measurements isthe lack of su�cient information on the change of turbulent power spectra caused bypolymer additives in pipe ow. Finally, we noted that most previous experiments havecompared drag-reduced pro�les (non-dimensionalised on inner variables) with Newtonianpro�les obtained at equal ow rate. We feel that it may be better to keep the wall shearstress constant in such a comparison, to avoid mingling of two e�ects, viz. the possibleinuence of the polymers on the shape of the velocity pro�le and changes in the scalingparameter, i.e. the friction velocity.

These considerations convinced us of the necessity of performing additional pipe owexperiments of polymeric drag-reduced ow, in which we attempted to improve on thede�ciencies of previous experiments as well as to answer the open questions mentionedabove.

We used only one type of polymer solution, and at one concentration, in our experi-ments. The type of polymer was selected after a comparison between various polymersin a preliminary study (den Toonder et al. 1995b). That means that we do not addresshere the issue of the dependence of the drag reduction e�ectiveness on molecular weight,chemical composition, concentration etc. It may be assumed that the drag reductionbehaviour of the polymer solution we used is representative of a solution of dilute, high

17

18 CHAPTER 3. INTRODUCTION

molecular weight, linear polymers dissolved in a Newtonian uid.

Chapter 4

Description of the experimental setup

4.1 The pipe ow loop

Figure 4.1: The pipe ow facility. The main part is a cylindrical perspex pipe with length 34 m andinner diameter 40 mm. It contains a test section for the LDA measurements.

The laboratory experiments were performed in the re-circulatory pipe ow facility of theLaboratory for Aero- and Hydrodynamics, a schematic diagram of which is shown in Fig.4.1. An extensive description of this setup can be found in Draad (1996). The main partof the facility consists of a cylindrical perspex pipe with a length of 34 m and an innerdiameter of 40 mm. After the pump and before entering the pipe, the uid passes �rstthrough a ow straightening device and then through the settling chamber, which consistsof a smooth contraction. The settling chamber contains another ow straightening deviceas well as several screens. At a distance of 1 m behind the settling chamber a so-called\trip ring" is inserted in the pipe to force transition to turbulence. This trip ring causesa sudden narrowing of the pipe diameter with a step of 6 mm, extending 5 mm in thestream-wise direction. To avoid secondary circulations due to free convection the entirepipe is insulated with 3 cm Climaex pipe insulation. In the whole setup no contact ofthe uid with metals is allowed. The reason for this is that the polymers are damagedby metal ions, i.e. zinc, copper or iron. The total volume of the system is 1.4 m3.

The pressure gradient along the pipe is measured with two membrane di�erential-

19

20 CHAPTER 4. DESCRIPTION OF THE EXPERIMENTAL SETUP

pressure transducers (Validyne Engineering Corp., type DP15-20), which are identicalexcept for having di�erent ranges (only one of these is sketched in Fig. 4.1). One trans-ducer with a range of 88 mmH2O measures the pressure di�erence between the positions20 m and 31.5 m downstream of the settling chamber. The second transducer has a rangeof 255 mmH2O and measures the pressure di�erence between 18 m and 28 m behind thesettling chamber. The pressure taps have a diameter of 1 mm. The ow rate is mea-sured with a magnetic inductive ow meter (Krohne Altometer, type SC 100 AS). Thetemperature of the uid is measured with a thermocouple in the free-surface reservoir.The measured pressure gradient, ow rate and temperature are acquired automaticallywith a personal computer.

The pump in the system is a so-called disk pump, manufactured by Begemann. Thispump was selected because it causes relatively little mechanical degradation of polymers(see Sec. 4.3 and den Toonder et al. 1995b). The number of revolutions of the pump canbe regulated by feeding either the measured pressure di�erence or the measured ow rateinto the pump control unit, hence keeping either the pressure gradient or the ow rateconstant during an experiment.

Figure 4.2: The test section.

The curved geometry of the pipe results in the di�culty to measure very close to thewall with Laser Doppler Anemometry, because of refraction of the laser beams by thepipe wall. This di�culty arises because of the di�erences in refractive index of the test

uid (i.e. water with n = 1:33) and the material of the pipe (i.e. n = 1:49). To minimisethis problem, we used a special test section, illustrated in Fig. 4.2. In the test section,located 30 m downstream of the inlet of the pipe, the pipe wall is partly replaced bya thin foil made of Teon FEP (uorised ethylene propylene) with a thickness of 190�m, kindly provided by Du Pont de Nemours. This material has a refractive index ofn = 1:344� 0:003, which is quite close to that of our test uid. The use of this foil incombination with the square perspex box �lled with water around the cylindrical foilminimises the refraction of the laser beams. As a result we can perform measurementsdown to a distance of 0.2 mm from the wall, as shown by Tahitu (1994). The innerdiameter of the pipe segment formed by the foil is 40.37 mm. The walls of the perspexbox have a thickness of 6 mm while the distance of these walls to the pipe centre is 60mm.

4.2. THE LDA SETUP 21

4.2 The LDA setup

Laser Doppler Anemometry (LDA) is a well-established technique for measuring turbu-lent ows. An exposition of the general principles can for instance be found in Durst etal. (1976) or Ruck (1987).

Figure 4.3: The LDA setup used is a two-colour backscatter system supplied by Dantec. The receivedsignal is processed by two Burst Spectrum Analyzers.

The LDA setup used in our experiments is sketched in Fig. 4.3. The measurementswere performed with a 2-component LDA system manufactured by Dantec. This systemuses two orthogonal pairs of laser beams with pairwise light of a di�erent wavelengthto measure the uid-velocity in two directions. Each of the pairs forms a so-called\measurement volume" at the position where the two beams intersect, and the lightthat is scattered by a particle travelling through the measurement volume is gatheredin the backscattered direction. The optics to focus the laser beams into the pipe andalso to receive the scattered light is built in one measuring probe (Dantec), having afocusing front lens with focal length 80 mm. This probe is attached to a 3-D traversingsystem supplied by Dantec. The probe is connected to an Argon-ion laser of SpectraPhysics (model 2020) via a �bre and a transmittorbox. The transmittorbox splits upthe light that is produced by the laser into two wavelengths, namely 514.5 nm and 488nm (one for each laser beam pair), and feeds it into the �bre through which the lighttravels to the measuring probe. At the same time, the �bre carries the scattered laserlight from the probe back to two photo-multiplier (PM) tubes, via the transmittorboxand a colour separator that separates the two di�erent colours. The output from thePM tubes goes to two \Burst Spectrum Analyzers" (Dantec, type Enhanced 57N20 andEnhanced slave 57N35), one for each laser beam pair, or velocity component. By means

22 CHAPTER 4. DESCRIPTION OF THE EXPERIMENTAL SETUP

of a spectral analysis of the signal, each BSA computes the Doppler shift between thetransmitted and the scattered light, that corresponds to the velocity component of theparticle perpendicular to both beams (see Tahitu (1994) for a detailed description).

The stream-wise (or axial) velocity component was measured using the 488 nm laserbeam pair, and the normal (or radial) component with the 514.5 nm pair. That meansthat the 514.5 nm pair lay in the plane perpendicular to the stream-wise direction (i.e.the \vertical pair"), and the 488 nm pair lay in the plane parallel to both the stream-wisedirection and the optical axis of the probe (i.e. the \horizontal pair"). Prior to the actualmeasurements the measurement volumes for both components were made to coincide atthe same position by using a pinhole with diameter 20 �m. The paths of all four laserbeams within the probe were adjusted so, that they all were focused on the pinhole ifthis was placed in the focal point of the probe front lens in air. The dimensions of themeasurement volumes are estimated to be 20, 20 and 100 �m in the stream-wise, normaland span-wise directions respectively.

In all measurements to be presented later in this thesis, the probe was traversed invertical direction, causing the measurement volumes to travel along the vertical centralaxis of the pipe, as indicated in Fig. 4.3.

We aligned the probe in the horizontal and vertical planes with respect to the pipeby observing the reections of the laser-beams coming from the test section. The probewas adjusted so, that these reections travelled along symmetric paths with respect tothe incoming beams. Furthermore, the probe was rotated about its optical axis and �xedin the position in which the mean velocity measured with the vertical laser beam pair inthe centre of the pipe was identically zero.

The actual location of the measurement volumes was determined by examining in-stantaneously the output of the BSA's, i.e. the Doppler signal, with a digital scope whiletraversing the probe. When the measurement volumes hit the foil, this shows up as asudden increase in the noise level of the observed signal. In this way, the location ofthe wall was determined at four positions with which the zero position of the probe wasde�ned. The accuracy of the wall determination using this procedure, is estimated tobe 0.1 mm. The �nal measurement results are corrected for a possible error in the walllocation with a procedure described in Sec. 4.4. These corrections always fall within theestimated accuracy of 0.1 mm.

In all experiments the uid was seeded with pigment on TiO2-basis to get su�cientlyhigh data rates (see Sec. 4.4). The power supplied by the laser was always 2.5 W,corresponding to a power of approximately 150 mW in each laser beam.

The BSA's and the traversing system are controlled by a PC, making fully automaticmeasurements possible.

4.3 The polymer solution

The polymer we used in our experiments is Superoc A110 (Cytec Industries), which isa partially hydrolysed polyacrylamide (PAMH). Superoc A110 has a molecular weightof 6{8�106 g/mol, according to the manufacturer. The advantage of this polymer overother types of polymer encountered in the literature (e.g. Polyox WSR301, Union Car-bide, or Separan AP273, Dow Chemical Company), is that it is relatively resistant tomechanical degradation (see den Toonder et al. 1995b). Mechanical degradation denotes

4.3. THE POLYMER SOLUTION 23

the breaking of the polymers by mechanical actions, which reduces the molecular weightof the polymers and thus reduces its drag-reducing capability (see Virk 1975). Thisis an important point, since our experimental setup is a re-circulatory one so that thepolymers are continuously subjected to deformations in the ow which might cause thescission of the polymers, especially in the pump. Severe mechanical degradation leads tounacceptable changes in the measurement conditions during an LDA measurement. Theuse of Superoc A110 in combination with the disk pump minimises this problem, asshown in den Toonder et al. (1995b). The structure of Superoc A110 is schematicallyas follows:

In all experiments we used a Superoc A110-water solution with a polymer concen-tration of 20 wppm. That means that only 28 g of polymer was dissolved in the entiresystem.

First a 1000 wppm master solution was prepared in an self-made stirring vessel asfollows. The vessel was �lled with 70 l of \Delft" tap water. A total amount of 70 g ofSuperoc A110 was divided into 7 equal portions. Each of the portions was suspended in0.3 l ethanol. The 7 suspensions thus obtained were then added to the tap water in thevessel while stirring, creating a large vortex in the vessel. For two subsequent hours thesolution was mixed gently. After this period, the solution contained small air bubbles,that were allowed to escape in the following 24 hours. At the end of this procedure, thesolution obtained was visibly clear and did not contain any detectable inhomogeneities.

The 1000 wppm solutions prepared in this way were used in the pipe setup within 1 to2 weeks. To obtain the desired concentration, i.e. 20 wppm, an amount of 28 l of the 1000wppm solution was added in the free-surface reservoir (see Fig. 4.1) and mixed by hand,while at the same time the pump was operated to obtain a ow rate of approximately3000 l/hr to mix the polymer in the entire setup.

Although, as mentioned, the use of Superoc A110 in combination with the disk pumpsu�ers much less from mechanical degradation than other polymer/pump combinations,it turned out to be impossible to completely avoid degradation. This is shown in Fig.4.4, in which we depict a typical degradation curve. It shows the time-evolution of thedrag reduction at a constant ow rate Q = 2850 l/hr. The drag reduction DRQ in thiscase is de�ned as in Eq. (2.15), viz.:

DRQ =�PN ��PP

�PN� 100%; (4.1)

where �PN is the pressure loss per unit length of the pipe for the Newtonian solventalone and �PP is the same quantity for the polymer solution. At the start the dragreduction is quite high, i.e. 70 %. After an initial fast decay, the degradation slowsdown. However, the drag reduction never reaches a plateau value, as was suggested bythe experiments of den Toonder et al. (1995b).

24 CHAPTER 4. DESCRIPTION OF THE EXPERIMENTAL SETUP

Figure 4.4: Typical degradation curve of a 20 wppm Superoc A110 solution in our setup. The ow rateis kept constant at Q = 2850 l/hr.

The typical length of the LDA experiments we performed is approximately 6 hours,so there is always some degradation during such measurements, even if they are startedconsiderable time after the addition of the polymer master solution in the setup. Be-cause of the fact that the degradation depends on the actual ow conditions during ameasurement (i.e. the ow rate and the pressure drop) which vary from experiment to ex-periment, the degradation was di�erent for each measurement. The precise experimentalconditions of our experiments are treated in the next section.

4.4 The experimental conditions

We performed two types of measurements: \pro�le measurements", from which turbu-lence statistics as a function of wall distance were determined, and \spectrum measure-ments" for the computation of power spectra. This section contains a description ofthe experimental conditions. Table 4.1 lists the experimental conditions of the measure-ments. In the �rst column the symbols are shown which will denote in the �gures in theremainder of this thesis the corresponding measurements.

A concentration CP = 0 denotes a measurement with pure water (i.e. \Delft" tapwater).

The friction velocity u� is de�ned with �u2� = �w, in which � is the density of the

uid and �w is the mean shear stress at the wall [see Eq. (2.10)]. This quantity wasdetermined from the pressure measurements according to:

u� =

sD

4�

�����P�z����; (4.2)

where D is the diameter of the pipe, �P is the measured pressure di�erence and �z isthe distance over which �P is measured. For the �rst four pro�le measurements (i.e. the

4.4. THE EXPERIMENTAL CONDITIONS 25

Table 4.1: Experimental conditions of the measurements. Each symbol (�rst column) corresponds toone full pro�le or spectrum measurement, and will also be used in the �gures in this thesis to denotethe corresponding measurement. The ow rate and the drag reduction for the polymer solution pro�lemeasurements are shown in Figs. 4.5 and 4.6, respectively.

Experimentalconditionsofthepro�lemeasurements

symbol

CP

(wppm)

u�

(mm/s)

�=u�

(mm)

Q(l/hr)

Re

Re�

1 2D+

y corr(mm)

�

0

9.2

0.120

618

4922

337.6

168.8

-0.017

N

0

16.9

0.0642

1235

9972

628.8

314.4

0.0

4

20

17.0

0.0647

(a)

{

623.9

312.0

0.049

5

20

16.9

0.0648

(a)

{

623.1

311.5

-0.049

�

0

27.8

0.0388

2191

17773

1039.2

519.6

0.0

2

20

27.7

0.0391

(a)

{

1033.5

516.8

-0.027

3

20

27.6

0.0391

(a)

{

1033.2

516.6

-0.020

+

0

37.3

0.0292

3058

24580

1381.5

690.7

-0.029

�

20

37.6

0.0289

(a)

{

1399.0

699.5

-0.009

(a)seeFig.4.5.

Thedi�erencebetweenthepro�lemeasurements4

and5

isthedegradationstageofthepolymersolutions.

Thesameappliestothemeasurements2

and3.

Experimentalconditionsofthespectrummeasurements

symbol

CP

(wppm)

u�

(mm/s)

�=u�

(mm)

Q(l/hr)

Re

Re�

DRP(%)

y+

N

0

15.6

0.0641

1143

10018

629.8

{

12.0

30.0

125.0

4

20

17.1

0.0649

1410

11129

621.9

11.2

10.7

29.5

124.4

�

0

27.7

0.0390

2180

17680

1035.4

{

11.9

29.8

125.0

3

20

27.6

0.0395

2469

19844

1022.2

12.2

11.7

29.2

122.6

+

0

35.0

0.0286

2886

25284

1413.0

{

11.9

30.1

126.0

�

20

38.0

0.0289

4433

35306

1394.6

29.6

11.7

29.8

123.9

26 CHAPTER 4. DESCRIPTION OF THE EXPERIMENTAL SETUP

two lowest Re� ) in Table 4.1 the 88 mmH2O pressure transducer was used, in the othercases the 255 mmH2O transducer.

The viscous length scale �=u� is composed of the kinematic viscosity of the uid,�, and the friction velocity. For �, the value which applies to water was used in thepolymer solution measurements as well, under the assumption that for the small polymerconcentration used the shear viscosity of the uid is not altered.

The Reynolds number Re is de�ned with the bulk velocity in the pipe, i.e. Re =UbD=�, and is determined from the measured ow rate Q. The Reynolds number Re� isde�ned with the friction velocity: Re� = u�D=�. D

+=2 denotes the centre of the pipe inwall units, and actually is equal to Re�=2.

ycorr is the correction for the position of the wall, and will be explained later.The variation of the temperature of the uid during each pro�le measurement was

less than 0:1�C, with a mean value between 16:3�C and 17:0�C.In all the measurements, the pressure gradient, and hence u� , was kept constant. Four

di�erent values of u� were used. One pro�le measurement consisted of a traverse throughthe pipe, measuring the velocity at a large number of positions (i.e. 35 for the lowestReynolds number, and 70 for the largest Reynolds number) starting from the wall andmoving toward the centre of the pipe. At each position samples were gathered at a certainmean data rate (fdr), by operating the BSA's in so-called dead-time mode, and duringa certain time T . In dead-time mode, the time separation between the samples can be�xed to a certain value (however only approximately, due to the stochastic nature of thesampling process); in this way, the mean data rate can be inuenced, provided there areenough samples in the �rst place. For the two lowest Reynolds numbers, fdr was around30 Hz, while T = 600 s. In the other cases fdr � 60 Hz and T = 300 s. That means,that typically 18000 samples were obtained per position. Close