web.fe.up.ptfpinho/pdfs/MSc_Thesis_AfonsoGhira_vfinal.pdf · Intense Vorticity Structures in...

Intense Vorticity Structures in Newtonian Turbulence and

Turbulent Dissipation in Viscoelastic Fluids Analyzed by

Direct Numerical Simulations

Afonso Silva Resende de Avelar Ghira

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisor: Prof. Carlos Frederico Neves Bettencourt da Silva

Examination Committee

Chairperson: Prof. Filipe Szolnoky Ramos Pinto CunhaSupervisor: Prof. Carlos Frederico Neves Bettencourt da Silva

Member of the Committee: Prof. José Manuel da Silva Chaves Ribeiro Pereira

July 2019

Dedicated to my Grandfathers!

iii

Acknowledgments

I would like to emphasize my thankfulness for the unswerving support and confidence in me of

Professor Carlos Silva during these years.

Some words of gratitude should be given to Professor Fernando Pinho for the useful advices he gave

me.

I want also to state my acknowledgment to my friend Hugo Abreu for the worthwhile discussions

about several subjects.

No words of acknowledgment should be left unsaid to my friend Nuno Pimentel, who showed aston-

ishing skills in Project Management when he helped me to schedule the writing process of my thesis.

Special thanks are naturally entangled with my deep love for my Mother Myriam, Dad Joao and my

Sister Ines.

v

Resumo

Este trabalho tem como objectivo o estudo das caracterısticas das estruturas de intensa vorticidade

(EIV) em turbulencia Newtoniana, assim como pretende estudar a dissipacao turbulenta em fluıdos vis-

coelasticos. Estas abordagens baseiam-se em simulacoes numericas directas (SND) de turbulencia

homogenea e isotropica (THI) forcada em condicoes estatisticamente estacionarias. O primeiro caso

explora-as numericamente para um espectro de numeros de Reynolds mais amplo do que aqueles en-

contrados na literatura, nomeadamente 88 < Reλ < 429. Em geral, confirmam-se resultados obtidos

anteriormente por outros autores, tais como as escalabilidades do raio daquelas estruturas, Rivs, com

a escala de comprimento de Kolmogorov η e da velocidade azimutal, a distancia Rivs, com o desvio-

padrao da velocidade u′. Contudo, descobriu-se que o comprimento das EIV escala com η, trazendo

um novo resultado a esta area de estudo. O ultimo estudo ilustra as diferencas, inerentes aos graus

de liberdade elasticos, que caracterizam os fluıdos viscoelasticos, comparando a dinamica espectral,

da equacao da evolucao da energia cinetica, de um fluıdo puramente viscoso, modelado com Carreau-

Yasuda, e um viscoelastico, simulado com FENE-P, impondo a mesma lei de reducao de viscosidade

por corte. Para alem disso, ainda no que se refere ao modelo reologico constitutivo FENE-P, foram

conduzidos testes que suportam a lei de escalabilidade entre a concentracao β e o numero de Weis-

senberg, no comeco da reducao de dissipacao, previsto pela teoria de de Gennes. Finalmente, um

estudo preliminar, sobre a possıvel presenca duma assımptota tipo a de Virk, foi desenvolvida em torno

das condicoes de reducao de dissipacao maxima.

Palavras-chave: Simulacoes Numericas Directas, Estruturas de Intensa Vorticidade, FENE-

P, Turbulencia Homogenea e Isotropica, Viscoelasticidade

vii

Abstract

This work intends to study the features of intense vorticity structures (IVS) in Newtonian turbulence

and also aims to provide a survey related to turbulent dissipation in viscoelastic turbulence. Both ap-

proaches are based in direct numerical simulations (DNS) of forced homogeneous and isotropic turbu-

lence (HIT) at statistically steady sate conditions. The former case explores numerically those structures

for a wider range of Reynolds numbers than those found in the literature, namely 88 < Reλ < 429. In

general, statistics confirm previous results from other authors, like the scalabilities of the radius of those

structures, Rivs, with Kolmogorov’s length scale η and of their azimuthal velocity, at a distance Rivs, with

the velocity root-mean-square u′. However, the length of IVS was found to be scalable with η, bringing

a new result in this field of research. The latter survey depicts the differences inherent to the elastic

degrees of freedom, characterizing viscoelastic fluids, by comparing the spectral dynamics of the kinetic

energy evolution equation of a purely viscous fluid, modeled with Carreau-Yasuda, and a viscoelastic

one, simulated with FENE-P, when imposing the same shear-thinning behavior. Furthermore, still re-

garding the FENE-P constitutive rheological model, tests were conducted to support the scalability law

between the β concentration and Weissenberg number, at the onset of dissipation reduction, predicted

by de Gennes’s theory. Finally, a preliminary study about a possible presence of a Virk’s-like asymptote

was developed near maximum dissipation reduction conditions.

Keywords: Direct Numerical Simulations, Intense Vorticity Structures, FENE-P, Homogeneous

and Isotropic Turbulence, Viscoelasticity

ix

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 3

2.1 What is Turbulence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Homogeneous and Isotropic Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Kolmogorov Hypothesis and Scales of Turbulent Motion . . . . . . . . . . . . . . . 5

2.2.2 Statistical Description of Turbulence and Two-point Correlation Functions . . . . . 8

2.2.3 Isotropic Correlation Tensors Rij and Φij . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.4 Other Characteristic Scales and Relations . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.5 Turbulent Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Coherent Vorticity Structures in Turbulent Field . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Intense Vorticity Structures (IVS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Burgers’ Vortex Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Enstrophy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.4 Velocity Gradient and Rate of Strain Invariants . . . . . . . . . . . . . . . . . . . . 21

2.3.5 IVS Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.6 Length of IVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Complex Fluids and Turbulence with Polymer Additives . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Purely Viscous and Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 Drag Reduction in Dilute Polymer Solutions . . . . . . . . . . . . . . . . . . . . . . 26

2.4.3 Numerical Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

xi

2.4.4 de Gennes’ Theory for Drag Reduction Prediction . . . . . . . . . . . . . . . . . . 31

3 Governing Equations and Numerical Methods 33

3.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Constitutive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Newtonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.2 Carreau-Yasuda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.3 FENE-P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Pseudo-Spectral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Velocity Field and Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.2 Navier-Stokes Equations in Fourier Space . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Conformation Tensor Transport Equation Discretization . . . . . . . . . . . . . . . . . . . 42

3.4.1 Convection Term: Kurganov-Tadmor (KT) method . . . . . . . . . . . . . . . . . . 42

3.4.2 Stretching Term: Central Finite Differences Method . . . . . . . . . . . . . . . . . . 44

3.5 Temporal Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.7 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8 Forcing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Results and Discussion 51

4.1 Intense Vorticity Structures Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.2 Post-processing Algorithm - Wormtracker Code . . . . . . . . . . . . . . . . . . . . 52

4.1.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.4 Results Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Carreau-Yasuda and FENE-P Comparative Survey . . . . . . . . . . . . . . . . . . . . . . 57


4.2.2 Kinetic Energy Equation for Carreau-Yasuda . . . . . . . . . . . . . . . . . . . . . 60

4.2.3 Kinetic Energy Equation for FENE-P . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.5 Results Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Dissipation Reduction Onset and Preliminary Study Near Maximum Dissipation Reduction 67


4.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3.3 Onset Of Drag Reduction Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3.4 Preliminary Study Near Maximum Dissipation Reduction . . . . . . . . . . . . . . . 72

5 Conclusions 75

5.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

xii

Bibliography 77

xiii

List of Tables

2.1 Statistics of IVS in several kinds of flows at several Reλ. The flow types are Forced

Isotropic Turbulence (F. I.), Decaying Isotropic Turbulence (D. I.), Homogeneous Shear

(S.), Mixing Layer (M. I.), Circular Duct Flow (D. F.), Boundary Layer Flow (B. L.), Chan-

nel Flow (C. F.), Round Jet (R. J.) and Plane Jet (P. L.). The quantities are the Taylor

microscale-based Reynolds number Reλ; mean radius of IVS 〈Rivs〉 non-dimensionalized

by the Kolmogorov’s length scale η; mean value of IVS radius non-dimensionalized by

the local Burgers’ radius 〈Rivs/RB〉; mean value of the azimuthal velocity 〈Uivs〉 non-

dimensionalized by the root-mean-square velocity u′; mean value of the azimuthal velocity

〈Uivs〉 non-dimensionalized by the Kolmogorov’s velocity uη; circulation-based Reynolds

numberReΓ normalized byRe1/2λ ; mean value of the length of IVS 〈Livs〉 non-dimensionlized

by the integral length scale L11; mean value of the length of IVS 〈Livs〉 non-dimensionlized

by the Kolmogorov’s length scale η. The Reynolds numbers Reλ in [16] were estimated

with the available data in the paper. Some authors use as definitionReΓ = Γ/ (2πν), while

the definition used in this work is ReΓ = Γ/ν. In those cases the values were converted

to the present definition of this work. In [23], the reference values λ, u′ and η are taken

from the conditional mean profiles for each instantaneous field, which are roughly con-

stant deep inside the turbulent region for the 11 fields used in the statistics. Table and

caption adapted from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Simulations Parameters. Mesh points in each direction N ; Taylor micro-scale based

Reynolds number Reλ; turbulent dissipation ε, which equals the power input P ; kine-

matic viscosity ν; integral length scale L11; velocity root mean square u′; mean of IVS

lengths, 〈Livs〉, non-dimensionalized by the Kolmogorov length scale η; mean of IVS tan-

gential velocity, 〈Uivs〉, non-dimensionalized by u′; Ωλ = 〈ω0〉 /(ω′Re

1/2λ

)is the mean of

IVS vorticity magnitude at the its axis, 〈ω0〉, non-dimensionalized by the vorticity magni-

tude root mean square ω′ and normalized by Re1/2λ ; circulation-based Reynolds number

RΓ normalized by Re1/2λ , where RΓ = 〈Γ〉 /ν. . . . . . . . . . . . . . . . . . . . . . . . . . 53

xv

4.2 Constitutive Parameters for comparison between FENE-P and Carreau-Yasuda models.

Non-dimensional polymer maximum extensibility L; polymer relaxation time τp; non-dimensional

polymer concentration β; solution solvent kinematic viscosity ν[s]; Carreau-Yasuda pa-

rameter a; Carreau-Yasuda parameter nc; Carreau-Yasuda equivalent relaxation time λc;

solution kinematic viscosity at zero-shear-rate ν0. . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Simulation Parameters for the Newtonian case. The Newtonian simulation constitutes the

reference simulation. All reference quantities are computed with Newtonian parameters.

All simulations have the same mesh size in each direction N and all have the same

power input, (forcing), P . Mesh size in each direction N ; Reynolds number (equal to the

reference one); power input P ; root-mean-square velocity u′. . . . . . . . . . . . . . . . . 62

4.4 Simulations Parameters for the FENE-P cases. Reference Weissenberg number, (com-

puted with Newtonian τη), Weref ; Weissenberg number, (computed with solvent τ [s]η ),

We; polymer relaxation time τp; solvent dissipation rate ε[s]; polymer dissipation rate ε[p];

dissipation reduction DR; root-mean-square velocity u′. . . . . . . . . . . . . . . . . . . . 63

4.5 Simulations Parameters for the Carreau-Yasuda cases. Carreau-Yasuda equivalent relax-

ation time λc; dissipation rate, based on viscosity when SijSij →∞, ε∞; dissipation rate,

based on fluctuating vescosity, ε; root-mean-square velocity u′. . . . . . . . . . . . . . . . 63

4.6 Simulation Parameters for the Newtonian cases. Mesh size in each direction N ; Taylor

microscale-based Reynolds number Reλ; power input, (forcing), P ; kinematic viscosity ν;

integral length scale L11; root-mean-square velocity u′; dissipation coefficient Cε. . . . . . 69

4.7 Simulations Parameters for the FENE-P cases at the onset of dissipation reduction. Mesh

size in each direction N ; reference Weissenberg number, (computed with total dissipation

rate and solvent kinematic viscosity), Weref ; Taylor microscale-based Reynolds number,

(computed with solvent properties and statistics), Reλ; polymer relaxation time τp; β con-

centration; power input P ; solvent kinematic viscosity ν[s]; polymer dissipation rate ε[p];

dissipation reduction DR; root-mean-square velocity u′. These simulations have the non-

dimensional maximum extensibility L = 100 and, as can be computed with given values,

(kmaxη)ref

= 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.8 Simulations Parameters for the FENE-P cases near maximum dissipation reduction. Mesh

size in each direction N ; reference Weissenberg number, (computed with total dissipa-

tion rate and solvent kinematic viscosity), Weref ; Weissenberg number, (computed with

solvent properties and statistics), We; Taylor microscale-based Reynolds number, (com-

puted with solvent properties and statistics), Reλ; polymer relaxation time τp; β concentra-

tion; power input P ; solvent kinematic viscosity ν[s]; solvent dissipation rate ε[s]; polymer

dissipation rate ε[p]; dissipation reductionDR; integral length scale L11; root-mean-square

velocity u′; dissipation coefficient Cε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xvi

List of Figures

2.1 Axial velocity component over time, U1 (t), measured on the centerline of a turbulent pla-

nar jet, from [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Time average of axial velocity component, 〈U1〉, as a function of radial position x2 from the

centreline, at a given stream perpendicular plane of a turbulent jet. The function ploted is

〈U1〉 normalized by its value at the centerline 〈U1〉0 and the coordinate x2 is normalized

by the distance of the given stream-perpendicular plane from the nozzle x1, from [11]. . . 4

2.3 Sketch of Q−R map showing the physical/topological features related to each zone, from

[23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Modified skin friciton f−1/2 as a function of modified Reynolds number log(Ref1/2

).

This are the so-called Prandtl-Karman coordinates. Prandtl-Karman Law is an analytical-

emprirical law, obtained for turbulent newtonian pipe flows, as a result of an integration of

the log-law region of the turbulent boundary layer. The laminar region, or Poiseuille’s Law,

results from the solution of laminar pipe flow. MDR asymptote is empirical. From [27]. . . 27

2.5 Stratification of a newtonian turbulent boundary layer. Velocity profile in wall units. From

[30]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Stratification of a viscoelastic turbulent boundary layer. Velocity profile in wall units. From

[31]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 For coordinates specification and plot information clarification see figure 2.4. Experimen-

tal results from [31] show some paths of viscoelastic fluids. Note that when MDR is

achieved all paths collapse in the same slope. From [31]. . . . . . . . . . . . . . . . . . . 29

4.1 Scalability tests for 〈Rivs〉 (a) with η and (b) with RB . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Scalability tests for 〈Uivs〉 (a) with u′ and (b) with uη. . . . . . . . . . . . . . . . . . . . . . 55

4.3 Scalability law for 〈Uivs〉 with η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Scalability tests for 〈Livs〉 (a) with L11 and (b) with λ. . . . . . . . . . . . . . . . . . . . . . 56

4.5 Scalability test for 〈Livs〉 with η. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 Scalability laws for (a) 〈Livs〉 /L11 and (b) 〈Livs〉 /λ. . . . . . . . . . . . . . . . . . . . . . . 57

4.7 Scalability law for 〈Livs〉 /η. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.8 PDF’s of (a) 〈ω0〉 /(ω′Re

1/2λ

)and (b) ReΓ/Re

1/2λ . . . . . . . . . . . . . . . . . . . . . . . . 58

xvii

4.9 Non-dimensionalized energy spectra E (k) /(u2ηη)

(a) for all FENE-P cases and (b) for all

Carreau-Yasuda cases. Both have the Newtonian non-dimensionalized energy spectra.

Parameters used to non-dimensionalize are from the Newtonian case. . . . . . . . . . . . 64

4.10 Non-dimensionalized dissipation spectra D (k) / (Pη) (a) for all FENE-P cases and (b)

for all Carreau-Yasuda cases. Both have the Newtonian non-dimensionalized dissipation

spectra. Parameters used to non-dimensionalize are the forcing P and η from the Newto-

nian case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.11 Non-dimensionalized cumulative transfer spectra Π (k) /P (a) for all FENE-P cases and

(b) for all Carreau-Yasuda cases. Both have the Newtonian non-dimensionalized dissipa-

tion spectra. Parameter used to non-dimensionalize is the forcing P . . . . . . . . . . . . . 65

4.12 Non-dimensionalized dissipation spectra D (k) / (Pη) and non-dimensionalized transfer

spectra T (k) / (Pη) for both cases F4 and C4. It is also provided the non-dimensionalized

dissipation spectraD (k) / (Pη) for the Newtonian case, which equals its non-dimensionalized

transfer spectra T (k) / (Pη) at steady state for kη > 2× 10−2, (non-dimensionalized wavenum-

ber beyond which the forcing spectra term equals zero). Parameters used to non-dimensionalize

are the forcing P and η from the Newtonian case. . . . . . . . . . . . . . . . . . . . . . . . 66

4.13 Dissipation reduction DR as a function of (1− β) /β at various reference Weissenberg

numbers Weref (a) for Rerefλ = 67 and (b) for Rerefλ = 91. These samples are at the

onset of dissipation reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.14 Scalability law for the onset of dissiaption reduction derived in subsection 2.4.4 and shown

in equation (2.115). The fitted slope indicates a value of ≈ −11/4, resulting in a stretching

exponent p ≈ 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.15 Dissipation statistics as a function of (1− β) /β at various reference Weissenberg num-

bers Weref , (a) represents the dissipation statistic DR and (b) represents the statistic

1/C1/2ε . These samples are near the maximum dissipation reduction. A sample near

Newtonian conditions, (DR ≈ 0), is also given. . . . . . . . . . . . . . . . . . . . . . . . . 73

4.16 Relation between dissipation coefficient Cε and Taylor microscale-based Reynolds num-

ber, (computed with solvent properties and statistics), Reλ at various reference Weis-

senberg numbers Weref and β concentrations, (a) in Prandtl-Karman coordinates and (b)

in a Cε −Reλ plot. These samples are near the maximum dissipation reduction. In (a), the

lines joining the samples follow an ascending order of (1− β) /β, being the lowest, (the

starting point), in the lower-right corner, (considering the set of points with Weref 6= 0). In

(b) it should be interpreted accordingly. Table 4.8 may also be used to identify the points. 73

xviii

. . .

xix

Nomenclature

Acronyms

CDF Cumulative Distribution Function.

CFL Courant-Friedrich-Levy number or Courant number.

CFSE Coherent Fine-Scale Eddies.

CY Carreau-Yasuda.

DNS Direct Numerical Simulations.

FENE-P Finitely Extensible Nonlinear Elastic model with Peterlin closure.

HTI Homogeneous Isotropic Turbulence

IVS Intense Vorticity Structures.

KT Kurganov and Tadmor.

MHD Magneto-Hydrodynamics.

PDF Probability Density Function.

PEO Polyethylene Oxide.

SPD Symmetric Positive Definite

Greek symbols

αp, βp Runga-Kutta 3rd−order accurate sheme coefficients for each time-step p.

α Strain rate for a Burgers’ Vortex.

β Non-dimensional concentration parameter.

γ Invariant measure of the rate-of-strain Sij .

Γ IVS circulation or Burgers’ Vortex circulation.

δ Dirac Delta function.

∆t Time-step length.

xx

∆x,∆y,∆z Grid spacing in each direction.

δ~k′;−~k Like Kronecker Delta but for vectors ~k′ and −~k.

δij Kronecker Delta.

δ Exponent of the onset of dissipation reduction scalability law.

ε[p] Sum of the polymer-solvent spectral transfer term,∑k Tp (k), (which is equal to the sum of the

polymer spectral dissipation term in statistically steady state conditions), for the FENE-P kinetic

energy evolution equation.

ε[s] Sum of the spectral dissipation term,∑kD (k), (which represents the sum of the solvent spectral

dissipation), for the FENE-P kinetic energy evolution equation.

ε∞ Sum of the spectral dissipation term,∑kD (k), (which represents the sum of the spectral dissi-

pation at constant ν∞), for the Carreau-Yasuda kinetic energy evolution equation.

ε Dissipation rate.

ε Sum of the spectral fluctating dissipation term,∑k L (k), for the Carreau-Yasuda kinetic energy

evolution equation.

ζSi Eigenvalues of Sij .

ζi Eigenvalues of Aij .

ζ Drag coefficient of a polymeric bead when interacting with the fluid.

η Kolmogorv’s characteristic length.

θ1, θ2 Uniformly distributed random real angles ∈ [0, 2π], for the forcing hi, generated at each wavenum-

ber for each time-step.

θ Azimuthal coordinate for cylindrical coordinates space vector.

λc Parameter for Carreau-Yasuda.

λ Taylor microscale, (equal to the transverse one).

µ[p] Zero-shear-rate polymeric equivalent dynamic viscosity.

µ0 Dynamic viscosity for Carreau-Yasuda when SijSij = 0.

µ∞ Dynamic viscosity for Carreau-Yasuda when SijSij →∞.

µeff Effective viscosity of a polymer solution defined as T [p]12 /S.

µ Dynamic viscosity.

ν0 µ0/ρ.

xxi

ν∞ µ∞/ρ.

νc µc/ρ.

ν Kinematic viscosity.

ν νc − ν∞.

ξ1, ξ2 Inner products uie1i and uie2i for the forcing hi.

Πp Spectral Cumulative polymer-solvent transfer term.

Π Spectral cumulative transfer term.

ρ Fluid density.

σ Stretching rate.

ς Polymer stretching as a function of a characteristic length scale `.

τ∗ Characteristic time scale which is equal to τp.

τ ′ Characteristic time scale correspondent to the characteristic length scale r′.

τp Polymer relaxation time.

τ Characteristic time.

Φij Fourier transform of Rij for continuous wavenumber.

φi Generic vectorial quantity i−component.

φ Uniformly distributed random real angle ∈ [0, π], for the forcing hi, generated for each wavenum-

ber for each time-step.

ψ Real angle ∈ [0, 2π], generated with uniform distribution and equal to θ2 − θ1.

ω0 Vorticity magnitude at the axis of IVS.

Ωij Rate-of-rotation tensor.

ωivs Vorticity magnitude threshold for IVS identification.

ωi Vorticity vector i-component.

ωz Vorticity vector z−component in cylindrical coordinates.

ω Vorticity magnitude.

~ω Vorticity vector.

Roman symbols

A Physical space (discrete).

xxii

A Parameter for the forcing.

A Spectral space (discrete).

A,B,C Arbitrary scalar-valued functions.

Aij Velocity gradient.

Aran, Bran Complex functions for the forcing hi.

a Exponent parameter for Carreau-Yasuda model.

A′ Continuous physical space.

~Bi Brownian force at bead i.

C C22 for a imposed uniform shear flow with a constant velocity gradient with one non-zero value,

∂u1/∂x2 = S.

C Conformation tensor.

H Flux tensor.

Cε Dissipation coefficient.

Cij Conformation tensor.

c Parameter for the forcing.

~Di Drag force at bead i.

DA Descriminant of Aij characteristic polynomial equation.

DR Dissipation reduction.

D Spectral dissipation term.

~eSi Eigenvectors of Sij .

~e1, ~e2 Vectors orthogonal to each other and to ~k for the forcing hi.

E Discrete energy spectra.

eijk Permutation symbol tensor.

E Continuous energy spectra.

e1i, e2i Vectors ~e1 and ~e2 i−components.

f Longitudinal auto-correlation function.

~Fi Spring force at bead i.

~F Spring force.

xxiii

FM M -point joint CDF.

fM M -point joint PDF

f Peterlin function.

F (·) , (·) Discrete Fourier Transform.

F−1 (·) Inverse Discrete Fourier Transform.

Fi Canonical vectorial functional i−component for a time-dependent problem.

Gk Nonlinear term of Navier-Stokes nomentum equation in Fourier space.

gA, gB Scalar real-valued function for the forcing hi.

g Transverse auto-correlation function.

hi Forcing vector i−component in Fourier space.

H Force spectrum.

hi Forcing vector i−component in physical space.

H Elastic modulus.

=· Imaginary part of a comlex number.

i Imaginary unit.

~k,~k′ Cartesian wavenumber vectors in Z.

k, k′ Norm of cartesian wavenumber vectors or shperical radial vector component.

k0 Norm of wavenumber vector locating the maximum of the energy spectrum E (k) for initial con-

ditions.

k1 Radial wavenumber component in spherical coordinates.

ka, kb Limits of the wavenumber radial component in spherical coordinates interval [ka, kb] where the

forcing is to be applied.

kB Boltzmann constant.

Kel Elastic energy stored by polymers per unit volume.

kh Radial component in shperical coordinates of the wavenumber vector locating the maximum of

the forcing spectrum H (k).

ki, k′i Cartesian wavenumber vectors i-components in Z.

kmax Maximum wavenumber norm considering de-aliasing.

xxiv

kNyquist Nyquist wavenumber.

K Averaged kinetic energy.

L Size of a cubic box.

`∗ Length scale which time scale corresponds to τp.

`′ Characteristic length scale below which polymers influence turbulence dynamics.

`0 Characteristic length of large scales.

`1 Characteristic length scale separating the energy-containg range from inertial subrange.

`2 Characteristic length scale separating the inertial subrange from the dissipation range.

` Characteristic lenght of an arbitrary scale.

L Spectral fluctuating dissipation term.

L11 Longitudinal integral length scale.

Lmax Dimensional maximum extensibility of a spring.

L Non-dimensional maximum polymer extensibility.

M ′i(1−β)β

ν[s]

τpPikikjF (f (Cmm)Ckj − δkj).

Mi PikikjF (2νSkj).

M Integer value representing number of points for statistics.

N Natural numbers set.

~n Normal to a surface.

Nr,Mθ Integers.

nr,mθ Integers.

nc Exponent parameter for Carreau-Yasuda.

N Mesh size.

n Number density of polymer molecules.

N (a, b) Normal distribution with mean a and variance b.

O (·) Order of magnitude.

P1, P2 Contributions of a forcing.

PA, Q,R Invariants of Aij .

Pij Projection tensor.

xxv

Px, Py, Pz Contributions of P1 in each direction.

P Averaged power input.

p Pressure.

P (·) Probability.

Qij Isotropic tensor

QS , RS Invariants of Sij .

q Stretching exponent.

R Real numbers set.

<· Real part of a comlex number.

r Space vector radial component in cylindical coordinates.

~R Spring end-to-end displacement vector.

~r Cartesian space vector.

Rij Fourier transform of Rij for discrete wavenumber, (components of the Fourier Series).

RB Burgers’ radius

Rij Two-point velocity correlation (physical space).

Rivs IVS radius.

Ri Spring end-to-end displacement vector i-component.

ri Cartesian space vector i-component.

ReΓ IVS circulation-based Reynold number.

Re0 Large length-scale-based Reynolds number.

Re` Arbitrary length-scale-based Reynolds number.

Reη Kolmogorov’s length-scale-based Reynolds number.

Reλ Taylor micro-scale-based Reynolds number.

Re Reynolds number.

r Cartesian space vector norm or spherical radial vector component.

S ∂u1/∂x2.

~s Cartesian space vector.

Sij Strain-rate tensor.

xxvi

S Surface.

s Slope of the energy spectrum E (k) for initial conditions.

T Absolute temperature.

Tij Deviatoric stress tensor.

Tp Polymer-solvent spectral transfer term which is equal to polymer spectral dissipation term in

statistically steady state conditions.

T Spectral transfer term.

t Time.

T0 Characteristic rate of transfer of kinetic energy at large scales.

T` Characteristic rate of transfer of kinetic energy at an arbitrary scale `.

u, v, w Area-averaged velocity components in Cartesian coordinates.

ui Fourier transform of ui.

~U Velocity vector (random variable).

~u Fluctuating velocity vector (random variable).

u′ Velocity root-mean-square.

u0 Characteristic velocity of a scale with a characteristic length `0.

u` Characteristic velocity of a scale with a characteristic length `.

uη Kolmogorov’s characteristic velocity.

Uivs Azimuthal velocity of IVS.

Ui Velocity vector i−component (random variable).

ui Fluctuating velocity vector i-component (random variable).

~ui Velocity vector at bead i.

∆V Finite volume.

~Vi Velocity vector of bead i.

~V Independent sample-space vector variable corresponding to ~U .

~v Velocity difference.

Vi Independent sample-space vector i−component variable corresponding to Ui.

V Volume.

xxvii

~Wi Wiener process at bead i.

~W Wiener process.

~w Cartesian wavenumber vector in R.

wi Cartesian wavenumber vector i-component in R.

We Weissenberg number.

w Cartesian wavumber vector norm or shperical radial vector component.

~x Cartesian space vector.

xi Cartesian space vector i-component.

~y Position difference.

Z Integer numbers set.

Subscripts

C Carreau-Yasuda.

eq Equilibrium conditions.

F FENE-P.

f Longitudinal.

G Generalized Newtonian.

g Transverse.

i, j, k, . . . Vector/tensor indexes. The p and n indexes might be used for parallel and normal components,

respectively.

ivs IVS.

N Newtonian.

r, θ, z Cylindircal coordinates.

sph Spherical.

x, y, z Cartesian space directions.

Superscripts

∗ Complex conjugate.

+ Interface limit value approaching from right side.

− Interface limit value approaching from left side.

xxviii

(i) At point i.

nt Time-step index.

[p] Polymeric.

p Runge-Kutta 3rd−order accurate method sub-step index.

ref Reference conditions.

[s] Solvent.

x, y, z Cartesian space directions.

xxix

Chapter 1

Introduction

This chapter intends to give emphasis to the relevance of the subjects explored in this work. Further-

more, the objectives to be achieved will be mentioned as well as the structure of this dissertation will be

summarized.

1.1 Motivation

The study of fluid motion is of particular interest since it is ubiquitous in nature and in a wide range

of engineering applications. Its dynamics is, in many cases, highly tree-dimensional and turbulent.

Although this is not necessarily a drawback, its understanding is complex, leading to a lot of demanding

research work towards its comprehension.

Mastering turbulent phenomenon is crucial for engineering conceptualization and, in particular, for

aerospace applications. A phenomenological research of turbulence by means of Direct Numerical

Simulations (DNS) is conducted in this work, focusing on two specific subjects, the first related with the

Intense Vorticity Structures (IVS) in Newtonian turbulence, and the second with the turbulent dissipation

in viscoelastic fluids.

The dynamics of vortical structures is inherently linked with the kinetic energy (production and dissi-

pation), mixing, diffusion, and transport of mass, heat and momentum. It has also an important role in

the enhancement of frictional drag reduction in turbulent boundary layers, [1]. The study of these struc-

tures is also encouraged by their simplicity, the knowledge of their mechanisms being a prerequisite for

flow prediction and control, [1, 2].

Since the work published by Toms [3] in solutions with polymer additives, the phenomenon of drag re-

duction has been investigated and became important, e.g., in long-distance liquid transporting in pipeline

facilities. Heat transfer of viscoelastic fluids has been studied in cooling applications of turbines [4, 5],

and it may become an important application in aerospace industry.

To simulate viscoelastic fluids, the Finitely Extensible Nonlinear Elastic model with Peterlin closure

(FENE-P) is used. An analogy may be made among this set of equations and that for Magneto-

Hydrodynamics (MHD), [6]. Plasma flows are now part of a set of new technologies that takes place in

1

hypersonic propulsion in the aerospace industry. Plasma flows simulated with MHD have inherent rel-

evance, [7–9], since these tools along with leading knowledge in Electromagnetics, Aerodynamics and

Chemical Kinetics may help in propulsion systems for improving aerospace vehicle performance, [10].

1.2 Objectives

This work aims to investigate the IVS as well as turbulent dissipation with polymer additives in forced

Homogeneous and Isotropic Turbulence (HIT), at statistically steady state conditions. They are analyzed

using DNS. The objectives can be summarized as follows:

1. To simulate Newtonian turbulence and analyze the IVS for several Reynolds numbers. This intends

to give statistical results for higher Reynolds numbers and a scalability law for the length of IVS.

2. To simulate turbulence with the Carreau-Yasuda (CY) model. This intends to assess the capability

of a purely viscous flow of reproducing the dynamics of viscoelastic turbulence, simulated with

FENE-P, when applied the same shear-thinning law. The analysis will be performed through the

comparison of the kinetic energy evolution equation for both models.

3. To simulate FENE-P at onset of dissipation reduction DR conditions. This intends to investigate

the scalability law between polymer concentration and Weissenberg number at those conditions.

4. To simulate FENE-P near maximum DR conditions. This intends to investigate preliminarily the

behavior of dissipation coefficient Cε and to assess a possible presence of a Virk’s-like asymptote.

1.3 Thesis Outline

This thesis is organized into five chapters.

The first chapter gives the motivation for this work with an overview on the topic. The objectives to

be achieved are also summarized as well as the organization of this dissertation.

The second chapter gives background details on the topics and tools used in the analysis. The

principles of HIT, the features of IVS and description of turbulence with polymers will be given.

The third chapter goes deep in the dynamic equations and numerical methods. The CY model is

presented and FENE-P is derived. Informations about the code implementation methods are also given.

The fourth chapter presents the results and discussion. The IVS are analyzed and a comparative

survey between CY and FENE-P is provided, concluding with the results about the turbulent dissipation

study with the FENE-P viscoelastic fluid.

The fifth chapter gives the conclusion where the achievements are summarized and some proposals

for future work will be given.

2

Chapter 2

Background

In this chapter some concepts about turbulence will be introduced. After a descriptive definition,

classical features of HIT will be developed. Then, how the coherent vorticity structures are studied in

turbulent fields will be mentioned and their main characteristics will also be described. At the end, it

is given some introductory remarks about complex fluids and also an overview about turbulence with

polymer additives.

Although Navier-Stokes equations will be referred to on chapter 3, references to them will be present

in this chapter, making it logical to present them in advance. Navier-Stokes equations, for a velocity field

with an average of zero, for incompressible Newtonian fluids with constant properties can be read as,

∂ui∂xi

= 0 , (2.1a)

∂ui∂t

+∂

∂xj(uiuj) = −1

ρ

∂p

∂xi+ 2ν

∂Sij∂xj

, (2.1b)

where ~x = (x1, x2, x3) is the space vector in Cartesian coordinates, ~u = (u1, u2, u3) is the fluctuating

velocity vector, ρ is the fluid density, p is the pressure and ν is the fluid kinematic viscosity. Here,

Sij =1

2

(∂ui∂xj

+∂uj∂xi

), (2.2)

is the strain-rate tensor.

2.1 What is Turbulence?

Turbulence is the term widely used to name a flow regime characterized by its unsteadiness, ir-

regularity, randomness, chaotic and unpredictable nature in some sense, as resumed by [11]. These

features can be identified when observing the velocity field variation in position and time. On the other

and, turbulent flows are very effective for transporting and mixing matter, momentum and heat.

To illustrate it consider the plot of figure 2.1 which displays the axial velocity component at the cen-

terline of a turbulent jet, see [11]. The behavior showed on figure 2.1 is irregular and difficult to treat,

3

Figure 2.1: Axial velocity component over time, U1 (t), measured on the centerline of a turbulent planarjet, from [11].

at least in such a form. A careful interpretation allows one to observe that U1 (t) might be seen as per-

turbation around a certain mean value. This introduce a statistical view point of the problem, which, in

fact, plays an essential role on turbulence description. In general, instantaneous values are not used

to report the phenomena, instead various statistical approaches lead to a delineation of a picture much

more comprehensive. In order to give a glance on that, consider the simple statistic of taking an average

over time of U1 (t), for each radial distance from the centerline x2 and for a given stream-perpendicular

plane located at a distance x1 from the nozzle. Say that U1 (t) average is 〈U1〉 and normalize the latter

by its own value when x2 = 0, 〈U1〉0. The plot of 〈U1〉 /〈U1〉0 as a function of x2/x1 is in figure 2.2. As it

Figure 2.2: Time average of axial velocity component, 〈U1〉, as a function of radial position x2 fromthe centreline, at a given stream perpendicular plane of a turbulent jet. The function ploted is 〈U1〉normalized by its value at the centerline 〈U1〉0 and the coordinate x2 is normalized by the distance of thegiven stream-perpendicular plane from the nozzle x1, from [11].

can be seen in figure 2.2, the data is smoother and more meaningful. Actually, it represents a behavior

similar to that found for an instantaneous (not-averaged) measure of the same kind in a laminar jet.

4

Turbulence is also characterized by nonlinearities and present great sensitiveness to perturbations.

Turbulence is a multi-scale process and a Fourier analysis might be used. If one considers the Fourier

Transform of the tensor representing the two-point correlation of the velocities, which will be seen in

section 2.2, and if one also considers the turbulence to be homogeneous and isotropic, that tensor in

Fourier space is uniquely determined by the energy spectrum E (k). This shows the contribution of each

turbulent scale to the averaged kinetic energy K. Furthermore, a deeper study of the dynamic equation

which rules the temporal evolution of K reveals that its nonlinear term, related to the nonlinear term

of equation (2.1), is responsible for the development of new scales and for the energy transfer through

them. This emphasizes the importance of non-linearity of turbulence.

In the section 2.2 the analysis of homogeneous and isotropic turbulence will be made.

2.2 Homogeneous and Isotropic Turbulence

2.2.1 Kolmogorov Hypothesis and Scales of Turbulent Motion

Turbulence description has its fundamental basis on the dimensional analysis of the phenomenon.

The most important idea first introduced was to interpret turbulence as a multi-scale process. The

turbulence phenomena was illustrated by the energy cascade concept and it will be quantified later.

Energy Cascade

The energy cascade idea was introduced by Richardson in 1922 [12]. He depicted turbulent process

as one composed by eddies of different sizes that go smaller as energy cascade proceeds. Given an

eddy of length scale `, velocity scale u` (`) and time scale τ` (`) = `/u`, consider the local Reynolds

number Re` ≡ u``/ν, where ν is the kinematic viscosity. For the largest eddies say that their charac-

teristic scales are `0, u0 and τ0, so that Re0 ≡ u0`0/ν. Now let Re0 be large enough to consider that

viscous forces are negligible. These eddies are anisotropic and unstable, but contain the root mean

square of the kinetic energy of the system u′. So it can be state that u0 ∼ u′. The instability of those

eddies makes them to break up and transfer their energy to smaller ones. A repetition of this process

takes place again until Re` become sufficiently small so the eddies can be stable. Viscous forces are

now effective in dissipating energy and the cascading process reaches its end. As it can be seen, the

picture drawn here suggests that the rate of kinetic energy dissipation ε is determined by the rate of

transfer of that energy which took place on the largest eddies at first, say that is T0. On other words,

ε ∼ T0. But T0 ∼ u30/`0 which is, in turn, independent of ν. So ε ∼ u3

0/`0 for high Re0.

Kolmogorov’s Theory

The idea of an energy cascade gave basis to Kolmogorov’s theory presented in 1941, see [13]. It

gave a quantitative perspective of the concept already developed and established a deeper view of the

mechanisms underlying on this multi-scale phenomenon. The main definitions and hypothesis, as well

his results, will be presented next.

5

Considering a set of points in space x(0), x(1), . . . , x(M) and letting,

~y (i) ≡ ~x (i) − ~x (0) , (2.3)

and,

~v(~y (i)

)≡ ~U

(~x (i), t

)− ~U

(~x (0), t

), (2.4)

where ~U is the velocity and i = 1, . . . , M , it is determined the joint probability density function (PDF) of

~v at the M points, denoted by fM . Furthermore we can state the following definitions.

• Definition of Local Homogeneity. The turbulence is said to be locally homogeneous if the

M−point PDF fM does not depend on x(0) and ~U(~x (0), t

).

• Definition of Local Isotropy. The turbulence is said to be locally isotropic if it is locally homoge-

neous and if in addition fM is invariant with respect to rotations and reflections of the coordinate

axes.

Based on these definitions Kolmogorov has constructed the main hypothesis his work is based on. He

verified that large scales are highly anisotropic and have a specific geometry as well as directional

information which are determined by boundary conditions and mean flow field. As the transfer of energy

begins to happen along with the break up of large scales, which dictates scale-reducing process, he

stated that information and geometry of large scales are lost during that, hypothesizing that small scales

are isotropic.

• Hypothesis of Local Isotropy. At sufficiently high Reynolds number, turbulence is locally isotropic

under a sufficiently small spatial region far from boundaries and flow singularities.

For scales ` < `0 where isotropy principle applies, statistics are identical and it is legitimate to name

that spectrum of scales the Universal Equilibrium Range. On this range, the dominant processes are the

energy transfer and dissipation rates. So it is dependent on ν and ε. It will be shown that T`, the energy

transfer rate for a given scale `, is somewhat related with ε. Now, the First Similarity Hypothesis arises.

• The First Similarity Hypothesis. In locally isotropic turbulence, the M−point PDF fM is uniquely

determined by ν and ε.

As it was stated, the range where turbulence is isotropic is the Universal Equilibrium Range. The notion

of equilibrium is straightly related to the concept that τ`, the characteristic time scale of eddies of size `,

is small enough so they can adapt quickly to maintain a dynamic equilibrium with T`. One of the greatest

consequences of the statements made so far is the discovery of the so-called Kolmogorv Scales. Due

to the tight dependence of the statistics on ν and ε, the unique characteristic scales that can be formed

with those are the Kolmogorov Scales, defined as,

η ≡(ν3

ε

)1/4

, (2.5)

uη ≡ (νε)1/4

, (2.6)

6

τη ≡(νε

)1/2

, (2.7)

where η, uη and τη are, respectively, the Kolmogorov length, velocity and time scales. To identify what

eddies are being characterized, its useful to verify that,

Reη ≡ηuην

= 1 , (2.8)

ν

τ2η

= ε . (2.9)

From (2.8) it can be seen that, for η, viscous effects are dominant, whereas from (2.9) it is verifiable that

velocity gradients for eddies scalable with η are in fact consistent with viscous dissipation. These facts

show that eddies of size comparable to η are the smallest ones.

If Re0 continue to increase, it is inevitable that might be encountered a range with Re` Re0 and

yet with Re` Reη = 1. Both of these suggest that such a region would be locally isotropic and vis-

cous effects could be neglected simultaneously. This idea inspired the Kolmogorov’s Second Similarity

Hypothesis.

• The Second Similarity Hypothesis. For Sufficiently high Reynolds number, statistics of motions

of size ` in the range `0 ` η are universal and uniquely determined by ε, independent of ν.

If one defines `1 and `2 such that η < `2 < ` < `1 < `0, the Universal Equilibrium Range (` < `1) can split

into two subranges, the Inertial Subrange (`2 < ` < `1) and the Dissipation Range (` < `2). The motion

dynamics in the Inertial Subrange is essentially inviscid, whereas in the Dissipation Range viscous

effects are dominant. The region where ` > `1 is called the Energy-Containing Range, where `0 is

included. In the Inertial Subrange, characteristic scales cannot be formed from ε alone. However, given

an eddy of size ` where `2 < ` < `1, a characteristic velocity u` and a characteristic time τ` can be

defined as,

u` (`) = (ε`)1/3

, (2.10)

τ` (`) =

(`2

ε

)1/3

. (2.11)

As a consequence, u` and τ` decrease while ` decrease as well as Re` also decrease. Still in the Inertial

Subrange, in order to understand how the energy transfer rate T` varies, one should start to estimate its

scalability. Once,

T` ∼u2`

`= ε , (2.12)

it is readily seen that T` does not depend on `. This suggests that the energy transfer rate is an ap-

proximately constant value in the Inertial Subrange for a high Re0 flow field. In addition it is consistent

with assumptions made in advance. If T0 = ε and the energy transfered to the Dissipation Range should

balance ε, saying that T` ≈ ε is a consistent picture for the energy cascade.

7

2.2.2 Statistical Description of Turbulence and Two-point Correlation Functions

It is relevant at this point to understand how turbulence should be treated to interpret meaningful

data. First, the random nature of turbulent phenomenon should be given proper context. Then what

methods are used to deal with it and what is actually done will be briefly explored here.

Random Nature of Turbulence

When an event is said to be random, it is being said that it is neither certain nor impossible. In fact

it may occur or it may but need not to occur, [11]. A random event is deterministically unpredictable.

Obviously, a deterministic system has predictability, but even non-explicit solutions of a deterministic

mechanical system have some sort of unpredictability if its initial and boundary conditions are not 100%

controlled, [11].

In practice, a turbulent regime of a flow field is exactly an example of a sequence of events which

sensitiveness to slight perturbations on initial or boundary conditions makes it unpredictable. Once it is

impossible to control at 100% those conditions, turbulence is random. In contrast, a laminar regime can

damp those perturbations more effectively for long periods, making solutions asymptoticly predictable.

So, with steady boundary conditions, Navier-Stokes equations can have steady solutions at laminar

regimes, but chaotic, turbulent solutions at turbulent ones.

The Navier-Stokes equations describe a deterministic system so, when simulating them directly,

some sort of stochastic perturbation should be embodied.

Characterization of a Random Field

A characterization of a random variable is related to the notion of probability distribution.

Consider ~U (~x, t) a velocity field random variable and ~U = (U1, U2, U3). Defining ~V = (V1, V2, V3) as

the independent sample-space velocity variable corresponding to ~U , the joint cumulative distribution

function (CDF), denoted by F1, is defined as,

F1 (Vi, ~x, t) ≡ P (Ui (~x, t) < Vi) , (2.13)

where P (·) denotes probability. The joint probability density function (PDF) f1 is defined as,

f1

(~V ; ~x, t

)≡∂3F1

(~V , ~x, t

)∂V1∂V2∂V3

. (2.14)

The mean velocity field can be defined as,

⟨~U (~x, t)

⟩≡∫ ∞−∞

~V f1

(~V ; ~x, t

)d~V , (2.15)

and the velocity fluctuating filed is,

~u (~x, t) ≡ ~U (~x, t)−⟨~U (~x, t)

⟩. (2.16)

8

In this work,⟨~U (~x, t)

⟩= 0.

The joint PDF in equation (2.14) just fully characterizes the velocity ~U at a specific point in space-

time, but has no additional information about the remaining surrounding field. That joint PDF represents

a one-point, one-time statistics. To characterize the whole velocity field an extension of the definition in

equation (2.13) should be made. For a set of M space-time points the M−point statistics would have a

joint CDF of the form,

FM

(V

(1)i , ~x(1), t(1); . . . ;V

(M)i , ~x(M), t(M)

)≡ P

(U

(1)i < V

(1)i ; . . . ;U

(M)i < V

(M)i

), (2.17)

and a joint PDF of the form,

fM

(V

(1)i , ~x(1), t(1); . . . ;V

(M)i , ~x(M), t(M)

)≡∂3MFM

(V

(1)i , ~x(1), t(1); . . . ;V

(M)i , ~x(M), t(M)

)∏Mk=1

∏3j=1 ∂V

(k)j

. (2.18)

As it can be seen, it is impossible to characterize an infinite set of points in space-time, so, in practice, a

random velocity field cannot be fully characterized.

Once it is impossible to compute the mean by its definition, see equation (2.15), some methods are

used to estimate it. For instance, for stationary flows a mean over time is used to estimate the ’real’

mean. In the case of homogeneous turbulence spatial means are used whereas ensemble averages

are applied for experiences that can be repeated. In either cases, the values obtained are themselves

random variables and can only be used to estimate the mean defined in equation (2.15), see [14] and

[11] for some details.

Two-point Correlation Functions

In subsection 2.2.1 it was presented Kolmogorov’s theory. The work he developed was based on

velocity differences statistics. He basically studied the covariance of the difference in velocity in two

points, the so-called second order velocity structure function. See [11] for details. On the other hand,

a view of turbulent scale-to-scale interactions can be developed considering a two-point correlation

function in homogeneous turbulence. The PDF’s defined for velocity differences may apply to any flow

field whereas a wavenumber interpretation follows the condition of homogeneity to be made on those

PDF’s in advance, [11]. If turbulence is homogeneous, a relation between the second order structure

function and the two-point correlation is well defined, see [11].

The two-point correlation mentioned above is denoted by Rij and is defined as,

Rij (~x,~r, t) = 〈ui (~x+ ~r, t)uj (~x, t)〉 . (2.19)

If turbulence is homogeneous equation (2.19) reduces to be just dependent on ~r so,

Rij (~r, t) = 〈ui (~x+ ~r, t)uj (~x, t)〉 . (2.20)

9

In subsection 2.2.3 a further condition will be imposed on Rij , the condition of isotropy. If the tensor

is homogeneous and isotropic, a Fourier Transform can be performed and the resulting isotropic tensor

is fully characterized by a scalar-valued function dependent on the wavenumber norm k. That scalar

function can be the so-called energy spectra, which provides the information of the contribution of each

wavenumber to the root mean square of the velocity fluctuation. This is the reason why great relevance

will be given on this subject in subsection 2.2.3.

2.2.3 Isotropic Correlation Tensors Rij and Φij

Apart from homogeneity, there are further restrictions that might be made relatively to the joint-

probability distribution of the values of the velocity at any n points of space. In particular, an isotropic

symmetry condition may be imposed. It is often referred to in turbulence studies the two-point joint

probability distribution, where it arises a second order tensor of averaged velocity products, see [14].

For a given system of coordinates ri, a isotropic second order tensor Qij is uniquely defined by,

Qij (~r) = A (r) δij +B (r) rirj , (2.21)

where A (r) and B (r) are scalar-valued functions dependent on r =√riri. Here δij is the Kronecker

Delta.

In the case of a turbulent field one has a specific isotropic velocity correlation tensor, Rij . It is defined

as [14], [11],

Rij (~r) ≡ 〈ui (~x+ ~r)uj (~x)〉 , (2.22)

where ~x ≡ (x1, x2, x3) is the positioning vector and ~r ≡ (r1, r2, r3) represents a vector displacement

relative to ~x. Here 〈·〉 represents ensemble averaging. Note that due to homogeneity, Rij is just a

function of ~r. The dependence on time was and will be omitted for simplicity however implicit. Now it is

defined,

f (r) =〈up (~x+ ~r)up (~x)〉

u′2, (2.23)

and

g (r) =〈un (~x+ ~r)un (~x)〉

u′2, (2.24)

where up and un stand for velocity components along directions parallel and normal to ~r, respectively,

with no summation convention applied. The definition of u′ follows the next observation,

Rij (0) = 〈uiuj〉 = u′2δij . (2.25)

In addition, an incompressible turbulent field gives, by the continuity equation, an additional condition on

Rij which is,∂Rij∂xj

= 0 . (2.26)

Applying the identity (2.21) and condition (2.26), along with definitions (2.23) and (2.24), it is known that

10

[14], [11],

Rij (~r) = u′2(g (r) δij + (f (r)− g (r))

rirjr2

), (2.27)

where it is written in a slightly different way, although equivalent. Furthermore, it provides the following

relation between f and g, [14],[11],

g (r) = f (r) +1

2r∂f

∂r, (2.28)

thus leaving Rij with just one unknown. By their construction, f is called the longitudinal autocorrelation

function and g the transverse one.

In order to make a relation to Fourier Space, the Fourier transform of the correlation Rij will be

presented. First it should be noted that a complex Fourier Series is wanted to describe Rij ,

Rij (~r) =∑~k

Rij

(~k)ei2πkmrm/L , (2.29)

where ~k ≡ (k1, k2, k3) ∈ Z3 is the wavenumber vector and Rij is assumed periodic with period L. The

Fourier Transform of Rij , Rij , is then,

Rij

(~k)

=1

L3

∫ L0

Rij (~r) e−i2πkmrm/L d~r . (2.30)

The next expressions demonstrate the form of Rij .

⟨ui

(~k′)uj

(~k)⟩

=

⟨1

L3

∫~x′ui (x′) e−i2πk′mx

′m/L d~x ′

1

L3

∫~x

uj (x) e−i2πkmxm/L d~x

⟩=

(1

L3

)2 ∫~x

∫~x′〈ui (x′)uj (x)〉 e−i2πk′mx

′m/Le−i2πkmxm/L d~x ′d~x

=

(1

L3

)2 ∫~x

e−i2π(k′m+km)x′m/L d~x

∫~r

Rij (r) e−i2πk′mrm/L d~r

=

(1

L3

)∫~r

Rij (r) e−i2πk′mrm/L d~r δ~k′;−~k .

(2.31)

The change of variable ~x′ = ~x+ ~r was made and the homogeneous property was used. Here, δ~k′;−~khas the same meaning of the Kronecker Delta but, instead for the subscripts i and j, it applies for the

vectors ~k′ and −~k. This implies that,

⟨ui

(~k)uj

(−~k)⟩

=

(1

L3

)∫~r

Rij (r) ei2πkmrm/L d~r

= Rij

(~k),

(2.32)

which means that ui at ~k only correlates with uj at wavenumber −~k.

If one wants to write a continuous fashion of Rij , a continuously-valued tensor Φij , one has the

following,

Rij (~r) =

∫ ∞−∞

Φij (~w) eiwmrm d~w , (2.33)

where ~w ≡ (w1, w2, w3) ∈ R3 is now a continuous variable. The Fourier Transform of Rij , Φij , is now

11

read as,

Φij (~w) =1

(2π)3

∫ ∞−∞

Rij (~r) e−iwmrm d~r . (2.34)

For a periodic function, with period L, both equations (2.29) and (2.33) are possible simultaneously. So,

if one plugs equation (2.29) in equation (2.34) one has,

Φij (~w) =1

(2π)3

∫ ∞−∞

∑~k

Rij

(~k)ei2πkmrm/Le−iwmrm d~r

=1

(2π)3

∑~k

Rij

(~k)∫ ∞−∞

ei(2πkm/L−wm)rm d~r =

=1

(2π)3

∑~k

Rij

(~k)

(2π)3δ(

2π~k/L − ~w)

=∑~k

Rij

(~k)δ(

2π~k/L − ~w),

(2.35)

where δ is the Dirac Delta function.

Due to the isotropic nature considered for turbulence, the isotropic tensor Φij has a unique expres-

sion of the form,

Φij (~w) = A (w) δij +B (w)wiwj , (2.36)

where A (w) and B (w) are scalar-valued functions dependent on w =√wiwi. The continuity condition

in spectral space dictates that,

wiΦij = 0 , (2.37)

which applied on equation (2.36) reduces the latter to be dependent on one unknown scalar multiply.

Introducing the definition for E (w),

E (w) =

∮S(w)

1

2Φii dS (w) , (2.38)

where dS (w) is an elementary surface area of a sphere with radius w, one can rewrite (2.36) in the form,

Φij =E (w)

4πw2

(δij −

wiwjw2

)=E (w)

4πw2Pij ,

(2.39)

where Pij is the projection tensor term. E (w) is the energy spectrum. Using equation (2.35) one has,

E (w) =∑~k

1

2Rii

(~k)δ (2πk/L − w)

=∑k

∑~k′

1

2Rii

(~k′)δk′;k δ (2πk/L − w)

=∑k

E (k) δ (2πk/L − w) .

(2.40)

12

Analogously to equation (2.38), E (k) was defined as,

E (k) =∑~k′

1

2Rii

(~k′)δk′;k . (2.41)

Note that the kinetic energy K = 12 〈uiui〉 is,

K =1

2Rii (0) =

1

2

∫ ∞−∞

Φii (~w) d~w

=

∫ ∞0

E (w) dw

=∑k

E (k) .

(2.42)

2.2.4 Other Characteristic Scales and Relations

Further references to characteristic scales widely used to describe turbulent motion are important to

give. From subsection 2.2.3, one has the longitudinal autocorrelation function f (r) and the transverse

one g (r), see equations (2.23), (2.24) and (2.27). Based on those, two definitions for length scales will

be given, the longitudinal integral length scale L11 and the Taylor microscale λ.

For a characteristic length, which large scales scale with, it is defined, [14], [11],

L11 =1

2

∫ L0

f (r) dr , (2.43)

and it is called the longitudinal length scale. It describes motions with velocities scalable with u′. Here,

periodicity with period equal to L was considered again. This scale can be determined directly from

spectral space. After some math one arrives to [14], [11],

L11 =π

2u′2

∫ ∞0

E (w)

wdw

=π

2u′2

∑k

E (k)

2πk/L.

(2.44)

See section 2.2.3 for details about the energy spectrum.

Another characteristic length scale often used is the Taylor microscale λ. First, f (r), which is even,

is approximated by its Taylor series expansion up to the second order term, [14], [11],

f (r) ≈ f (0) + f ′ (0) r +1

2f ′′ (0) r2

= 1 +1

2f ′′ (0) r2 ,

(2.45)

and then the longitudinal Taylor microscale is defined,

f (r) ≈ 1− r2

λ2f

, (2.46)

13

λf =

(−1

2f ′′ (0)

)−1/2

. (2.47)

On the other hand,

f ′′ (0) = limr→0

∂2

∂r2

〈u1 (~x+ e1r)u1 (~x)〉u′2

=1

u′2

⟨∂2u1

∂x21

(~x+ e1r)u1 (~x)

⟩=

1

u′2

⟨∂2u1

∂x21

u1

⟩= − 1

u′2

⟨(∂u1

∂x1

)2⟩.

(2.48)

So, ⟨(∂u1

∂x1

)2⟩

=2u′

2

λ2f

. (2.49)

A similar microscale can be also derived, the transverse Taylor microscale, defined by,

λg =

(−1

2g′′ (0)

)−1/2

, (2.50)

which is related to the longitudinal Taylor microscale λf according to,

λf =√

2λg . (2.51)

Relation (2.51) can be derived through equation (2.28), [14], [11]. In section (2.2.5) it is shown that,

ε = ν

⟨∂ui∂xj

∂ui∂xj

⟩. (2.52)

Through the specification of the 4th-order tensor,

⟨∂ui∂xj

∂ul∂xm

⟩= Aδijδlm +Bδilδjm + Cδimδjl , (2.53)

where such a form is unique apart from the arbitrary constants A, B and C, see [14] and [11], one is

able to relate the term in 〈·〉 on equation (2.52) with the one in 〈·〉 on equation (2.49). Showing that,

⟨∂ui∂xi

∂ul∂xm

⟩= 0 , (2.54)

⟨∂ui∂xj

∂uj∂xm

⟩= 0 , (2.55)

applying the continuity,∂ui∂xi

= 0 , (2.56)

and the homogeneity conditions, it is left just one unknown to be determined. Being that unknown B,

14

one can write it in the following way, ⟨(∂u1

∂x1

)2⟩

=1

2B . (2.57)

Then, ⟨∂ui∂xj

∂ui∂xj

⟩= 15

⟨(∂u1

∂x1

)2⟩. (2.58)

So, equation (2.52) becomes,

ε = 15ν

⟨(∂u1

∂x1

)2⟩. (2.59)

Using the identity (2.51), equation (2.49) now reads,

λ2g =

15u′2ν

ε. (2.60)

The transverse Taylor microscale is often just referred to as Taylor microscale and is represented by λ.

The Taylor microscale based Reynolds number is commonly used in turbulence, defined as,

Reλ =u′λ

ν. (2.61)

To finalize some useful relations will be presented which can easily be derived with the material given

so far.λ2

η2=√

15Reλ , (2.62)

λ2

η2= 15

u′2

u2η

, (2.63)

Reλ =√

15u′

2

u2η

, (2.64)

L11

η∼ 15−3/4Re

3/2λ . (2.65)

2.2.5 Turbulent Dissipation

This subsection aims to give an overview of the term that specifies what turbulent dissipation is. The

theory developed was based on dimensional analysis and statistical assumptions made upon the de-

scription of a turbulent velocity field. It is clear that much more information should be derived from the

equations, however what was told gives the basic tools for analyzing turbulent motion. Turbulent dissipa-

tion arises from the equation which describes the evolution of the averaged kinetic energy, K = 〈uiui〉 /2.

Multiplying equation (2.1b) by ui gives,

∂ui∂t

ui +∂

∂xj(uiuj)ui = −1

ρ

∂p

∂xiui + ν

∂2ui∂xj∂xj

ui . (2.66)

15

Rewriting equation (2.66) and using the incompressibility condition, equation (2.1a), one has,

∂

∂t

(uiui2

)+

∂

∂xj

(uiuj2

ui

)= −1

ρ

∂

∂xi(pui) + 2ν

∂

∂xj(Sijui)− 2νSijSij . (2.67)

Now, averaging and applying the homogeneity principle, equation (2.67) becomes,

∂K

∂t= −2ν 〈SijSij〉 . (2.68)

Equation (2.68) describes the evolution of averaged kinetic energy in decaying turbulence. The term on

the right hand side is always negative and its symmetric is called the turbulent dissipation ε. So,

ε = 2ν 〈SijSij〉 . (2.69)

In homogeneous turbulence an interesting result can be derived. First note that,

SijSij =1

2

(∂ui∂xj

∂ui∂xj

+∂ui∂xj

∂uj∂xi

). (2.70)

On the other and, the vorticity ~ω ≡ (ω1, ω2, ω2) where,

ωi = eijk∂uk∂xj

, (2.71)

and eijk is the permutation symbol, one has,

ωiωi = eijk∂uk∂xj

eilm∂um∂xl

= (δjlδkm − δjmδkl)∂uk∂xj

∂um∂xl

=

(∂um∂xl

∂um∂xl

− ∂ul∂xm

∂um∂xl

),

(2.72)

where δij is the Kronecker Delta. So, applying the homogeneous principle,

2 〈SijSij〉 = 〈ωiωi〉 , (2.73)

and ε is also,

ε = ν 〈ωiωi〉 . (2.74)

If the turbulence is forced with an averaged power input P , equation (2.68) reads,

∂K

∂t= P − ε . (2.75)

For statistically stationary turbulence one has,

P = ε . (2.76)

16

2.3 Coherent Vorticity Structures in Turbulent Field

In turbulence, coherent vorticity structures have been studied for some years. They consist in regions

with concentrated vorticity and with a lifetime compared with that of large scales. They represent vortical

motions that, despite of their shape simplicity, influence complicated fluid dynamics. Their study is

motivated by the role played on transporting, mixing and diffusing mass, momentum and scalars, [1, 2].

In general, to identify vortical structures, many methods may be applied. Some are discussed in [1].

The use of isosurfaces of vorticity have been widely used in homogeneous and isotropic turbulence and

it was the method used by those who first brought this motions to attention, e.g. [15]. Nevertheless,

this method may not be sufficient as in the case of channel flows and mixing layers, [16, 17]. Along

with vorticity magnitude, an analysis through velocity gradient invariants is used to infer flow topology,

[1, 16, 17], namely to recognize between swirling tubular vortices (tube-like structures) and shear flat

vortices (sheet-like structures), which appear simultaneously in high vorticity regions with strong back-

ground/mean shear [16, 17].

This work centers specifically on the intense vorticity structures (IVS) and the next subsections intend

to provide an overview about the work developed in this area, accounting for the main issues investi-

gated. Along with a definition for IVS, some previous results will be summarized. In order to give some

support to subsection about ’worm’ formation 2.3.5, where it will be resumed what some researchers

have concluded, subsections 2.3.3 and 2.3.4 will introduce the enstrophy equation and the invariants

of velocity gradient and rate of strain tensors, respectively. To finalize, some notes about what was

obtained about the length of IVS will be given.

The work developed in this thesis recovers some results previously obtained but gives a deeper

support to them due to the high Reynolds number of the simulations. In addition, this investigation have

concluded a different result on what concerns the length of IVS, introducing a new content in this field of

research.

2.3.1 Intense Vorticity Structures (IVS)

In homogeneous and isotropic turbulence, intense vorticity structures (IVS) are particular coherent

structures which have strong vorticity. Topologically they are characterized by swirling slender tubular

vortices, reason for being known as ’worms’. These structures are identified by means of a vorticity

magnitude threshold ωivs. Jimenez defined that as being the vorticity magnitude above which the flow

points with the highest enstrophy represent 1% of the total, [18]. A study about the features of these

structures in isotropic turbulence have been given relevance by many researchers, e.g. [15, 18–21].

The IVS, known as well as coherent fine-scale eddies (CFSE) have been studied in other types

of flows such as mixing layers [17], channel flows [16] and jets [2, 22]. So far, similar statistics were

presented for these type of structures independently of the nature of the flow considered. It has been

seen that the radius of those is 〈Rivs〉 ≈ 5η in isotropic turbulence [18, 21], mixing layers [17], channel

flow [16] and in jets [2, 22]. These results have been achieved in spite of different tracking methods have

been used [2], as can be seen when comparing, e.g., [16] and [18].

17

Some usual statistics are defined to characterize the ’worms’. When a point along the axis of a worm

has been detected, it is sampled the vorticity distribution oriented perpendicular to the plane, which in

turn, is perpendicular to the vorticity vector at that point and passes at that same point as well. Then

a Gaussian distribution profile is assumed and used to fit the computed distribution. That Gaussian

distribution is written,

ωz (r) = ω0e−r2/R2

ivs , (2.77)

where ω0 is the vorticity at the axis and Rivs is the worm radius estimated with the fit. Then, through the

relation given by the Stokes’ Theorem,

∮∂Sivs

~u · d~l =

∮Sivs

~ω · ~ndSivs , (2.78)

where ∂Sivs denotes the boundary of the surface Sivs, which is a circle with radius Rivs, and ~n is the

normal to it, an equivalent azimuthal velocity is defined as,

Uivs = 0.316ω0Rivs , (2.79)

where it has the implicit definition, ∮~u · d~l = Uivs2πRivs . (2.80)

The characteristic circulation of the worm is then,

Γ = Uivs2πRivs , (2.81)

and the worm circulation-based Reynolds number is defined as,

ReΓ =Γ

ν. (2.82)

The length of worm Livs is defined by the end-to-end distance that the direct detection method computes.

Details about the tracking method are given in subsection 4.1.2.

With these quantities defined, the mean values are taken and the statistics are then 〈Rivs〉, 〈Uivs〉

and 〈Livs〉.

The results obtained by some authors are given in table 2.1 and are used for comparison. In table

2.1 it is used the so-called Burgers’ radius RB which will be defined on subsection 2.3.2 with specific

reference to equation (2.88).

2.3.2 Burgers’ Vortex Model

In order to give simple explanations and predictions about the dynamics of IVS, the Burgers’ Model is

often used. It describes a steady and axisymmetric vortex tube subject to a strain field oriented towards

its vorticity vector. Considering cylindrical coordinates and writing the imposed z−component of the

18

Table 2.1: Statistics of IVS in several kinds of flows at several Reλ. The flow types are Forced IsotropicTurbulence (F. I.), Decaying Isotropic Turbulence (D. I.), Homogeneous Shear (S.), Mixing Layer (M. I.),Circular Duct Flow (D. F.), Boundary Layer Flow (B. L.), Channel Flow (C. F.), Round Jet (R. J.) andPlane Jet (P. L.). The quantities are the Taylor microscale-based Reynolds number Reλ; mean radiusof IVS 〈Rivs〉 non-dimensionalized by the Kolmogorov’s length scale η; mean value of IVS radius non-dimensionalized by the local Burgers’ radius 〈Rivs/RB〉; mean value of the azimuthal velocity 〈Uivs〉 non-dimensionalized by the root-mean-square velocity u′; mean value of the azimuthal velocity 〈Uivs〉 non-dimensionalized by the Kolmogorov’s velocity uη; circulation-based Reynolds number ReΓ normalizedby Re1/2

λ ; mean value of the length of IVS 〈Livs〉 non-dimensionlized by the integral length scale L11;mean value of the length of IVS 〈Livs〉 non-dimensionlized by the Kolmogorov’s length scale η. TheReynolds numbers Reλ in [16] were estimated with the available data in the paper. Some authors useas definition ReΓ = Γ/ (2πν), while the definition used in this work is ReΓ = Γ/ν. In those cases thevalues were converted to the present definition of this work. In [23], the reference values λ, u′ and η aretaken from the conditional mean profiles for each instantaneous field, which are roughly constant deepinside the turbulent region for the 11 fields used in the statistics. Table and caption adapted from [2].

Flow Ref. Reλ 〈Rivs〉 /η 〈Rivs/RB〉〈Uivs〉 /u′ 〈Uivs〉 /uη ReΓ/Re1/2λ 〈Livs〉 /L11 〈Livs〉 /η

F. I.

[18] 36–168 3.8–4.2 — — — 16.5–21.1 2.60–3.16 94.80–570.24

[21] 37–168 4.8–4.9 0.92–1.05 0.86–0.99 — — 2.2–2.8 58.6–310.9

[1] 46 3.2–3.8 — — — 14.7–32.4 — —[2] 111 4.6 0.99 0.68 9.0 28.8 — —

D. I. [21] 62 4.8 0.82 1.21 — — 3.2 184.0

S. [1] — 4.9–5.2 — — — — — —

M. L. [17] 80–100 4.5 — 0.50 — — — —

D. F. [24] 719–1934 5.5–6.2 1 0.59–0.64 8.7–14.0 10.5–12.3 — —

B. L. [24] 332–1304 5.2–6.2 1 0.68–0.82 7.6–12.8 13.6–13.7 — —

C. F. [16] 200–380 4.0–5.0 — — 1.2–2.0 — — —

R. J. [22] 150 3.0–7.5 — — — — — —

P. J. [23] 120 4.6 0.97 0.76 7.15 28.3 — —

velocity field as,

uz = αz , (2.83)

where α is the rate of strain, through the Navier-Stokes equations the solution is readily obtained. The

velocity field is then given by,

ur = −1

2αr , (2.84)

uθ = − Γ

2πr

(1− e−r2/R2

B

), (2.85)

uz = αz , (2.86)

19

where,

Γ = 2π

∫ ∞0

ωzrdr , (2.87)

and RB is the Burgers’ radius given by,

RB = 2

√ν

α. (2.88)

The vorticity z−component is,

ωz =αΓ

4πνe−r2/R2

B , (2.89)

being the other components equal to 0.

The relevance of using this model is related to the fact that for an applied strain rate α the smallest

flow features one should expect to generate are of the order of Burgers’ radius RB [18]. To assess

if this model can describe IVS, their local radius are compared to the local RB considering for α the

local stretching rate. So far, it has been concluded for IVS that Rivs ∼ RB and that their azimuthally

averaged vorticity profiles are approximately Gaussian, (consistent with Burgers’ model), showing that

those structures can be modeled as equilibrium Burgers’ vortices [18, 21].

2.3.3 Enstrophy Equation

To understand how vorticity evolves and can be characterized, its transport equation may give some

insight. It is important to know what produces vorticity to interpret how it is formed. This section intends

to introduce the vorticity and enstrophy equations as well as the source term responsible for stretching.

The vorticity, defined in equation (2.71), has its temporal evolution equation, which can be derived

performing the curl of Navier-Stokes equation (2.1b), given by,

∂ωi∂t

+ uj∂ωi∂xj

= ωj∂ui∂xj︸︷︷︸

V ortex Stretching Term

+ν∂2ωi∂xj∂xj

. (2.90)

On the other hand, the enstrophy, defined as ωiωi/2, has its evolution equation, which can be derived

multiplying equation (2.90) by ωi, given by,

∂(

12ωiωi

)∂t

+ uj∂(

12ωiωi

)∂xj

= ωiωjSij︸︷︷︸Stretching

+νωi∂2ωi∂xj∂xj

. (2.91)

The stretching rate is then defined as,

σ =ωiωjSijω2

. (2.92)

The stretching term in equation (2.91) accounts for the portion of the the strain aligned with the

vorticity and represent an inviscid source term. Thus, for an inviscid fluid that is the only mechanism

available for enstrophy production. In addition, if the fluid is inviscid and two-dimensinal it conserves its

enstrophy.

20

2.3.4 Velocity Gradient and Rate of Strain Invariants

In order to analyze worm formation it is commonly studied the relations between vorticity and strain,

leading to a evaluation of enstrophy production and the predominance of the invariants of the velocity

gradient tensor. Moreover, it also allows to distinguish flow topology and its relation with enstrophy

stretching/compression. This subsection intends to give a brief introduction to this kind of study. For a

detailed description it is recommended to read [23], reference which this subsection will be based on,

and subsequent references.

The velocity gradient is,

Aij =∂ui∂xj

, (2.93)

and it can be split into a symmetric and a skew-symmetric components,

Aij = Sij + Ωij , (2.94)

where Sij is the already defined rate-of-strain tensor and Ωij is the rate-of-rotation tensor,

Ωij =1

2

(∂ui∂xj− ∂uj∂xi

). (2.95)

The invariants of Aij are the coefficients of its characteristic equation,

ζ3i + PAζ

2i +Qζi +R = 0 , (2.96)

where ζi are the eigenvalues of Aij and,

PA = 0 , (2.97)

Q =1

4(ωiωi − 2SijSij) , (2.98)

R = −1

3

(SijSjkSki +

3

4ωiωjSij

)= −1

3

(SijSjkSki +

3

4ω2σ

).

(2.99)

The coefficient PA is zero due to incompressibility condition and σ is the already defined stretching rate,

see equation (2.92) for the latter.

On the other hand, the invariants of Sij can be readily found by setting Ωij to zero, (which is equiv-

alent to set ωi to zero). The same can be done to find the eigenvalues of Sij through equation (2.96).

Denoting the second and third invariants of Sij by QS and RS , respectively, one has,

QS = −1

2SijSij , (2.100)

RS = −1

3SijSjkSki . (2.101)

The term which defines RS is related to the source term present in the strain product SijSij/2 transport

21

equation,

∂(

12SijSij

)∂t

+ uj∂(

12SijSij

)∂xj

= −SijSjkSki︸︷︷︸=3RS

−1

4ωiωjSij − Sij

∂2p

∂xi∂xj+ νSij

∂2Sij∂xj∂xj

. (2.102)

In order to illustrate the meaning of this quantities some notes should be given. The invariants Q and

R allows one to infer about the relation between flow topology and enstrophy production/dissipation.

If Q 0, ω2 0, (enstrophy dominates), and R ∼ −ωiωjSij/4. Then, if R < 0 implies a predomi-

nance of vortex stretching and if R > 0 vortex compression dominates.

On the other hand, if Q 0, SijSij 0, (strain product dominates), and R ∼ −SijSjkSki/3 = RS .

Then, R < 0 is associated with a tube-like structure and R > 0 with a sheet-like structure.

In a Burgers’ vortex, its centre is characterized by Q > 0 while the regions surrounding it have Q < 0,

being the latter, therefore, strain product-dominated.

Consider the figure 2.3 which shows the Q−R map displaying physical/topological features of the

flow related to each region of the plot. The curved line is the isoline of the discriminant DA = 0. The

discriminant is,

DA =27

4R2 +Q3 . (2.103)

The sketch resumes what was said above.

Figure 2.3: Sketch of Q−R map showing the physical/topological features related to each zone, from[23].

In homogeneous and isotropic turbulence, regions of intense enstrophy are more likely to be found

in tube-like structures, whereas intense viscous dissipation is concentrated in sheet-like or ribbon-like

structures, [23]. In addition, the joint PDF of Q and R ploted in the Q−R map of figure 2.3, shows

a strong anti-correlation in the region R > 0 and Q < 0. The region R < 0 and Q > 0, related to a

predominance of vortex stretching, shows also an anti-correlation although not as strong, [23]. The

region where Q ∼ 0 with both terms ωiωi and SijSij representing high values and of the same order, or,

in other words, ωiωi = ω2 ∼ SijSij , the topology related with the dissipation is defined by vortex sheet

structures, [23].

22

Analyzing the orientation of the vorticity related to the eigenvectors of Sij is a useful tool to under-

stand the dynamics of vortex structures. In the next subsection, this will be referenced as a tool to

assess IVS formation.

2.3.5 IVS Formation

The process of worm formation was assessed by Vincent and Meneguzzi in [19] and [20]. They

had performed in [19] a statistical study of the structure of homogeneous turbulence before their deeper

study about the dynamics of vorticity tubes in [20].

In these studies the eigenframe of Sij is used for characterization purposes and later to understand

worm formation. So, let ζS3 > ζS2 > ζS1 be the eigenvalues of Sij and ~eS3 , ~eS2 and ~eS1 the corresponding

eigenvectors, respectively.

In the study of forced homogeneous and isotropic turbulence in [19], they showed that, at steady

state conditions, in strong vorticity regions, ~eS3 and ~eS1 are perpendicular to the vorticity vector, being ~eS2

generally aligned with that. Although this result was known at that time, their work allowed them to think

that worm formation is a process which begins by an instability of the Kelvin-Helmholtz type, [19]. The

work developed by Jimenez in [18] and [21] found that high stretching is not related with high vorticity

regions, showing, at least, that the more intense vorticity structures do not self-stretch. Jimenez, still in

[18], also conclude the same as Vincent in [19] about the vorticity alignment with ~eS2 .

Later, Vincent and Meneguzzi in [20], by revisiting their previous work, they argued that vorticity

tubes are formed by shear instabilities, which occur in zones where the fluid is submitted to strong

stretching and strain, [20]. By visual inspection, they have seen some vortices merging but, more often,

the process seemed to go through sheets rolling-up to form tubes. In Jimenez’s work [18], it was seen

shorter segments interacting strongly purposing that the formation of IVS might be linked to an accretion

phenomenon.

To clarify the process, Vincent and Meneguzzi have performed decay turbulence simulations. The

first structures seen appeared in pancake-like zones, which flatten with time leading to the appearance

of sheets that tend to roll-up forming vorticity tubes. This along with the increasing vorticity during

the process, made them to concluded that shear instability is accompanied by vortex stretching, [20].

Using again the eigenframe of Sij , now on the decay results, they have seen that, in the beginning, ~eS1

is perpendicular to the vorticity pancakes, while the vorticity is parallel to them and sometimes to ~eS3 .

Later, ~eS1 remains perpendicular to the pancakes, but the vorticity became aligned with ~eS2 . Then the

pancake became a vorticity sheet which began to bend while the vorticity was still aligned with ~eS2 . They

have concluded that an initial straining phase lead to a formation of a vorticity sheet with a later rolling-

up phase. In addition, they stated that the alignment of the vorticity with ~eS2 happens long before the

vorticity tube production, [20]. Furthermore, they argued that if the mechanism behind worm formation

is shear instability of strained vorticity sheets, when the instability occurs the vorticity vector should be

aligned with ~eS2 , fact that they observed, [20].

In general Vincent and Meneguzzi have concluded that tubes are more stable than sheets, reason

23

why the former type of vorticity structures are dominant in steady state turbulent flows. They also said

that the alignment of the vorticity vector with ~eS2 is a result of vorticity sheet production by strong strain

rather than a consequence of tube formation, [20].

2.3.6 Length of IVS

The length of IVS has not been neither studied over a large range ofReλ nor systematically measured

over the available range. The majority of the references to it are mostly based in visual inspection of

those structures embedded in the flow.

Vincent have seen those structures to have O (Livs) = O (L11), [19].

Some quantitative measurements were given by Jimenez in [18], based on a direct detection method,

and he said that Livs scales with L11, [18]. In a later work, Jimenez has also given measurements by a

direct detection method, see table 2.1, saying that Livs scales with L11 and that O (Livs) = O (L11), [21].

Based on an integration of normalized autocorrelation functions of properties of the filaments, Jimenez

in [21] gave also other measurements.

Mouri, in duct flow and boundary layer experiments, has registered O (Livs) = O (L11), [24].

In this work a systematic study about the length of IVS is done over a wider range of Reλ. The

method of measurement is based on direct detection of the structures. It is worthwhile to distinguish

between a measurement being of the same order than a given variable and being scalable with that. In

fact, Livs is approximately of the same order of L11 but it does not scale with that. It was found that Livs

scales with the Kolmogorov length scale η.

2.4 Complex Fluids and Turbulence with Polymer Additives

This section intends to give a brief introduction to complex fluids. Although it goes beyond the scope

of this thesis to give a deep development on the subject, it is important to provide, at least, some context.

The idea is to understand from where does viscoelasticity come from and then explore what has been

done concerning experimental and numeric research on turbulence with polymer additives (viscoelastic

turbulence). At the end a theoretical result will be derived which will give the basis towards the objectives

of this work.

So far, in the previous section, the classical Newtonian constitutive model for the stresses was con-

sidered, which is a linear relationship between those and velocity gradients. The question is, how far

can we go modeling fluids with this approach? The answer relies on how the dynamics of individual con-

stituents of the fluid are altered, given a velocity gradient applied on it. More precisely, it is equivalent to

understand how comparable are the length and time scales of the flow relative to those characterizing

the dynamics of individual constituents of the fluid.

In Newtonian fluids, like water or liquid argon, typical intermolecular distances or velocity distributions

for a given realistic flow are the same as at rest and, hence, the energy dissipation mechanism, which

is represented by viscosity, is not affected by the flow, [25]. On the other hand, it becomes different

24

for solutions of colloidal particles, long flexible polymers, worm-like micelles and similar complex fluids.

These particles are much larger than individual molecules of typical Newtonian fluids, and time scales of

stress relaxation in complex fluids are longer, (than their Newtonian counterparts), and are comparable

to those of real-life flows, [25].

Considering a complex fluid, if one writes the maximum stress relaxation time of the solution τt, it

can be defined a non-dimensional group as,

We ≡ τtγ , (2.104)

the so-called Weissenberg number, where γ ≡√SijSij/2 is an invariant measure of the rate-of-strain

in the fluid. Based on what was said earlier, if We 1 the fluid is expected to behave like a Newtonian

one at the same Reynolds number. If We ≥ 1 a complex fluid has a non-Newtonian behavior and

obey complicated constitutive models, often involving nonlinear dependence of the local stresses on the

velocity gradient and the deformation history of the fluid, [25].

The next subsection 2.4.1 will give a brief insight about the simplest extension of the Newtonian

model, valid when the flow only influences the instantaneous viscosity of the fluid, the so-called gener-

alized Newtonian model. In addition, it will be introduced the history-dependent feature of viscoelastic

fluids.

2.4.1 Purely Viscous and Viscoelastic Fluids

A complex fluid which can still be modeled as having a viscous response to forces is purely viscous.

A generalized Newtonian fluid is a model which assumes that the applied flow only changes the dissi-

pation rate in the fluid (i.e., its viscosity), but does not change the tensorial structure of the Newtonian

constitutive model, [25]. The deviatoric stress tensor can have the form,

Tij = µG (SijSij)Sij , (2.105)

where µG is the variable dynamic viscosity of the fluid. The simplest possibility of a monotonic function

for µG gives name to the shear-thickening or shear-thinning fluids, whenever the viscosity increases

or decreases as the strain-product increases, respectively. One of the most popular models for shear-

thinning fluids is the Carreau-Yasuda viscosity model, [25]. This model approaches to a Newtonian

behavior for We 1 and tends to a (shifted) power-law as We increases. Details on this model can be

seen in subsection 3.2.2. For further reading see e.g. [25].

Many complex fluids behave quite differently from purely viscous kind of responses. In the case of

viscoelastic fluids, the presence of memory changes the picture. Stresses in such fluids depend on

the flow history and are anisotropic. Generally, a viscoelastic fluid generates stresses that are absent

in a Newtonian fluid subjected to the same deformation history, [25]. Viscoelastic fluids, as the name

suggests, have both viscous and elastic responses to forces.

In [25] it is developed a mathematical framework that allows one to incorporate both memory and

25

stress anisotropy into differential constitutive equations for viscoelastic fluids. One of the models refered

is the FENE-P (Finitely Extensible Nonlinear Elastic with Peterlin closure), the model used to simulate

polymer additives in this work. Although never dissociated from differential constitutive equations derived

in [25], a kinetic theory can also be used to derive such constitutive models. Based on that, this work

presents a derivation for the FENE-P model through a kinetic theory arguments in subsection 3.2.3.

2.4.2 Drag Reduction in Dilute Polymer Solutions

The discovery of drag reduction phenomenon in solutions with polymer additives is attributed to

Toms [3] in 1948 and, for that reason, it is sometimes known as Toms’ Phenomenon. Since then, many

experiments have been performed and theories have been given in order to confirm and study such

effect.

The many experiments available allowed Virk to start to compile and analyze them more system-

atically. His work in [26], centered in data of pipe flows with polymer additives and smooth surfaces,

resulted in a characterization based on three flow regimes.

• Laminar Regime. In this regime no drag reduction is observed and the flow obeys Poiseuille’s

Law.

• Polymeric Regime. This regime is turbulent and drag reduction effects begin to be observed. Virk

stated that the onset of drag reduction occurs at a universal characteristic shear stress. This shear

stress was found to be, along with a dimensional analysis, related to the random-coiling effective

diameter of the polymer. In addition, the amount of drag reduction is a function of the polymer

properties.

• Maximum Drag Reduction (MDR). A regime characterized by a universal asymptote, independent

of the system and polymer properties, limiting the amount of drag reduction.

In figure 2.4 it is shown where it would lie the laminar, polymeric and MDR regimes. Polymeric regime

would be any path between the Prandtl-Karman Law (turbulent newtonian regime) and the MDR asymp-

tote.

Lumley [28] observed that drag reduction occur in thermodynamically dilute solutions of long, flexible

and expanded high-molecular-weight linear polymers such as the polyethylene oxide (PEO).

In 1973, Lumley gave some theoretical explanations about drag reduction phenomenon. In [29] he

stated that in regions where polymers are elongated viscosity is enhanced. To clarify this picture he

said that, at sufficiently high wall shear stresses, the fluctuating strain rate expands polymer molecules,

thus increasing viscosity. He theorized that strain would elongate polymers while vorticity would diminish

that. Based on that, he explained his conclusion that, in wall turbulent flows, elongated polymers are

found in the wall layer, (log-law region), and not in the viscous sub-layer, where vorticity and strain rate

are not correlated and there is only simple shear in this latter layer. With this observation, he stated that

the increased viscosity in the wall layer will lead to dissipation of turbulence fluctuations. As a result,

small eddies will be damped in the wall layer and the lower Reynolds stresses at the buffer layer will

26

Figure 2.4: Modified skin friciton f−1/2 as a function of modified Reynolds number log(Ref1/2

). This are

the so-called Prandtl-Karman coordinates. Prandtl-Karman Law is an analytical-emprirical law, obtainedfor turbulent newtonian pipe flows, as a result of an integration of the log-law region of the turbulentboundary layer. The laminar region, or Poiseuille’s Law, results from the solution of laminar pipe flow.MDR asymptote is empirical. From [27].

thicken the viscous sub-layer. The large eddies on the viscous sub-layer will expand generating higher

streamwise fluctuating velocities in that region. At MDR conditions large eddies will be dominant. See

figure 2.5 for boundary layer stratification.

Figure 2.5: Stratification of a newtonian turbulent boundary layer. Velocity profile in wall units. From [30].

Later, in 1975, Virk expanded his work and analyzed more deeply what effects polymers induce in

27

the boundary layer of solutions with polymer additives in pipe flows. In [31] he confirmed his previous

results. He concluded that MDR occurs when polymers affect all turbulent scales causing the buffer layer

to thicken to the whole boundary layer extent. His boundary layer analysis led to a further stratification

when compared with the newtonian one. He identified three main zones, the standard viscous sub-layer,

the elastic layer and the newtonian log layer. The elastic and the newtonian log layers are a subdivision

of the ’old’ wall layer. See figure 2.6.

• Standard Viscous Sub-Layer. The same as the ’old’ newtonian viscous sub-layer.

• Elastic Layer. This is the first subdivision of the ’old’ wall layer, right after the viscous sub-layer and

the buffer layer. In this region the effects of the polymer begin to be felt and so it is characteristic of

drag reduction. The velocity profile follows a slope greater than that of the newtonian log-law. That

slope is the ultimate profile. Its extent increases as drag reduction increases up to the maximum

drag reduction. When MDR is achieved, the ultimate profile spreads through the whole cross

section.

• Newtonian Log Layer. This is the second subdivision of the ’old’ wall layer, right after the elastic

layer. It is also known as ”Newtonian Plug”. Here the velocity profile recovers the universal slope

of the newtonian law of the wall. The difference is that it is shifted upwards.

Figure 2.6: Stratification of a viscoelastic turbulent boundary layer. Velocity profile in wall units. From[31].

Figure 2.7 shows some paths of viscoelastic fluids in a Prandtl-Karman chart. When MDR is achieved

all paths collapse in the same slope.

Some years later, an alternative theory relative to Lumley’s was purposed by de Gennes [32]. His

idea is based on the elastic theory arguments. He postulated that the elastic energy stored by partially

stretched polymers is an important variable for drag reduction and, thus, for turbulence suppression. He

also stated that the increase in the effective viscosity is fairly low. Briefly, he said that drag reduction

effects begin to occur when the elastic energy stored by polymers matches the turbulent kinetic energy

of the buffer layer. As a consequence, small scales are damped and buffer layer thickness increases

inducing reduced drag. Contrary to Lumley, de Gennes introduced concentration of polymer additives

28

Figure 2.7: For coordinates specification and plot information clarification see figure 2.4. Experimentalresults from [31] show some paths of viscoelastic fluids. Note that when MDR is achieved all pathscollapse in the same slope. From [31].

as variable for the onset of drag reduction, where the total elastic energy stored is, in fact, a function

of concentration. This theory will be mathematically formulated in subsection 2.4.4 once a prediction of

that will be used in this work.

Some recent developments lead to a new theory about polymer effects on turbulence. The ”energy

flux theory” says that turbulence energy flux through its cascade process is gradually reduced due to

the energy transfer to the elastic motion of polymers, which plays a dominant role in small scales. An

experimental work using this theory is presented in [33].

2.4.3 Numerical Developments

Lately, with technological advancements, Direct Numerical Simulations (DNS) have brought impor-

tant support to laboratory experiments. In what concerns turbulence with polymers, it has been no

exception. Some issues about numerical features have challenged researchers since the beginning,

but solving them was mandatory. Indeed, the main advantages of computational experiments are, un-

doubtedly, the description of polymers micro-structure and velocity field as well as the possibility of full

experiment control.

There are some different approaches to numerically simulate polymers behavior, as can be seen in

[34]. One of them, which is widely used, consists in a polymer orientation description through a 2nd-

order tensor, the so-called conformation tensor Cij . The Navier-Stokes equations are solved along with

the transport equation for that tensor. Momentum and Cij transport equations are coupled via the stress

components resulting from the interaction among polymers and velocity field. The constitutive model

is based on two massless beads connected by a spring. The description of the spring force and its

29

extensibility give closure to the constitutive model. Two typical constitutive equations are the Oldroyd-B

and FENE-P (Finitely Extensible Nonlinear Elastic with Peterlin Closure) as descibed in [35]. FENE-P

formulation has been widely used to study turbulence with polymer additives in DNS in works such as

[36–38]. As FENE-P will be used in this work, a derivation of it is provided in subsection 3.2.3.

The Cij tensor is symmetric and positive definite (SPD) and it should remain so when transported.

Although the equations themselves conserve those properties, they may be lost by cumulative numerical

errors. In fact, first attempts to solve those equations had problems with numerical instabilities.

Sureshkumar and Beris [39] tried to solve those instabilities by means of an introduction of a stress

diffusion term into the transport equation of Cij . As a result, in 1997 they presented the first DNS of

channel flow [36] showing the drag reduction phenomenon. Although it was a great development, the

Reynolds numbers were still lower than those of laboratory.

Later, Vaithianathan [40] exploited the SPD property of Cij in order to derive independent equations

for the eigenvectors and eigenvalues of Cij . Apart from the SPD property which should be guaranteed,

the sum of the eigenvalues should remain less or equal the square of the non-dimensional polymer

maximum extensibility. Although their formulation preserved the extensibility condition, the compact

finite-differences they used, based on [41], did not guarantee the eigenvalues to remain positive. They

tried to overcome this problem by setting negative eigenvalues to zero, ensuring stability. Nevertheless,

it resulted in a lost of the overall conservation of the conformation tensor. In addition, spatial averages of

Cij contained spurious contributions from the convective term. Thus, the method used by [40] ensured

stability but did not guarantee conservation.

The problems researchers were having were related to the hyperbolic nature of the transport equation

for Cij in the Oldroyd-B, FENE-P and Giesekus models, which admits shocks (discontinuities) in the

polymer stress tensor, [42]. This discontinuities, which mathematically are step-like variations, cannot

be fully resolved by a discrete finite grid. The challenge of a solver is to predict the jump magnitude,

satisfying the overall conservation balance to guarantee a meaningful elastic wave propagation.

The Gibbs phenomenon, typical in a spectral representation through a discontinuity, may be attenu-

ated by introducing an artificial diffusivity. However, it can also reduce the jump magnitude. Alternatively,

specific finite-differences schemes have been designed to compute the magnitude of the jump while

avoiding an excessive spread of the discontinuity.

A finite-difference scheme method was given by Kurganov and Tadmor (KT) in [43]. In [44] the KT

method was used. Their second-order scheme made positive scalars to maintain their positiveness. The

KT method was generalized to guarantee that SPD properties are maintained. The method dissipates

less elastic energy than artificial diffusion, allowing stronger polymer-flow interactions.

This latter approach has had largely acceptable results being used in DNS of Shear Flow, Decaying

and Forced Homogeneous and Isotropic Turbulence, in [45–47]. As so, it will be used in this work.

30

2.4.4 de Gennes’ Theory for Drag Reduction Prediction

It is important to note that there is no ’universal’ theory that explains drag reduction mechanisms. In

subsection 2.4.2 they were cited Lumley’s and de Gennes’ theories for drag reduction prediction. The

former is based on the premise that when polymers are elongated viscosity is enhanced, while the latter

is based on elastic arguments. In contrast to Lumley’s, de Gennes’ theory says that, in turbulent flows

with randomly fluctuating strain rates, polymers are only moderately stretched and thus they do not pro-

duce measurable change in viscosity. In addition, de Gennes relates drag reduction to the interference

of stored elastic energy in the Newtonian energy cascade. Furthermore, as the stored elastic energy is

a function of polymer concentration, this new variable takes place in de Gennes’ theory.

The reason why it is being given relevance to de Gennes’ theory in this subsection is the fact that it

has produced plausible results, at least for the onset of drag reduction in pipe flows. In this sense, this

work aims to test its agreement with dissipation reduction in Homogeneous and Isotropic Turbulence.

However, before proceeding mathematically, some notes should be made.

Sreenivasan in [48] used this theory to test its prediction, for the onset of drag reduction, when

adapted to pipe flows. It should be kept in mind that de Gennes based his premises away from walls.

Sreenivasan clearly stated that experiments do not provide unequivocal support that drag reduction

begins before polymer reaches the wall, [48]. Nevertheless, it does not negate the essentials of elastic

theory, [48]. He argued that the usage of de Gennes’ theory might be justified at least for two reasons.

First, measurements have shown that drag reduction depends systematically on polymer concentration,

e.g. [49], and secondly, experiments have suggested that partial stretching is perhaps the rule, other

premise of elastic theory. So, the fact that de Gennes’ theory is based on homogeneous turbulence and

those reasons, given by [48], are taken to support this work.

Consider a high Reynolds number homogeneous flow, away from walls, and a length scale `∗ in the

inertial subrange. Now let the corresponding characteristic time scale be equal to the polymer relaxation

time τp, τ∗ = τp. Through the relation (2.11) one gets,

τp =

(`∗2

ε

)1/3

. (2.106)

The elastic theory postulates that polymers can be stretched by scales ` < `∗. Furthermore, it states that

there is a range of scales `′ < ` < `∗ where polymers are stretched little. Even more, although scales

` < `∗ affect polymers, scales `′ < ` remain unaffected by polymers. The scale `′ is such that the elastic

energy Kel stored by polymers (per unit volume) match the turbulent kinetic energy (per unit volume) at

that scale,

ρu2`′ ∼ Kel (`

′) . (2.107)

Below `′ turbulence scales are strongly affected by the elastic forces. On the other hand, Kel (`) depends

on the stretching at that scale ς (`). The theory states that,

ς (`) =

(`∗

`

)q, (2.108)

31

where q depends on the dimensionality of the stretching, being 1 in two dimensions and 2 in three

dimensions. For the elastic energy Kel one has,

Kel (`) = nkBTς5/2 = nkBT

(`∗

`

)5q/2

, (2.109)

where n is the number density of polymer molecules, kB is the Boltzmann constant and T is the absolute

temperature. So, according to equations (2.10), (2.106), (2.107) and (2.109) one has,

ρ(ε`′)2/3 ∼ nkBT

((τ3p ε)1/2`′

)5q/2

. (2.110)

Using again equation (2.11) and defining τ ′ as the characteristic time scale correspondent to the scale

`′, equation (2.110) can be rewritten as,

ρ(ε`′)2/3 ∼ nkBT

(τpτ ′

)15q/4

. (2.111)

On the onset conditions one should expect the polymers to begin to influence the turbulence scales.

This is the same as saying that `′ = η. When this relation holds, τ ′ = τη. Introducing the Weissenberg

number,

We ≡ τpτη, (2.112)

on the onset relation (2.111), the latter becomes,

ρ(εη)2/3 ∼ nkBTWe15q/4 . (2.113)

Introducing the polymer concentration parameter β and referring to the equation (3.36), (see subsection

3.2.3), equation (2.113) becomes,

ρ(εη)2/3 ∼ (1− β)

β

µ[s]

τpWe15q/4 , (2.114)

where µ[s] is the solvent dynamic viscosity. Noting that the solvent kinematic viscosity ν[s] = µ[s]/ρ,

equation (2.114) can be rearranged to give,

(1− β)

β∼We1−15q/4 . (2.115)

Relation (2.115) represents a scalability law that should be observed at onset conditions of drag reduc-

tion and will be used in this work.

32

Chapter 3

Governing Equations and Numerical

Methods

In this chapter the main equations needed towards the accomplishment of the objectives for this work

will be described.

3.1 Navier-Stokes Equations

Navier-Stokes equations, for a velocity field with an average of zero, for incompressible fluid at con-

stant properties in its generalized form can be read as,

∂ui∂xi

= 0 , (3.1a)

∂ui∂t

+∂

∂xj(uiuj) = −1

ρ

∂p

∂xi+

1

ρ

∂Tij∂xj

. (3.1b)

The stress tensor Tij is then prescribed according with the three constitutive equations needed in the

present work.

3.2 Constitutive Models

In this section the constitutive models relevant to this work will be described. These equations solve

the closure problem yet present in the generalized Navier-Stokes equations. Along with this mathemati-

cal feature, such models are intrinsic descriptors in what concerns the physical behavior of a given fluid.

The key here is to relate the stress tensor with the deformation one and with other sources of stress. The

three closures treated are descriptors of the so-called Newtonian, Carreau-Yasuda or polymeric fluids.

33

3.2.1 Newtonian

A Newtonian fluid is represented by a proportional relation between Tij and Sij , the latter represent-

ing the rate of strain tensor,

Tij = 2µSij . (3.2)

Here µ is the dynamic viscosity of the fluid and Sij is given by,

Sij =1

2

(∂ui∂xj

+∂uj∂xi

). (3.3)

3.2.2 Carreau-Yasuda

A Carreau-Yasuda type of fluid exhibits a relation similar to a Newtonian fluid but µ is no longer

constant, it depends on the strain rate tensor contraction SijSij . The function is a power-law-like one

but it has limit when SijSij → 0 and 0 ≤ limx→∞ SijSij <∞. So,

Tij = 2µc(SrmSrm)Sij , (3.4a)

µc (SijSij) = µ∞ + (µ0 − µ∞)(

1 +(λc√

2SijSij

)a)nc−1a

. (3.4b)

Here µc is again the kinematic viscosity, µ0 is that when SijSij = 0 while µ∞ when SijSij →∞. On the

other hand, λc, a and nc are parameters defining the curve.

3.2.3 FENE-P

Here, importance will be given to a light derivation of FENE-P constitutive model to simulate polymer

additives within a solvent. To a more detailed one, reference like [34] can be seen.

To start, consider a schematic of a dumbbell with two massless beads and one spring embedded in

velocity field. Performing a force balance of the system at bead 1 it can be written,

(~F1 − ~D1 − ~B1

)dt = 0 , (3.5)

where ~F1 is the spring force, ~D1 is the drag force and ~B1 is the Brownian force due thermal fluctuations.

The drag force can be expanded in the form,

~D1 = ζ(~u1 − ~V1

), (3.6)

where ζ is the drag coefficient, ~u1 the flow velocity and ~V1 the bead one. The Brownian forces can be

modeled via a Wiener process increment d ~W1 so ~B1dt can be expressed as

~B1dt =√

2ζkBT d ~W1 , (3.7)

where kB is the Boltzmann constant and T is the absolute temperature.

34

Now, in order to eliminate the dependence on ~V1, one should subtract the balance equation at bead

2 from the first one. Thus it follows,

(~F1 − ~F2 − ζ

(~u1 − ~u2 −

(~V1 − ~V2

)))dt−

√2ζkBT

(d ~W1 − d ~W2

)= 0 . (3.8)

It is assumed that the velocity field is locally linear so that the next relation holds,

(~u1 − ~u2) = Rj∂ui∂xj

, (3.9)

where ~R = (R1, R2, R3) is the spring end-to-end displacement. On the other hand, the relation

(~V1 − ~V2

)dt = d~R , (3.10)

also stands once it represents the relative velocity between beads. After noting that ~F2 = −~F1 = −~F ,

the stochastic dynamic equation is now of the form,

2~Fdt− ζ(Rj

∂ui∂xj

dt− d~R

)−√

2ζkBT(

d ~W1 − d ~W2

)= 0 . (3.11)

Remembering that dW is equal in distribution to a normal with zero mean and variance equal to dt,

N (0,dt), and that both Wiener process increments are independent, the following result takes place,

d ~W1 − d ~W2 =√

2 d ~W , (3.12)

so equation (3.11) becomes, now fully in index notation,

dRi =

(Rj

∂ui∂xj− 2

Fiζ

)dt+

√4kBT

ζdWi . (3.13)

This stochastic differential equation can be reduced to a deterministic one through its second order

moment equation, d 〈RiRj〉. Here, 〈·〉 means ensemble averaging the random variables. According with

the next operation,

〈dRiRj +RidRj〉 , (3.14)

one finally gets,

d 〈RiRj〉 =

(∂uj∂xk〈RiRk〉+

∂ui∂xk〈RjRk〉 − 2

〈FiRj〉+ 〈FjRi〉ζ

+4kBT

ζδij

)dt . (3.15)

Now a model for the spring force is required. As suggested, a finite extensible relation is given,

Fi =HRi

1− RkRkL2max

, (3.16)

being H the elastic modulus and Lmax the dimensional maximum extensibility of the spring, and then,

35

the force term gives,

2〈FiRj〉+ 〈FjRi〉

ζ= 4

⟨HRiRj

ζ(

1− RkRkL2max

)⟩ . (3.17)

In order to simplify this expression, Peterlin made the following approximation,⟨HRiRj

ζ(

1− RkRkL2max

)⟩ ≈ H 〈RiRj〉

ζ(

1− 〈RkRk〉L2max

) . (3.18)

At this stage one can write the dimensional dynamic equation of FENE-P model,

d 〈RiRj〉dt

=∂uj∂xk〈RiRk〉+

∂ui∂xk〈RjRk〉 − 4

H 〈RiRj〉

ζ(

1− 〈RkRk〉L2max

) +4kBT

ζδij . (3.19)

Equation (3.19) is often presented in a non-dimensional way. To find the parameter needed to nor-

malize 〈RiRj〉, it is defined the equilibrium condition as the solution of the next equation,

0 = 4H〈RiRj〉eq

ζ(

1− 〈RkRk〉eqL2max

) +4kBT

ζδij , (3.20)

with respect to 〈RiRj〉eq, which is,

〈RiRj〉eq =1

3〈RkRk〉eqδij , (3.21)

with,

〈RkRk〉eq =3kBTH

1 + 3kBTHL2

max

. (3.22)

The conformation tensor Cij is then defined as,

Cij ≡〈RiRj〉

13 〈RkRk〉eq

, (3.23)

in order to, by convention,

(Cij)eq = δij . (3.24)

Finally, the non-dimensional version of equation (3.19) is,

dCijdt

=∂uj∂xk

Cik +∂ui∂xk

Cjk −1

τp(f (Ckk)Cij − δij) , (3.25)

where f(Ckk) is the Peterlin function given by,

f (Ckk) ≡ L2 − 3

L2 − Ckk(3.26)

and L2 by,

L2 ≡ L2max

13 〈RkRk〉eq

. (3.27)

36

The variable τp is the Zimm relaxation time of the polymer, which is defined as,

τp ≡ζ

4H

(L2 − 3

)L2

=ζ

4H + 12kBTL2max

. (3.28)

Note that equation (3.22) can be rewritten to obtain the following result,

kBTH

13 〈RkRk〉eq

=L2

L2 − 3, (3.29)

which is used to simplify, in equation (3.25), the coefficient of δij to end up with 1/τp.

Equation (3.25) should now be transformed from a Lagrangian to a Eulerian frame of reference,

substituting d/dt by the material derivative D/Dt. The conformation tensor transport equation is then

given by,∂Cij∂t

+ uk∂Cij∂xk

=∂uj∂xk

Cik +∂ui∂xk

Cjk −1

τp(f (Ckk)Cij − δij) . (3.30)

To couple equation (3.30) with the system (3.1) it is mandatory to give a closure for Tij . So,

Tij = T[s]ij + T

[p]ij , (3.31)

with the superscript [s] standing for solvent and [p] for polymer stresses. Solvent stresses are modeled

as Newtonian,

T[s]ij = 2µ[s]Sij , (3.32)

and polymer stresses are given by,

T[p]ij = nkBT (f (Ckk)Cij − δij) , (3.33)

where n is the number density of polymer molecules. Now defining a new parameter,

β =µ[s]

µ[s] + µ[p], (3.34)

where µ[p] is the zero-shear-rate polymeric equivalent dynamic viscosity given by,

µ[p] = nkBTτp , (3.35)

and then nkBT can be rewritten as,

nkBT =(1− β)

β

µ[s]

τp. (3.36)

So equation (3.33) becomes,

T[p]ij =

(1− β)

β

µ[s]


37

3.3 Pseudo-Spectral Method

In this section the Pseudo-Spectral Method will be described and how does it arise in this context.

First, being spectral means that it leads with variables described in wavenumber space, the Fourier

space.

3.3.1 Velocity Field and Spatial Discretization

Turbulence is often characterized and interpreted in wavenumber space. The reason is that a si-

nusoidal decomposition of correlated quantities is convenient to study the concept of scales and this

phenomenon is essentially multi-scale. On the other hand, Homogeneous and Isotropic Turbulence is

studied to describe dynamic processes allover a wide range of scales and Reynolds numbers, mainly

to search for universal laws and to discover general phenomenology features about turbulent flows. At

the same time, to simulate these states, a cubic box with periodic boundary conditions is widely used for

congruent results, once it is capable to reproduce reliable information when trying to develop turbulent

motions far from walls. Merging together these needs, a logic representation of the velocity field, as a

solution assumed for Navier-Stokes equations, is,

ui (x1, x2, x3, t) =

N/2∑k3=−N/2+1

N/2∑k2=−N/2+1

N/2∑k1=−N/2+1

ui (k1, k2, k3, t) e2πikjxjL , (3.38)

where N is the number of grid points in each direction and L is the length of box side, the latter being

set always to 2π meters just for convenience. In the exponent i denotes the imaginary unity. Physical

space coordinates are ~x = (x1, x2, x3) and for Fourier space one has ~k = (k1, k2, k3). Note that physical

quantities are continuous and spectral ones are discrete, with ki = −N/2 + 1, . . . , N/2 and i = 1, 2, 3.

Both k and i are integers. To simplify the notation, it will be omitted in which variables ui and ui depend,

being related to the physical and Fourier spaces, respectively. The triple sum will be represented solely

by∑~k. Thus, equation (3.38) becomes,

ui =∑~k

uieikjxj . (3.39)

This is the so-called Fourier Series. In general, this representation may lead to an infinite sum of terms.

However, for a finite numerical representation, the Fourier Series is always truncated.

Due to this statements, spatial discretization is then straightforward, spectral space is the set A

defined for a given N ∈ N and even as,

A =

~k :

(~k)i

= ki , ki ∈ Z ,−N2

+ 1 ≤ ki ≤N

2, i = 1, 2, 3

, (3.40)

and continuous physical space A′ as,

A′ = ~x : (~x)i = xi , xi ∈ R , i = 1, 2, 3 . (3.41)

38

Actually, physical space is not always considered continuous. In general, statistics defined in physi-

cal space use the continuous description of the velocity, leading to exact discrete operations in Fourier

space. Nevertheless, when performing numeric computations of Navier-Stoker equations, the Pseudo-

Spectral Method interchanges between physical and spectral space. The goal is to lead with local

operations as much as possible to optimize parallel computations. Numerically, derivatives are not lo-

cal operations once their discretization deal with information from neighbors. On the other hand, their

Fourier space counterparts are products, being therefore locally accessible. For local products in physi-

cal space one has convolutions in Fourier space. So the rule of thumb here is whenever the proceeding

operation becomes non-local, the space transformation is performed. It is important to note that space

transformations are non-local as well however, in a proper optimized code, these transformations save

computational time when compared to other non-local operations and, in addition, allow a maximization

of the order of spatial accuracy. As it can be readily seen, physical representation of the velocity field

is needed during computations and it should be discrete. Although it is continuous, velocity field can be

fully known with a discrete set of points if its Fourier series is truncated. So, local operations in physical

space preserve local information. The discrete set of physical points A numerically needed is, for a

given N ∈ N and even,

A =

~x : (~x)i = xi , xi ∈ R , xi =

2π

N(n− 1) , i = 1, 2, 3 , n ≤ N ,n ∈ N

. (3.42)

To transform from physical to spectral space the following operation is needed, the so-called Discrete

Fourier Transform,

F (ui) = ui =1

N3

∑~x

uie−ikjxj , (3.43)

The inverse Discrete Fourier Transform is then,

F−1 (ui) = ui =∑~x

uieikjxj . (3.44)

As a final remark in this subsection, it should be noted that nonlinearities in Navier-Stokes equations

can lead to aliasing errors if one keeps the maximum wave number close to the Nyquist wavenumber. In

order to eliminate these errors, the solution for ui is truncated considering the maximum representative

wavenumber, according to the 2/3 rule, the following,

kmax =2

3kNyquist =

2

3

N

2. (3.45)

Here k =∣∣∣~k∣∣∣. As it can be seen, spatial discretization do not contribute to discretization errors. It is

important to check however if the minimum length scale measurable with a given solution is physically

compatible with the estimated Kolmogorov length scale. In general one should have kmaxη ≥ 1.5 in order

to ensure that all scales are resolved. Two other important errors should be kept in mind, those from

machine truncation and temporal discretization. Actually, these are the sources of errors that might be

found in the solution. For DNS purposes, the latter can be tolerated with a 3rd−order explicit Runge-

39

Kutta method, which will be described in section 3.5.

3.3.2 Navier-Stokes Equations in Fourier Space

As it was seen in the previous subsection, an approach regarding a flow description based on the

spectral space will be used. To complete the picture, Navier-Stokes equations and the constitutive

relations needed to complete them will be transformed to Fourier space through this subsection.

Considering the following relation,

F(∂ui∂xj

)= ikj ui , (3.46)

and transforming each side of both equations in system (3.1) one gets,

kiui = 0 , (3.47a)

∂ui∂t

+ ikj uiuj = −ikip

ρ+

ikj Tijρ

. (3.47b)

Multiplying equation (3.47b) by ki and using equation (3.47a) one has,

kikj uiuj = −k2 p

ρ+kikj Tijρ

, (3.48)

where k2 = kiki. Rewriting equation (3.48) with respect to p/ρ yields,

p

ρ=kkkjk2

(−ukuj +

Tkjρ

). (3.49)

Substituting (3.49) in (3.47b) and simplifying gives,

∂ui∂t− Pik

ikj Tkjρ

= −Pikikj ukuj , (3.50)

where,

Pik =

(δik −

kikkk2

). (3.51)

Defining,

Gk = ikj ukuj , (3.52)

the momentum equation in spectral space (3.47b) reads,

∂ui∂t− Pik

ikj Tkjρ

= −PikGk . (3.53)

The term Pikikj Tkj/ρ will depend on the constitutive equation used for Tij . The final closures will be

shown next.

40

Newtonian

Transforming equation (3.2) and recalling (3.3) the former becomes,

Tkj = µ (ikj uk + ikkuj) . (3.54)

Using the continuity, equation (3.47a), one has,

Pikikj Tkjρ

=µ

ρk2ui = νk2ui , (3.55)

where ν is the dynamic viscosity. Thus, substituting this result in (3.53) one gets,

∂ui∂t

+ νk2ui = −PikGk , (3.56)

the momentum equation in spectral space for an incompressible Newtonian fluid with constant proper-

ties.

Carreau-Yasuda

For this case, as Tkj becomes nonlinear, once computed Skj according with,

Skj =1

2F−1 (ikj uk + ikkuj) , (3.57)

Tkj is computed, (after have calculated µc (SkjSkj)), by the following procedure,

Tkj = F (2νc (SrmSrm)Skj) , (3.58)

where,

νc =µcρ, (3.59)

is the kinematic viscosity for a Carreau-Yasuda fluid. Note that ν0 and ν∞ are defined under the same

idea and represent kinematic viscosities. Remember from subsection 3.2.2, relatively to the statements

on µ0 and µ∞, that one has, by analogy, ν0 as the kinematic viscosity when SijSij = 0 and ν∞ when

SijSij →∞. So, the momentum equation in spectral space for a Carreau-Yasuda fluid reads,

∂ui∂t− PikikjF (2νc (SrmSrm)Skj) = −PikGk . (3.60)

FENE-P

Here there are one linear component and one nonlinear in Tkj , (see subsection 3.2.3 for details

about the nomenclature). Following the procedure developed in equations (3.54) and (3.55), the term

41

Pikikj Tkj/ρ on equation (3.53) becomes,

Pikikj Tkjρ

=Pikikj

(T

[s]kj + T

[p]kj

)ρ

= −µ[s]

ρk2ui +

Pikikj T[p]kj

ρ= −ν[s]k2ui +

Pikikj T[p]kj

ρ, (3.61)

where ν[s] is the kinematic viscosity of the solvent. According to equation (3.37), Pikikj T[p]kj /ρ is,

Pikikj T[p]kj

ρ=

(1− β)

β

ν[s]

τpPikikjF (f (Cmm)Ckj − δkj) . (3.62)

So, for the FENE-P case, the momentum equation (3.53) becomes,

∂ui∂t

+ ν[s]k2ui −(1− β)

β

ν[s]

τpPikikjF (f (Cmm)Ckj − δkj) = −PikGk . (3.63)

3.4 Conformation Tensor Transport Equation Discretization

The conformation tensor transport equation is a hyperbolic differential equation. Those of this nature

allow discontinuities on their solutions. Due to this aspect, using a spectral scheme to solve a hyperbolic

differential equation would lead to a loss of convergence [44]. So, in this case, the spatial discretization

will be done using a second-order central differences scheme for the stretching term and the Kurganov-

Tadmor (KT) scheme for the convection term. Note that the coupling term in equation (3.47) is discretized

on subsection 3.3.2 in its subsubsection ’FENE-P’.

∂Cij∂t

+ uk∂Cij∂xk︸︷︷︸

Convection

=∂uj∂xk

Cik +∂ui∂xk

Cjk︸︷︷︸Stretching

− 1


Through the next subsections it will be given the details about the spatial discretization of equation

(3.64). The notation will change a little for subsection 3.4.1, tensors like Cij will be denoted as C and

subscripts i, j and k will stand for the position of a generic grid point.

3.4.1 Convection Term: Kurganov-Tadmor (KT) method

Two of the fundamental properties of C are its symmetric and positive definite (SPD) characteristics.

To ensure these properties everywhere the Kurganov-Tadmor method arises. The nodal value for the

convection term is approximated by its mean value allover a finite cell surrounding a generic grid point

i, j and k. Thus, this approach follows a finite volume method. As it is described in [44], second-

order accurate spatial derivatives are used whenever SPD is guaranteed. If not, the method changes to

firs-order accurate. Further details are given next. To start one has,

uk∂Cij∂xk

= ~u ·∇C ≈ 1

∆V

∫uk∂Cij∂xk

dV =1

∆V

∫~u ·∇C dV , (3.65)

42

~u ·∇C ≈Hxi+1/2,j,k −Hx

i−1/2,j,k

∆x+

Hyi,j+1/2,k −Hy

i,j−1/2,k

∆y+

Hzi,j,k+1/2 −Hz

i,j,k−1/2

∆z, (3.66)

where the flux tensor H in each direction is given by,

Hxi+1/2,j,k =

1

2ui+1/2,j,k(C+

i+1/2,j,k + C−i+1/2,j,k)− 1

2

∣∣ui+1/2,j,k

∣∣ (C+i+1/2,j,k −C−i+1/2,j,k) , (3.67)

Hyi,j+1/2,k =

1

2vi,j+1/2,k(C+

i,j+1/2,k + C−i,j+1/2,k)− 1

2

∣∣vi,j+1/2,k

∣∣ (C+i,j+1/2,k −C−i,j+1/2,k) , (3.68)

Hzi,j,k+1/2 =

1

2wi,j,k+1/2(C+

i,j,k+1/2 + C−i,j,k+1/2)− 1

2

∣∣wi,j,k+1/2

∣∣ (C+i,j,k+1/2 −C−i,j,k+1/2) . (3.69)

Here, the velocities ui+1/2,j,k, vi,j+1/2,k and wi,j,k+1/2 are area-averaged velocities along the respective

face of the finite volume cell. The superscripts ‘+’ and ‘–’ represent the values of C at the interface,

obtained in the limit when approaching the point of interest from the right (+) or left (–) sides. The

following second-order linear approximations are used to compute those values,

C±i+1/2,j,k = Ci+1/2±1/2,j,k ∓(

∆x

2

)(∂C

∂x

)i+1/2±1/2,j,k

, (3.70)

C±i,j+1/2,k = Ci,j+1/2±1/2,k ∓(

∆y

2

)(∂C

∂y

)i,j+1/2±1/2,k

, (3.71)

C±i,j,k+1/2 = Ci,j,k+1/2±1/2 ∓(

∆z

2

)(∂C

∂z

)i,j,k+1/2±1/2

. (3.72)

These central differences are suitable for this discretization as they are able to capture sharp variation

in the conformation field as well as they guarantee the conservation of mean conformation [44]. One is

left to define spatial approximations for the C presented in the last equations above. To maintain SPD

properties for C±, some possibilities should be assessed. The potential options are,

(∂C

∂x

)i,j,k

=

Ci+1,j,k−Ci,j,k

∆x

Ci,j,k−Ci−1,j,k

∆x

Ci+1,j,k−Ci−1,j,k

2∆x

. (3.73)

The possibility that yields SPD for C+i−1/2 and C−i+1/2 is then chosen. If more than one meet the criterion,

it is chosen that one which maximizes the minimum eigenvalue for both tensors. If none agree with the

criterion, the derivative is set to zero, thus reducing the approximation to first-order accurate. In order

to compute H it is necessary the area-averaged velocities along the faces of a cell finite control volume.

A simple average would not preserve validity of the continuity equation, as described in [44], which can

be assessed by writing that equation according to the finite volume method. These errors can than be

propagated into the conformation tensor transport equation and consequently to the estimates of C.

Fortunately, spectral methods can easily find those averaged velocities through an exact expression.

43

So, for the following quantities,

ui±1/2,j,k =1

∆y∆z

∫ yj+∆y/2

yj−∆y/2

∫ zk+∆z/2

zk−∆z/2

u

(xi ±

∆x

2, y, z

)dydz , (3.74)

vi,j±1/2,k =1

∆x∆z

∫ xi+∆x/2

xi−∆x/2

∫ zk+∆z/2

zk−∆z/2

v

(x, yi ±

∆y

2, z

)dxdz , (3.75)

wi,j,k±1/2 =1

∆x∆y

∫ xi+∆x/2

xi−∆x/2

∫ yj+∆y/2

yj−∆y/2

w

(x, y, zk ±

∆z

2

)dxdy , (3.76)

which are the area-averaged velocities to be determined, have the following exact expression,

ui±1/2,j,k = F−1

(u(kx, ky, kz)e

±ikx∆x/2 sin (ky∆y/2)

ky∆y/2

sin (kz∆z/2)

kz∆z/2

), (3.77)

vi,j±1/2,k = F−1

(v(kx, ky, kz)e

±iky∆y/2 sin (kx∆x/2)

kx∆x/2

sin (kz∆z/2)

kz∆z/2

), (3.78)

wi,j,k±1/2 = F−1

(w(kx, ky, kz)e

±ikz∆z/2 sin (kx∆x/2)

kx∆x/2

sin (ky∆y/2)

ky∆y/2

). (3.79)

Note that (kx, ky, kz) = (k1, k2, k3). At this point the convective term is fully discretized.

3.4.2 Stretching Term: Central Finite Differences Method

The stretching term is discretized spatially using a second-order central differences method to ap-

proximate velocity gradients. So it reads,

∂uj∂xk

Cik +∂ui∂xk

Cjk ≈uj (~x+ ~s)− uj (~x− ~s)

2∆xkCik +

ui (~x+ ~s)− ui (~x− ~s)2∆xk

Cjk , (3.80)

where,

~s = (x1 + δ1k∆x1;x2 + δ2k∆x2;x3 + ∆x3) . (3.81)

Note that here the notation switched back to its default.

3.5 Temporal Integration

In this section it will be described the temporal integration method both for equations (3.47b) and

(3.64). The method used is a 3rd order Runge-Kutta scheme. For a generic vectorial quantity φi and

time-dependent problem of the form,

∂φi (~x, t)

∂t= Fi

(~φ (~x, t) , t

), (3.82)

a 3rd order Runge-Kutta scheme is given, for each time sub-step p = 1, 2, 3, as follows,

φi (~x, tp)− φi(~x, tp−1

)∆t

= αpFi

(~φ(~x, tp−1

), tp−1

)+ βpFi

(~φ(~x, tp−2

), tp−2

), (3.83)

44

where the coefficients αp and βp are given in equation (3.84), see also [50],

α1 = 8/15 β1 = 0

α2 = 5/12 β2 = −17/60

α3 = 3/4 β3 = −5/12

. (3.84)

It should be noted that for computational purposes, the Conformation tensor is first updated in the new

sub-step so the momentum equation can proceed in its numeric integration.

3.6 Stability

In what stability concerns, there is a specific arbitrary parameter that controls it, ∆t, the time-step

of the numerical scheme. In convection-dominant problems, a non-dimensional parameter arises as

the limiting relation which guarantees stability, the Courant-Friedrich-Levy number or simply the Courant

number, CFL, which is defined along x−direction as,

CFL ≡ |u1|∆t∆x1

, (3.85)

where |u1| is the absolute value of the velocity and ∆x1 is the grid spacing along the same direction. If

one deals with a 3D problem with constant grid spacing in each direction, it can be written that the most

restrictive CFL is,

CFL ≡ max

max |u1|

∆x1,

max |u2|∆x2

,max |u3|

∆x3

∆t . (3.86)

For Newtonian and Carreau-Yasuda simulations CFL < 3/5, [51]. On the other hand, for FENE-P simu-

lations, not only for stability purposes, but also to guarantee simetry and positive definitiveness, Courant

number is kept CFL < 1/6, see [44] for proof of SPD conservativeness.

3.7 Initial and Boundary Conditions

For boundaries, as described in section 3.3 and subsection 3.3.1, one should have periodic condi-

tions. By construction, a Fourier Transform uses the periodicity of a function over its period L, equal to

2π meters in this case, to derive the transformation itself. Furthermore, it should be remembered that

such derivation uses the orthogonality property of the complex exponential over the period L. So, it is

not necessary further constraints to guarantee periodicity as velocities are described by their Fourier

modes. In this sense, when transforming back into physical space, the function is periodic. In the case

of the Conformation tensor transport equation, whenever the points necessary are not in the grid, the

generic periodic condition is used. For a generic vectorial quantity φi it reads,

φi (~x, t) = φi (~x+ L (l,m, n) , t) , (3.87)

45

where the vector (l,m, n) is such that,

(l,m, n) ∈ Z . (3.88)

Initial condition are as well set for the velocity and conformation tensor fields. For the velocity one

has a field generated based on a desired energy spectrum. In this case that is of the form,

E (k) = kse− s2(kk0

)2

, (3.89)

where s is the slope of the spectrum at the large-scales region while k0 is the wavenumber location of

the maximum of the energy spectrum. For Cij (~x, t) at t = 0 one has,

Cij (~x, 0) = δij . (3.90)

3.8 Forcing Method

In this work, the simulations performed are forced in order to develop a turbulent field. The forcing

method follows that described by [52]. It is a volume random forcing made divergence free, so it does

not influence the pressure directly, delta-autocorrelated in time, once it is considered as white noise, and

uncorrelated with the velocity, the latter due its random nature, see [52]. Here it will be mentioned the

way of determining the forcing field hi, which is implemented in spectral space, such that it lies on the

Navier-Stokes momentum equation (3.47b) as,

∂ui∂t

+ ikj uiuj = −ikip

ρ+

ikj Tijρ

+ hi . (3.91)

An expression like equation (3.53) can be derived yielding,

∂ui∂t− Pik

ikj Tkjρ

= −PikGk + Pikhk . (3.92)

To prescribe a forcing divergence free one has,

hiki = 0 , (3.93)

and equation (3.92) becomes,∂ui∂t− Pik

ikj Tkjρ

= −PikGk + hk . (3.94)

Now considering the kinetic energy uiui/2, the evolution of its mean, K = uiui/2, where (·) denotes

spatial mean, has two contributions when forced, P1 and P2. When discretizing the equation of that

evolution and taking the spatial mean for a time-step length ∆t, these contributions are,

P1 =1

2hnti h

nti ∆t , (3.95)

46

and,

P2 = unti hnti , (3.96)

where nt is the current time-step. Note that the velocity and forcing were written in physical space. The

expression for P1/∆t as a function of spectral variables is,

P1

∆t=

1

2hihi =

1

2

∫hih∗i d~k , (3.97)

where (·)∗ represents the complex conjugate of the respective variable. This expression can be rewritten

as,P1

∆t=

1

2

∫hih∗i d~k =

∫2πk2

⟨hih∗i

⟩sph

dk =

∫ ∞0

H (k) dk , (3.98)

where 〈·〉sph means an average over a spherical shell with radius k in Fourier space. This implies that,

⟨hih∗i

⟩sph

=H (k)

2πk2. (3.99)

This function H (k) is a force spectrum that will be prescribed. Now it will be finally developed the

functional form for hi.

Recalling the condition (3.93), to ensure that one can define hi as,

hi ≡ Aran(~k, t)e1i

(~k)

+Bran

(~k, t)e2i

(~k), (3.100)

where ~e1 and ~e2 are vectors orthogonal to each other and to ~k. Aran and Bran are complex numbers

which will be determined. Applying now equation (3.99) one gets,

〈AranA∗ran〉sph + 〈BranB∗ran〉sph =H (k)

2πk2. (3.101)

Equation (3.101) is satisfied if one chooses,

Aran =

(H (k)

2πk2

)1/2

eiθ1gA (φ) , (3.102a)

Bran =

(H (k)

2πk2

)1/2

eiθ2gB (φ) , (3.102b)

where gA and gB are two real valued functions related to each other by,

g2A + g2

B ≡ 1 . (3.103)

The sets θ1, θ2 ∈ [0, 2π] and φ ∈ [0, π] contain random real variables uniformly distributed which are

generated at each wavenumber for each time-step.

As it is intended to make the forcing uncorrelated with the velocity, P2 is set ro zero for each time-step.

47

For each ~k it is possible to ensure as long as 2<h∗i ui

= 0. This is equivalent to say that,

<A∗ranξ1 +B∗ranξ2 = 0 , (3.104)

with,

ξ1 = uie1i , (3.105a)

ξ2 = uie2i . (3.105b)

Note that <· denotes the real part of a complex number. Instead of generating both θ1 and θ2, to

apply this constraint one defines ψ ≡ θ2 − θ1 and generate it randomly with a uniform distribution such

that ψ ∈ [0, 2π]. At this point θ1 and θ2 are found by solving equation (3.104) along with the definition

of ψ, which is resumed and simplified in the following set of equations,

tan θ1 =gA (φ)<ξ1+ gB (φ) (sinψ=ξ2+ cosψ<ξ2)−gA (φ)=ξ1+ gB (φ) (sinψ<ξ2 − cosψ=ξ2)

, (3.106)

ψ = θ2 − θ1 , (3.107)

where =· represents the imaginary part of a complex number.

What are left to be chosen are the spectrum H (k), the unit vectors ~e1 and ~e2, as well as the functions

gA and gB . For the spectrum one might have,

H (k) ≡ Ae−(k−kh)2

c , (3.108)

where,

A =P

∆t

1∫ kbkae−

(k−kh)2

c dk. (3.109)

Note that the prescription of the spectrum obeys a gaussian function, so kh is the wavenumber where

the maximum value of the forcing is applied and c is a variance-like parameter. The parameter A is a

constant while P is the power input. The limits of the integral, ka and kb, are the bounds of the region

where the forcing is to be applied. For the unit vectors mentioned, see condition (3.93) and definition

(3.101), one defines,

~e1 =

k2

(k21+k22)1/2

− k1

(k21+k22)1/2

0

, (3.110)

and,

~e2 =

k1k3

k(k21+k22)1/2

k2k3

k(k21+k22)1/2

− (k21+k22)1/2

k

. (3.111)

48

With this choices made so far, the correlation tensor,

1

2

⟨hihj

⟩∆t =

〈Px〉 0 0

0 〈Py〉 0

0 0 〈Pz〉

, (3.112)

shows zero off-diagonal elements. Furthermore the following relations hold,

〈Px〉 = 〈Py〉 , (3.113)

〈Px〉〈Pz〉

=1 + 2

⟨g2A

⟩4 (1− 〈g2

A)〉. (3.114)

Note that,1

2

⟨hihi

⟩∆t = 〈Px〉+ 〈Py〉+ 〈Pz〉 = P = 〈P1〉 . (3.115)

To make the forcing isotropic one only need to ensure that,

〈Px〉〈Pz〉

= 1 . (3.116)

For that the following choice of gA solves the problem,

gA = sin (2φ) , (3.117)

and, remembering (3.103), it yields,

gB = cos (2φ) . (3.118)

The functional form for hi at each time-step is then,

hi

(~k)

=

(Pe−(k−kh)2/c

∆t 2πk2∫ kbkae−(k−kh)2/c dk

)1/2 (eiθ1 sin (2φ) e1i

(~k)

+ ei(ψ−θ1) cos (2φ) e2i

(~k))

. (3.119)

For implementation purposes, P , kh, c, ka and kb are input parameters of the simulations. On the

other hand, ψ and φ are generated randomly at each time-step and for each wavenumber, while θ1 is

determined accordingly with the equation (3.106).

49

Chapter 4

Results and Discussion

In this chapter the results are presented and respective discussions are done. First, the results for

the IVS are presented for several Reynolds numbers. The various statistics are reported with special

emphasis to the scalability law of the mean length of those structures. Then, a comparative survey

is performed to analyze the capability of a purely viscous fluid, simulated with CY, of reproducing the

spectral dynamics of the kinetic energy evolution equation of a viscoelastic fluid, simulated using the

FENE-P model. Finally, a study focusing on turbulent dissipation of viscoelastic fluids using the FENE-

P model is made in order to determine the scalability law between the polymer concentration and the

Weissenberg number, at onset of dissipation reduction conditions. Additionally, a preliminary discus-

sion about a possible presence of a Virk’s-like asymptote is done near maximum dissipation reduction

conditions.

4.1 Intense Vorticity Structures Characterization

The first step of this work was to characterize Intense Vorticity Structures (IVS) embedded in a

homogeneous and isotropic turbulent field. As it was already mentioned, it follows the works developed

by Jimenez [18, 21], specially in what concerns the length of these structures, a feature left unclear

for higher Reynolds on those works. It was purposed by those researchers that such a length might

be scalable with the integral length scale of turbulent motion, however it will be shown here that the

mean length of IVS is indeed scalable with the Kolmogorov length scale. In the first two subsections the

problem will be summarized and an overview about the in-house post-processing Wormtracker code will

be made. Then the main results of the simulations are shown. Finally, an analysis is performed and the

scalability of the mean length of IVS is depicted.

4.1.1 Problem Description

Intense vorticity is nothing more than the maximum values of the vorticity field, mathematically speak-

ing. Physically, the regions are topologically coherent structures that are almost tubular, remembering

51

’worms’, the reason why this term is sometimes used. A discussion about physical consequences and

implications that these ’worms’ might have in turbulent phenomena was already told in section 2.3.

With this work one aims to clarify the main features of these structures and how those quantities are

related with turbulent descriptors. To understand how the characteristics evolve with Reynolds, several

simulations were made, where 88 < Reλ < 429. The resolution used was kmaxη = 2.

Several statistics were performed after achieving statistically steady state conditions, where the

power input P is balanced by viscous dissipation rate ε. The conditional statistics made on IVS were

made using the Wormtracker code. The main features of IVS that code computes are the length Livs,

the azimuthal velocity Uivs, the vorticity at the axis ω0 and circulation based Reynolds number Γ/ν.

These statistics were already introduced in section 2.3. The description of the code is given in the next

subsection 4.1.2.

4.1.2 Post-processing Algorithm - Wormtracker Code

This code was based on that described by Jimenez in [18]. It was implemented by da Silva and Reis

with results presented in [2]. Some modifications were made concerning dynamic memory manage-

ment. The code is serial and thus, it was necessary to rearrange data accessibility due to the biggest

simulations used.

The first consideration that must be made is the assumption made to define IVS. As did Jimenez [18]

and da Silva [2], it is defined as intense vorticity as the highest covering 1% of the total volume. The

threshold ωivs is defined so that ω > ωivs in the intense vorticity regions.

The points with ω > ωivs are then assigned an axis. This procedure takes the following steps.

1. The point with the highest ω that has not been assigned yet identifies a new axis.

2. Build the axis ’upwards’ along ~ω.

(a) To build the axis, it is made an intersection between the direction defined by ~ω and the first

plane corresponding to the space discretization. The four nearest points, (with respect to the

point resultant from that intersection), are chosen as candidates.

(b) From those four candidates, the point with the highest ω, if that has neither been assigned yet

nor has ω < ωivs, constitutes the new point of the identified axis.

(c) From this new ’upward’ point, if chosen, the algorithm proceeds at sub-step 2a. Otherwise,

these sub-steps stop when none of the candidates can be chosen, either due to the fact that

had been previously assigned an axis or due to ω < ωivs.

3. Build the axis ’downwards’ along −~ω. This procedure is equivalent to building the axis ’upwards’

along ~ω, the only difference is considering the direction −~ω at each stage.

4. If there is any point left to be assigned an axis, algorithm proceeds at step 1, otherwise it stops.

After this assignment, axis that have less than 20 points are discarded. In addition, clustered axis are

eliminated, leaving only that one with points with highest enstrophy.

52

For statistics, such as worm radius and circulation, the vorticity profile on a plane perpendicular

to ~ω, at each point of each axis, is considered. That vorticity profile is interpolated using the least

squares method at each point at that plane. To discretize a set of points at that plane it is considered

a cylindrical coordinate system (r; θ) were r = 0 at the axis. The set of points taken for interpolation

are (nr∆x; 2πmθ/Mθ), where nr = 0, . . . , Nr and mθ = 0, . . . ,Mθ, with Nr and Mθ being the radial and

azimuthal number of points, respectively. For each point the vorticity distribution is averaged radially. To

compensate for the noise induced by the discretization, the radial distribution is filtered over triples of

consecutive axial location using a [1/4; 1/2; 1/4] mask.

The worm radius Rivs is computed at each axial point by an iterative fitting of the detected vorticity

distribution to an assumed Gaussian shape as described in subsection 2.3.1 and written in equation

(2.77). The points where the fitting was unable to converge are discarded, like those with Rivs > 30η.

The azimuthal velocity Uivs computation is also described in subsection 2.3.1 and its expression is

given in equation (2.79).

The worm length Livs is the summation of all dLivs detected during the building of the axis procedure.

4.1.3 Simulations

In table 4.1 the results obtained for IVS are shown for various Reλ. The averaged values shown are

integrations, (for mean value), of the PDF’s analyzed in the next subsection 4.1.4.

Table 4.1: Simulations Parameters. Mesh points in each direction N ; Taylor micro-scale based Reynoldsnumber Reλ; turbulent dissipation ε, which equals the power input P ; kinematic viscosity ν; integrallength scale L11; velocity root mean square u′; mean of IVS lengths, 〈Livs〉, non-dimensionalized bythe Kolmogorov length scale η; mean of IVS tangential velocity, 〈Uivs〉, non-dimensionalized by u′;Ωλ = 〈ω0〉 /

(ω′Re

1/2λ

)is the mean of IVS vorticity magnitude at the its axis, 〈ω0〉, non-dimensionalized

by the vorticity magnitude root mean square ω′ and normalized by Re1/2λ ; circulation-based Reynolds

number RΓ normalized by Re1/2λ , where RΓ = 〈Γ〉 /ν.

Case N Reλε ν L11 u′ 〈Livs〉 /η 〈Uivs〉 /u′ Ωλ ReΓ/Re

1/2λ

[m2/s3

] [m2/s

][m] [m/s]

A 256 88 10.00 0.0145 0.89 2.98 49.24 0.97 0.41 13.739

B 512 145 10.00 0.0057 0.84 2.99 52.18 0.89 0.35 12.078

C 768 198 10.55 0.0034 0.89 3.11 54.44 0.83 0.31 11.365

D 1024 248 9.66 0.0022 0.91 3.07 57.96 0.84 0.30 11.033

E 1536 312 10.57 0.0014 0.85 3.10 58.06 0.90 0.28 11.304

F 2048 429 9.90 0.0009 0.88 3.23 57.18 0.79 0.23 10.141

G 2048 292 9.90 0.0009 0.60 2.67 57.81 0.84 0.28 10.844

4.1.4 Results Analysis

In this subsection the results analysis are given as well as comparisons to previous results from other

authors.

53

Scalability Law for 〈Rivs〉 and Assessment of Burgers’ Model Prediction Capability

One of the primary results ever looked at is the scalability of Rivs with η. In this case Rivs ≈ 4η was

obtained with convergence with higher Reynolds, in agreement with other results, see table 2.1. In figure

4.1(a) one can see the PDF’s at several Reynolds. These results confirm that for higher Reynolds the

scalability of Rivs with η is still present.

(a) (b)

Figure 4.1: Scalability tests for 〈Rivs〉 (a) with η and (b) with RB .

Other result of major importance when modeling such structures is the capability of doing so by a

Burgers’ Vortex model, see subsection 2.3.2 for details about this model. To assess that capability, it is

usually seen how a Burgers’ radius RB predicts Rivs. In this survey, the average value of local Rivs with

respect to local RB is 〈Rivs/RB〉 ≈ 0.98, in agreement with other authors, see table 2.1. In figure 4.1(b)

it is shown the PDF’s for several Reynolds. The accuracy of this model to predict the radius allows one

to infer that IVS might be seen as locally steady Burgers’ Vortexes. Even for values away from the mean,

those PDF’s show that extreme values are still close to unity.

Scalability Law for 〈Uivs〉

These structures are revealing of a radial characteristic length of the order and scalable with η. On

the other hand, the characteristic azimuthal velocity Uivs has been found to be of the order and scalable

with the large scale characteristic velocity u′. In order to confirm this latter result for higher Reynolds,

attention was also paid to that statistic.

In figure 4.2 are shown the PDF’s for 〈Uivs〉 non-dimensionalized by u′ and uη, respectively. Although

not fully collapsed, by visual inspection it is reasonable to think that u′ is a strong candidate for scalability.

On the other hand, uη does not seem to be the scalable variable for 〈Uivs〉. A direct approach to assess

these statements simultaneously is the evaluation of the plot of 〈Uivs〉 /uη as a function of Reλ. Using

the relation (2.64), if 〈Uivs〉 ∼ u′, one should expect that 〈Uivs〉 /uη ∼ Re1/2λ . In figure 4.3 this relation is

explored. It might be seen that 〈Uivs〉 does not scale with uη but rather with u′. Integration, (for mean

value), of PDF’s of figure 4.2(a), shows that 〈Uivs〉 /u′ ≈ 0.85, in agreement with previous results, see

54

(a) (b)

Figure 4.2: Scalability tests for 〈Uivs〉 (a) with u′ and (b) with uη.

Figure 4.3: Scalability law for 〈Uivs〉 /uη.

table 2.1.

Scalability Law for 〈Livs〉

One of the answers this work intends to provide is the scalability law for 〈Livs〉. As it was already said

in subsection 2.3.6, all authors whom referred to that length stated it is of the order of the integral length

scale. However, Jimenez in [18] and [21] said also that 〈Livs〉 scales with L11. The following results will

assess this question.

First consider figure 4.4. For several Reynolds, two plots of the PDF’s are shown, one of them in

figure 4.4(a) for 〈Livs〉 /L11 and other, in figure 4.4(b), for 〈Livs〉 /λ. Inspection of these two figures, due

to the lack of collapsing, there is no great support to the scalability either with L11 or with λ. Exploring

a possible scalability with an even smaller characteristic length, the PDF’s for severeal Reynolds for

〈Livs〉 /η is considered and shown in figure 4.5. This good collapsing shows a strong possible scalability

with η. To clarify this, one can simultaneously test the scalability with respect to one variable and the lack

of it with respect to another. Considering the relation (2.65), (which holds for Reλ > 120), if 〈Livs〉 ∼ η,

55

(a) (b)

Figure 4.4: Scalability tests for 〈Livs〉 (a) with L11 and (b) with λ.

Figure 4.5: Scalability test for 〈Livs〉 with η.

one should expect that 〈Livs〉 /L11 ∼ Re−3/2λ . On the other hand, as λ/η ∼ Re1/2

λ , if 〈Livs〉 ∼ η, one

should expect that 〈Livs〉 /λ ∼ Re−1/2λ . Figure 4.6 shows these behaviors, confirming that 〈Livs〉 scales

with η. To give a picture to the scalability trend, directly with η, figure 4.7 shows that behavior. It is

very important to note that convergence is only attained at Reλ > 250. The other statistics shown so far,

usually converge for Reλ > 120. It was found that 〈Livs〉 ≈ 58η. The simulation in black in these figures

indicate the result for simulation G. The importance of this simulation arises to confirm that there are

no confinement influences. In order to reach higher Reynolds, the forcing was concentrated at lower

wavenumbers, which in turn increases L11. When L11 is too high, due to the periodicity of the box,

confinement can influence the solution. Simulation G, with low L11 and thus, low Reλ, represents a

situation with no confinement. As G confirms the trend, it is safe to say that the other simulations, (with

higher L11), are confinement-free. Note as well that simulation G is right after the convergence threshold

for this statistic. If there was confinement in the solutions, in order to achieve higher Reynolds, with a

lower L11, bigger mesh sizes would be needed. Note that simulation G is also depicted in figure 4.3.

56

(a) (b)

Figure 4.6: Scalability laws for (a) 〈Livs〉 /L11 and (b) 〈Livs〉 /λ.

Figure 4.7: Scalability law for 〈Livs〉 /η.

PDF’s for 〈ω0〉 /(ω′Re

1/2λ

)and ReΓ/Re

1/2λ

Other quantities of interest are the distributions of 〈ω0〉 /(ω′Re

1/2λ

)andReΓ/Re

1/2λ , whereReΓ = 〈Γ〉 /ν.

These PDF’s are shown in figure 4.8 for several Reynolds, respectively.

Integration of the PDF’s, (for mean values), shows that 0.23 < 〈ω0〉 /(ω′Re

1/2λ

)< 0.41, in agreement

with Jimenez’s results in [21]. On the other hand, 10.1 < ReΓ/Re1/2λ < 13.7, which is similar with previ-

ous results, see table 2.1.

4.2 Carreau-Yasuda and FENE-P Comparative Survey

In this section a comparative survey is performed in order to understand the impact on the kinetic

energy evolution equation when applying the same shear-thinning law with two different mechanisms,

one purely viscous and another viscoelastic. The idea is to emphasize the elastic parallel mechanism of

viscoelastic fluids for a given Reynolds number and different Weissenberg numbers.

57

(a) (b)

Figure 4.8: PDF’s of (a) 〈ω0〉 /(ω′Re

1/2λ

)and (b) ReΓ/Re

1/2λ .


The approach to this problem will consist in the direct comparison of the terms underlying in the

kinetic energy evolution equation. The selection of the simulations to be compared follows the intention

to have a high Reynolds number and different Weissenberg numbers. A high Reynolds number will allow

to impose a large separation of scales, a condition which is preferable when analyzing small scales, (high

Reynolds numbers reduce the influence of large scales in the universal equilibrium range). On the other

hand, different We′s will allow to control the range of scales influenced by the polymers.

The rheological parameters for the FENE-P are set first. For convenience, the maximum non-

dimensional extensibility L and the β concentration are kept constant. Then, for constant given mesh

size and power input, to maintain the reference resolution, the solvent viscosity is determined. Refer-

ence conditions are those for a Newtonian simulation, e.g., to determine Weref , τ refη is computed using

the solvent viscosity and the total dissipation rate. Then, when these referred parameters are fixed, the

various τp are chosen to have Weref numbers from low to moderately-high values.

With the FENE-P parameters chosen, the rheological parameters for the Carreau-Yasuda constitutive

model may be determined. This is done by describing the viscosity induced by the polymer through

the Carreau-Yasuda shear-dependent relation. The method uses the solution of FENE-P equations

subjected to a imposed uniform shear flow. This method given by Pinho, (private communication), is

described next.

Considering a uniform laminar shear flow with,

∂ui∂xj

=

0 S 0

0 0 0

0 0 0

, (4.1)

one can determine the analytic solution with the FENE-P model, see [53] and [34]. From that solution

one can write the conformation tensor as a function of C22, which in turn is determined by the rheological

58

parameters. Writing C22 = C one has,

Cij =

C(1− L2

)+ L2 τpC2S 0

τpC2S C 0

0 0 C

, (4.2)

and the Peterlin function is,

f (Ckk) =1

C. (4.3)

The polymeric stress tensor is then,

T[p]ij =

(1− β)

βµ[s]

(L2 − CL2

)/ (τpC) CS 0

CS 0 0

0 0 0

. (4.4)

From equations (3.35) and (3.36), equation (4.4) can be rewritten as,

T[p]ij = µ[p]

(L2 − CL2

)/ (τpC) CS 0

CS 0 0

0 0 0

. (4.5)

Writing the effective polymer viscosity µeff with the definition,

µeff ≡T

[p]12

S, (4.6)

one has,

µeff = µ[p]C . (4.7)

As C is a function of SijSij , the total equivalent viscosity in a polymeric solution is µ[s] + µeff and is a

function of the strain product as well. The idea now is to describe it with the Carreau-Yasuda model.

Equating,

µc = µ[s] + µeff

= µ[s] + µ[p]C ,(4.8)

where µc is the viscosity of the Carreau-Yasuda model, one can relate equations (4.8) and (3.4b) to

state, by comparison, that,

µ[s] = µ∞ , (4.9)

and,

µ[p]C = (µ0 − µ∞)(

1 +(λc√

2SijSij

)a)nc−1a

. (4.10)

As the polymer induces an additional stress when comparing to the Newtonian one, and reaches its

59

maximum at zero-shear-rate, one has,

µ[p] = µ0 − µ∞ , (4.11)

leading to,

µ[s] + µ[p] = µ0 . (4.12)

So, with these statements one has,

µc = µ[s] + µ[p](

1 +(λc√

2SijSij

)a)nc−1a︸︷︷︸

C

. (4.13)

At this stage the problem reduces to a curve fitting to determine the rheological parameters for the

Carreau-Yasuda model in order to,

C (SijSij) ≈(

1 +(λc√

2SijSij

)a)nc−1a

. (4.14)

The dissipation reduction DR is defined as,

DR =ε[p]

ε[s] + ε[p]. (4.15)

The analysis will be made based on the spectra of the kinetic energy evolution equation. The follow-

ing subsections will show the different terms underlying on the respective equation for each model.

4.2.2 Kinetic Energy Equation for Carreau-Yasuda

First, the kinetic energy equation for the Carreau-Yasuda model is introduced.

For convenience, νc, see equation (3.4b), is decomposed according to,

νc = ν∞ + ν (SijSij) . (4.16)

Considering the equation (3.60), the evolution equation for the Fourier mode ui, and plugging in equation

(4.16), it becomes,∂ui∂t

= −PikGk + PikikjF (2νSkj)︸︷︷︸Mi

−ν∞k2ui . (4.17)

To derive the evolution equation for E (k), see equation (2.41), one needs to write the evolution equations

for ui and u∗i . Now, by doing the summation,

∂ui∂t

u∗i +∂u∗i∂t

ui = −PikGku∗i − PikG∗kui + Miu∗i + M∗i ui − ν∞k2uiu

∗i − ν∞k2u∗i ui , (4.18)

60

equation (4.18) becomes, after ensemble averaging and summing spherically for each k,

∂E (k)

∂t= −Pik<

Gku

∗i

︸︷︷︸

T (k)

+<Miu

∗i

︸︷︷︸

L(k)

− 2ν∞k2E (k)︸︷︷︸

D(k)

. (4.19)

For forced turbulence, a forcing term must be added. So, equation (4.19), for forced turbulence, be-

comes,∂E (k)

∂t= T (k) + L (k)−D (k) + H (k) . (4.20)

Summing the terms in equation (4.20) over all k gives,

∂K

∂t= −ε− ε∞ + P , (4.21)

where∑k E (k) = K,

∑k T (k) = 0,

∑k L (k) = −ε,

∑kD (k) = ε∞ and

∑k H (k) = P . In physical

space, 〈2νSijSij〉 = ε, 2ν∞ 〈SijSij〉 = ε∞.

It is also defined the cumulative transfer term Π (k) as,

Π (k) ≡ −k∑k′

T (k′) . (4.22)

4.2.3 Kinetic Energy Equation for FENE-P

For the FENE-P kinetic energy evolution equation the derivation method is the same. Rewriting

equation (3.63) in the form,

∂ui∂t

= −PikGk +(1− β)

β

ν[s]

τpPikikjF (f (Cmm)Ckj − δkj)︸︷︷︸

M ′i

−ν[s]k2ui , (4.23)

and doing the same as in equation (4.18), the kinetic energy evolution equation for the FENE-P consti-

tutive model reads, after ensemble averaging and summing spherically for each k,

∂E (k)

∂t= −Pik<

Gku

∗i

︸︷︷︸

T (k)

+<M ′iu

∗i

︸︷︷︸

Tp(k)

− 2ν[s]k2E (k)︸︷︷︸D(k)

. (4.24)

Note that the term D (k) is indeed the same in this case, once it was made, by comparison, ν[p] = ν∞.

The term Tp (k) represents the power transfered from the polymer to the solution.

Again, for forced turbulence, equation (4.24) should be read as,

∂E (k)

∂t= T (k) + Tp (k)−D (k) + H (k) . (4.25)

Summing the terms in equation (4.25) over all k gives,

∂K

∂t= −ε[p] − ε[s] + P , (4.26)

61

where∑k Tp (k) = −ε[p] and

∑kD (k) = −ε[s]. The other quantities where referred to previously. The

term∑k Tp (k) was defined as −ε[p] once, in statistically steady state, the net power transfer between

the polymer and solvent is directed from the solvent to the polymer and is, in turn, dissipated by the

polymer. In physical space,⟨T

[p]ij Sij

⟩/ρ = ε[p] and 2ν[s] 〈SijSij〉 / = ε[s].

The term Π (k) has the same definition as in equation (4.22).

4.2.4 Simulations

In this subsection they are presented the parameters chosen for FENE-P and the results of the curve

fitting mentioned in subsection 4.2.1. The average results of the simulations are also summarized.

In what concerns the parameters used in the simulations, table 4.2 gives an overview. For all simu-

lations the viscosities were set to,

ν = ν[s] = ν∞ = 0.0023 . (4.27)

The viscosity ν refers to that used in the Newtonian reference simulation.

Table 4.2: Constitutive Parameters for comparison between FENE-P and Carreau-Yasuda models. Non-dimensional polymer maximum extensibility L; polymer relaxation time τp; non-dimensional polymerconcentration β; solution solvent kinematic viscosity ν[s]; Carreau-Yasuda parameter a; Carreau-Yasudaparameter nc; Carreau-Yasuda equivalent relaxation time λc; solution kinematic viscosity at zero-shear-rate ν0.

FENE-P Carreau-Yasuda

Case Lτp

βν[s]

Case a ncλc ν0

[s] [m2/s] [s] [m2/s]

F1 100 0.025 0.8 0.0023 C1 1.9422 0.5163 0.0007 0.0029

F2 100 0.050 0.8 0.0023 C2 1.8758 0.4767 0.0013 0.0029

F3 100 0.100 0.8 0.0023 C3 1.7903 0.4416 0.0023 0.0029

F4 100 0.200 0.8 0.0023 C4 1.7084 0.4156 0.0043 0.0029

The Newtonian reference simulation has its characteristics summarized in table 4.3. In all simulations

the mesh size in each direction N was kept constant, as well as the power input P .

Table 4.3: Simulation Parameters for the Newtonian case. The Newtonian simulation constitutes thereference simulation. All reference quantities are computed with Newtonian parameters. All simulationshave the same mesh size in each direction N and all have the same power input, (forcing), P . Mesh sizein each direction N ; Reynolds number (equal to the reference one); power input P ; root-mean-squarevelocity u′.

N ReλP u′

[m2/s3] [m/s]

768 181 10 2.7

As described in subsection 4.2.1, reference conditions are computed with results from the Newtonian

simulation.

Table 4.4 summarizes the results from the FENE-P simulations.

62

Table 4.4: Simulations Parameters for the FENE-P cases. Reference Weissenberg number, (computedwith Newtonian τη), Weref ; Weissenberg number, (computed with solvent τ [s]

η ), We; polymer relaxationtime τp; solvent dissipation rate ε[s]; polymer dissipation rate ε[p]; dissipation reduction DR; root-mean-square velocity u′.

Case Weref Weτp ε[s] ε[p] DR u′

[s][m2/s3

] [m2/s3

][%] [m/s]

F1 2 1 0.025 7.05 2.95 30 2.7

F2 3 2 0.050 3.87 6.13 61 2.6

F3 7 3 0.100 2.00 8.00 80 2.5

F4 13 5 0.200 1.19 8.81 88 2.3

Table 4.5: Simulations Parameters for the Carreau-Yasuda cases. Carreau-Yasuda equivalent relaxationtime λc; dissipation rate, based on viscosity when SijSij →∞, ε∞; dissipation rate, based on fluctuatingvescosity, ε; root-mean-square velocity u′.

Caseλc ε∞ ε u′

[s][m2/s3

] [m2/s3

][m/s]

C1 0.0007 8.03 1.97 2.6

C2 0.0013 7.79 2.21 2.7

C3 0.0023 7.83 2.17 2.6

C4 0.0043 8.04 1.96 2.7

The results from Carreau-Yasuda simulations can be checked in table 4.5.

The results analysis follows in the next subsection. It will be essentially explored the spectra of the

terms of the kinetic energy evolution equation for statistically steady state conditions.

4.2.5 Results Analysis

Here, the results from the simulations performed will be shown. The idea is to characterize the

differences between two types of fluids with shear-thinning behavior, one purely viscous and another with

equivalent shear-thinning induced by polymer additives. All simulations represent forced homogeneous

and isotropic turbulence.

First consider the energy spectra at statistically steady state conditions, which can be seen on figure

4.9.

Figure 4.9(b) shows the energy spectra for the Carreau-Yasuda cases. It is fairly evident that dif-

ferences from the Newtonian case only appear at small scales. In fact, even for a variable viscosity

case, it is expected that Kolmogorov’s Second Similarity Hypothesis still holds, once, with sufficiently

high Reynolds number, a universal region with local Re` Reη might be encountered such that it is

independent from νc. On the other hand, at small scales, when viscous forces begin to have visible

effect, differences arise. Once there are regions where viscosity is greater than ν∞, for a given scale to

maintain its dissipation, its characteristic energy should be less than that for a lesser viscosity.

Figure 4.9(a) shows what happens when FENE-P is considered instead. It is clearly evident the influ-

ences in the cascade are increasingly felt up through the Newtonian inertial range as Weref increases.

63

(a) (b)

Figure 4.9: Non-dimensionalized energy spectra E (k) /(u2ηη)

(a) for all FENE-P cases and (b) for allCarreau-Yasuda cases. Both have the Newtonian non-dimensionalized energy spectra. Parametersused to non-dimensionalize are from the Newtonian case.

In contrast to what was said for the purely viscous case, there is no longer reason to believe that the

inertial range may remain intact. With low Weref , polymers are at most moderately stretched and de

Gennes theory, (see subsection 2.4.4), might be used to interpret the phenomenon. It is expected that

for scales ` < `∗ turbulence influences polymers but for scales `∗ < ` < `′ polymers do influence turbu-

lent motion. That influence in the energy spectra might be seen as a cut-off at a scale O (`) = O (`′).

For high Weref the picture is different. While in the first place the flux of energy occurs mainly in the

solvent, for high Weref that flux is redirected through the polymer. When it happens, the slope of −3 in

the energy cascade appears.

So far it was not made a direct evaluation of the terms underlying in the kinetic energy evolution

equation. To assess deeply the impact of polymer phenomenon, in order to take conclusions, (some

were already given in advance), one needs to interpret those terms.

First it is important to refer that the forcing spectra H (k) is concentrated at low wavenumbers, being

zero for kη > khη = 2× 10−2. Then, for the Newtonian case, at statistically steady state conditions, the

kinetic energy equation is just,

∂E (k)

∂t= 0 = T (k)−D (k) + H (k) . (4.28)

Spectra with respect to each type of fluid will be denoted with a subscript C, N or F , whenever it refers

to Carreau-Yasuda, Newtonian or FENE-P type of fluid. Numbers that might appear in the subscript as

well refer to each case for each type of fluid, see tables 4.4 and 4.5.

In figure 4.10, which shows the dissipation spectra D (k), it should be noted that integration of the

Newtonian curve gives the power input and, for kη > khη, DN (k) = TN (k).

In figure 4.11, which shows the cumulative transfer spectra Π (k), it should be noted that max ΠN = P

after H (k) becoming zero. It reflects the conservativeness of the power input before it becomes to be

dissipated. As long as ΠN ≈ P , DN ≈ 0. This remarks will allow to take some conclusions next.

In figure 4.10(b) they are depicted the purely viscous cases. This plot helps one to confirm what

64

(a) (b)

Figure 4.10: Non-dimensionalized dissipation spectraD (k) / (Pη) (a) for all FENE-P cases and (b) for allCarreau-Yasuda cases. Both have the Newtonian non-dimensionalized dissipation spectra. Parametersused to non-dimensionalize are the forcing P and η from the Newtonian case.

(a) (b)

Figure 4.11: Non-dimensionalized cumulative transfer spectra Π (k) /P (a) for all FENE-P cases and(b) for all Carreau-Yasuda cases. Both have the Newtonian non-dimensionalized dissipation spectra.Parameter used to non-dimensionalize is the forcing P .

was said earlier. In fact, differences are essentially present at small scales. Nevertheless, it is only true

if the term T (k) is identical in Newtonian and Carreau-Yasuda cases for larger scales. Figure 4.11(a)

shows that a Newtonian-like inertial range is present in Carreau-Yasuda cases. This shows a Newtonian

behavior of purely viscous flows at high Reynolds until viscous forces become dominant. Furthermore,

it is clear that∫

(DN (k)−DC (k)) = ε.

In figure 4.10(a) they are shown the viscoelastic cases. These spectra, especially for high Weref , al-

most do not show an inertial range like Newtonian fluids. Along with figure 4.11(a) some conclusions are

straightforward in what concerns the energy transfer to polymer motion. Consider k1 as the wavenum-

ber where a FENE-P curve in figure 4.11(a) departs from the Newtonian one or, if it does not reach the

65

Newtonian curve, take k1 = kh. Now, considering the summation,

0 =

k1∑k

TF (k) +

k1∑k

Tp (k)−k1∑k

DF (k) +

k1∑k

H (k)

= −ΠF (k1)︸︷︷︸≤P

+−Πp (k1)−k1∑k

DF (k)︸︷︷︸≤∑k1k DN (k)≈0

+P

≥ −P −Πp (k1) + P ,

(4.29)

one has,

Πp (k1) ≥ 0 . (4.30)

This relation (4.30) indicates that k1 is the wavenumber where the flux of energy is being redirected

through polymer motion. In case F4, the highest Weref , ΠF4 does not reach P , indicting that the flux

of energy, at the largest scales, is being redirected through polymer motion before it may be directed

through the solvent inertial range. In this case a slope of −3 appears in the energy spectra in figure

4.9(a).

Figure 4.12 shows the case F4 in comparison with the case C4, along with the Newtonian curve.

This time, the spectra D (k) are shown along with spectra T (k).

Figure 4.12: Non-dimensionalized dissipation spectra D (k) / (Pη) and non-dimensionalized transferspectra T (k) / (Pη) for both cases F4 and C4. It is also provided the non-dimensionalized dissipa-tion spectra D (k) / (Pη) for the Newtonian case, which equals its non-dimensionalized transfer spectraT (k) / (Pη) at steady state for kη > 2× 10−2, (non-dimensionalized wavenumber beyond which the forc-ing spectra term equals zero). Parameters used to non-dimensionalize are the forcing P and η from theNewtonian case.

It shows the case C4 behaving like a Newtonian fluid for scales from the largest down to the inertial

range. In small scales the differences arise as it was said earlier. The small difference between TC4 (k)

and DC4 (k) represents −LC4 (k).

In the case F4 it is clear that the high value of TF4 (k) indicates that the energy flux given to the largest

scales is being almost all redirected to polymer motion. Indeed, for kη ≥ khη, TF4 (k)−DF4 (k) = −Tp (k)

66

and, for TF4 −DF4 > 0, Tp < 0, which means that the energy flux flows from solvent to polymer. On the

other hand, when TF4 −DF4 < 0, Tp > 0, which means that some flux of energy is flowing from polymer

back to the solvent. This happens at small scales in this case F4.

In general it can be said that the Carreau-Yasuda model, although reproducing the same shear-

thinning law, is not capable to simulate the FENE-P energy flux cascade mechanism. This is primarily

due to the elastic mechanism that, even for high Reλ, might be felt from the inertial range up to the

energy-containing range, becoming, this influence, specially stronger as Weref increases. Even though,

integral quantities, like ε, could somewhat give an estimate of, e.g., εp. The reason that this is not verified

is that the strain correlation structure is largely different from Carreau-Yasuda cases to FENE-P cases,

as can be assessed in dissipation spectra of figure 4.10.

4.3 Dissipation Reduction Onset and Preliminary Study Near Max-

imum Dissipation Reduction

This section will provide an additional survey in what concerns polymer induced phenomenon. In

the literature there is a lot of work developed focusing on polymer behavior, specially based in pipe

flow experiments, such as [31] and [48], and in numerical studies, like in channel flow [36], shear flow

[46] and isotropic turbulence [47]. The idea here is to make an analogy between drag reduction and

dissipation reduction, being the latter analyzed in isotropic turbulence. Once identified this effect, it will

be explored the onset of dissipation reduction and the presence of a limiting Virk’s-like asymptote.


The main frame of this approach is to consider forced homogeneous and isotropic turbulence. The

model to simulate polymer additives will be the FENE-P. Then, to explore a dissipation reduction law,

de Gennes theory analyzed in subsection 2.4.4 will be used. By varying the rheological parameters, β

concentration and relaxation time, the presence of a dissipation reduction limiting asymptote will also be

assessed. In all simulations, the non-dimensional maximum extensibility L is kept constant.

As a reference for this study, as usual, Newtonian simulations are also performed. The Newtonian

reference simulation is that with resolution equal to the reference resolution, (kmaxη) = (kmaxη)ref , and

equal mesh size. In a FENE-P simulation, (kmaxη)ref is computed with the total dissipation rate, (which

balances the total power input at statistically steady state conditions), and solvent kinematic viscosity

so,

(kmaxη)ref

=N

3

(ν[s]3

P

)1/4

. (4.31)

Likewise, τ refη is determined accordingly to define Weref . On the other hand, Rerefλ is that resultant

from the Newtonian simulation. For all FENE-P simulations, (kmaxη)ref

= 1.5. So, for given power input

P and mesh size in each direction N , the viscosity is set.

For the onset of dissipation reduction they will be used two Rerefλ . Then, a set of (β; τp) pair values

67

is chosen. Different values of τp are defined in order to cover from low to moderately-high Weref and,

for each Weref , β values are varied to simulate from low to higher polymer concentrations. The mesh

size in each direction N , power input P and solvent viscosity ν[s] are equal to those used for each RerefλNewtonian simulation.

For the maximum dissipation reduction, it will be used one Rerefλ . Then, as well, a set of (β; τp) pair

values is chosen. This time, too low values of polymer concentration are, naturally, avoided. So, this

set of parameters differs from the previous one. The parameters N , P and ν[s] are chosen accordingly

as before. The plot of the results for maximum dissipation reduction will be based on a chart with

Prandtl-Karman coordinates, as in figure 2.7. To do that, the skin friction will be replaced by,

Cε =ε[s]L11

u′3, (4.32)

the dissipation coefficient, and the Reynolds number used will be Reλ.

4.3.2 Simulations

Newtonian simulations are performed not only to constitute reference simulations for the FENE-P

cases but also to determine Cε for different Reλ. That information will be used in subsection 4.3.4. These

simulations have different resolutions, (kmaxη), in order to obtain, for the same mesh size, different

Reynolds numbers. As said before, those simulations with (kmaxη) = (kmaxη)ref

= 1.5 and the same

mesh size constitute the reference simulations for the FENE-P cases. A compilation of Newtonian

simulations is provided in table 4.6, where Newtonian reference simulations have their case identification

in bold.

FENE-P simulations for the study of the onset of dissipation reduction have L = 100. In what it

concerns the reference resolution, (kmaxη)ref

= 1.5. Other parameters and statistics are summarized in

table 4.7.

For the cases which give basis to the maximum dissipation reduction survey, L = 100 and also

(kmaxη)ref

= 1.5. The remaining parameters and main statistics are given in table 4.8.

The results analysis follows in the next subsections. First it will provided the results for the onset

of dissipation reduction, where it will be tested the theory described and derived in subsection 2.4.4,

specifically by means of the comparison of the results with the scalability law shown in equation (2.115).

The last subsection will provide the preliminary study of how the dissipation coefficient behaves with

Reynolds, specially near maximum dissipation reduction conditions.

4.3.3 Onset Of Drag Reduction Law

In subsection 2.4.4 it was considered the de Gennes’ theory to predict the scalability law at the

onset of dissipation reduction involving β concentration and We. That theory, as explained earlier, is

suitable for onset conditions as it treats partially stretched polymers. Furthermore, it is not expected

measurable influence of the polymers in turbulent structure as `′ ∼ η. As so, the flow might be seen

68

Table 4.6: Simulation Parameters for the Newtonian cases. Mesh size in each direction N ; Taylormicroscale-based Reynolds number Reλ; power input, (forcing), P ; kinematic viscosity ν; integral lengthscale L11; root-mean-square velocity u′; dissipation coefficient Cε.

Case N ReλP ν L11 u′

Cε[m2/s3

] [m2/s

][m] [m/s]

N1 192 50 3.3 0.0285 1.32 2.0 0.56

N2 192 57 3.3 0.0235 1.32 2.0 0.53

N3 128 67 10.0 0.0248 1.24 2.9 0.49

N4 192 68 3.3 0.0173 1.27 2.0 0.49

N5 192 73 3.3 0.0147 1.22 2.0 0.48

N6 192 79 3.3 0.0128 1.21 2.0 0.47

N7 192 85 3.3 0.0113 1.22 2.1 0.46

N8 192 91 10.0 0.0145 1.20 3.0 0.45

N9 256 96 3.3 0.0088 1.17 2.1 0.45

N10 256 100 3.3 0.0080 1.16 2.0 0.45

N11 256 107 3.3 0.0071 1.16 2.1 0.44

N12 256 112 10.0 0.0099 1.19 3.0 0.43

N13 256 112 3.3 0.0065 1.16 2.1 0.44

N14 256 120 3.3 0.0060 1.19 2.1 0.43

N15 384 143 10.0 0.0057 1.11 3.0 0.42

N16 512 178 10.0 0.0039 1.13 3.0 0.41

as Newtonian with passive scalars embedded and We ≈Weref . With increasing β or Weref , `′ may

eventually become greater than η and polymers have an active role in turbulent dynamics. To measure

this transition arbitrariness cannot be avoided but a method can be ruled by inspection of the results.

Consider figure 4.13. For two Rerefλ , dissipation reduction is plotted as a function of (1− β) /β and

Weref . Those plots are bi-logarithmic and show regions with a constant slope for each Weref . That

(a) (b)

Figure 4.13: Dissipation reduction DR as a function of (1− β) /β at various reference Weissenbergnumbers Weref (a) for Rerefλ = 67 and (b) for Rerefλ = 91. These samples are at the onset of dissipationreduction.

slope is expected once ε[p] =⟨T

[p]ij Sij

⟩/ρ and T [p]

ij is proportional to (1− β) /β, see equation (3.37). For

the same Weref and small (1− β) /β, tiny variations of the latter do not influence the polymer stress

69

Table 4.7: Simulations Parameters for the FENE-P cases at the onset of dissipation reduction. Meshsize in each direction N ; reference Weissenberg number, (computed with total dissipation rate and sol-vent kinematic viscosity), Weref ; Taylor microscale-based Reynolds number, (computed with solventproperties and statistics), Reλ; polymer relaxation time τp; β concentration; power input P ; solvent kine-matic viscosity ν[s]; polymer dissipation rate ε[p]; dissipation reduction DR; root-mean-square velocity u′.These simulations have the non-dimensional maximum extensibility L = 100 and, as can be computedwith given values, (kmaxη)

ref= 1.5.

Case N Weref Reλτp

βP ν[s] ε[p]

DRu′

[s][m2/s3

] [m2/s

] [m2/s3

][m/s]

DR1 128 0.5 73 0.05 0.80000000 1 0.0115 1.8× 10−1 1.8× 10−1 1.4

DR2 128 0.5 67 0.05 0.95000000 1 0.0115 4.4× 10−2 4.4× 10−2 1.3

DR3 128 0.5 67 0.05 0.98000000 1 0.0115 1.8× 10−2 1.8× 10−2 1.4

DR4 128 0.5 67 0.05 0.99389000 1 0.0115 5.3× 10−3 5.3× 10−3 1.4

DR5 128 0.5 67 0.05 0.99938562 1 0.0115 5.5× 10−4 5.5× 10−4 1.4

DR6 128 0.5 67 0.05 0.99993853 1 0.0115 5.4× 10−5 5.4× 10−5 1.4

DR7 128 0.5 67 0.05 0.99999385 1 0.0115 5.4× 10−6 5.4× 10−6 1.4

DR8 128 0.5 68 0.05 0.99999939 1 0.0115 5.0× 10−7 5.0× 10−7 1.4

DR9 128 2.3 81 0.25 0.80000000 1 0.0115 4.4× 10−1 4.4× 10−1 1.3

DR10 128 2.3 77 0.25 0.98000000 1 0.0115 3.2× 10−1 3.2× 10−1 1.3

DR11 128 2.3 73 0.25 0.99389000 1 0.0115 2.3× 10−1 2.3× 10−1 1.3

DR12 128 2.3 70 0.25 0.99938562 1 0.0115 5.7× 10−2 5.7× 10−2 1.4

DR13 128 2.3 68 0.25 0.99993853 1 0.0115 7.1× 10−3 7.1× 10−3 1.4

DR14 128 2.3 67 0.25 0.99999385 1 0.0115 7.5× 10−4 7.5× 10−4 1.4

DR15 128 2.3 68 0.25 0.99999939 1 0.0115 6.6× 10−5 6.6× 10−5 1.4

DR16 128 4.7 84 0.50 0.99389000 1 0.0115 5.1× 10−1 5.1× 10−1 1.3

DR17 128 4.7 72 0.50 0.99938562 1 0.0115 1.6× 10−1 1.6× 10−1 1.3

DR18 128 4.7 68 0.50 0.99993853 1 0.0115 2.1× 10−2 2.1× 10−2 1.4

DR19 128 4.7 66 0.50 0.99999385 1 0.0115 2.3× 10−3 2.3× 10−3 1.4

DR20 128 4.7 66 0.50 0.99999939 1 0.0115 2.1× 10−4 2.1× 10−4 1.4

DR21 192 1.0 103 0.04 0.98000000 10 0.0145 2.0× 10−1 2.0× 10−2 3.0

DR22 192 1.0 90 0.04 0.99700000 10 0.0145 3.5× 10−2 3.5× 10−3 3.0

DR23 192 1.0 89 0.04 0.99997000 10 0.0145 3.7× 10−4 3.7× 10−5 3.0

DR24 192 1.0 91 0.04 0.99999700 10 0.0145 3.5× 10−5 3.5× 10−6 3.0

DR25 192 1.0 90 0.04 0.99999970 10 0.0145 3.3× 10−6 3.3× 10−7 3.0

DR26 192 1.0 89 0.04 0.99999997 10 0.0145 3.3× 10−7 3.3× 10−8 2.9

DR27 192 10.0 88 0.38 0.99997000 10 0.0145 1.8× 10−1 1.8× 10−2 2.9

DR28 192 10.0 90 0.38 0.99999700 10 0.0145 1.8× 10−2 1.8× 10−3 3.0

DR29 192 10.0 91 0.38 0.99999970 10 0.0145 1.7× 10−3 1.7× 10−4 3.0

DR30 192 10.0 89 0.38 0.99999997 10 0.0145 1.8× 10−4 1.8× 10−5 3.0

tensor with measurable impact on the strain rate field. So that slope, at onset condition, is ≈ 1, as visible

in those plots.

These statements help identifying the onset region. If one is trying to find a power law between

(1− β) /β and Weref while expecting a power law when relating DR with (1− β) /β, a region on those

plots showing parallel straight lines indicates the onset region. So, taking a set of points at constant DR

in the onset region one can test the scalability law (1− β) /β ∼Weδ.

70

Table 4.8: Simulations Parameters for the FENE-P cases near maximum dissipation reduction. Meshsize in each direction N ; reference Weissenberg number, (computed with total dissipation rate andsolvent kinematic viscosity), Weref ; Weissenberg number, (computed with solvent properties and statis-tics), We; Taylor microscale-based Reynolds number, (computed with solvent properties and statistics),Reλ; polymer relaxation time τp; β concentration; power input P ; solvent kinematic viscosity ν[s]; solventdissipation rate ε[s]; polymer dissipation rate ε[p]; dissipation reduction DR; integral length scale L11;root-mean-square velocity u′; dissipation coefficient Cε.

Case N Weref We Reλτp

βP ν[s] ε[s] ε[p] DR L11 u′

Cε[s]

[m2/s3

] [m2/s

] [m2/s3

] [m2/s3

][%] [m] [m/s]

MRa 192 7.1 2.7 142 0.3 0.5000 10 0.0145 1.43 8.57 86 1.54 2.3 0.19

MRb 192 7.1 2.8 145 0.3 0.8000 10 0.0145 1.59 8.41 84 1.56 2.4 0.19

MRc 192 7.1 3.8 143 0.3 0.9800 10 0.0145 2.89 7.11 71 1.45 2.7 0.20

MRd 192 7.1 5.2 113 0.3 0.9970 10 0.0145 5.35 4.65 47 1.26 2.8 0.29

MRe 192 10.0 3.4 127 0.4 0.5000 10 0.0145 1.19 8.81 88 1.58 2.1 0.21

MRf 192 10.0 3.6 130 0.4 0.8000 10 0.0145 1.27 8.73 87 1.56 2.2 0.21

MRg 192 10.0 4.7 145 0.4 0.9800 10 0.0145 2.21 7.79 78 1.49 2.6 0.19

MRh 192 10.0 6.9 121 0.4 0.9970 10 0.0145 4.78 5.22 52 1.32 2.9 0.27

MRi 192 13.9 4.8 102 0.5 0.5000 10 0.0145 1.17 8.83 88 1.58 1.9 0.30

MRj 192 13.9 4.8 112 0.5 0.8000 10 0.0145 1.17 8.83 88 1.60 2.0 0.26

MRk 192 13.9 6.4 144 0.5 0.9800 10 0.0145 2.11 7.89 79 1.51 2.5 0.19

MRl 192 13.9 10.4 125 0.5 0.9970 10 0.0145 5.52 4.48 45 1.29 2.8 0.25

MRm 192 20.0 8.3 85 0.8 0.5000 10 0.0145 1.72 8.28 83 1.52 1.9 0.41

MRn 192 20.0 7.6 94 0.8 0.8000 10 0.0145 1.44 8.56 86 1.57 1.9 0.35

MRo 192 20.0 8.3 142 0.8 0.9800 10 0.0145 1.73 8.27 83 1.53 2.4 0.19

MRp 192 20.0 12.9 126 0.8 0.9970 10 0.0145 4.19 5.81 58 1.35 2.8 0.25

Figure 4.14 shows the results for both Rerefλ . The plot shows (1− β) /β as a function of We for

a constant DR = 1× 10−3. The curves shown represent linear regressions for the set of points used.

Figure 4.14: Scalability law for the onset of dissiaption reduction derived in subsection 2.4.4 and shownin equation (2.115). The fitted slope indicates a value of ≈ −11/4, resulting in a stretching exponentq ≈ 1.

Both slopes are approximately equal and are taken equal to δ ≈ −11/4. In equation (2.115), where

δ = 1− 15q/4, a q in the exponent was left to be determined. With the present result, q ≈ 1, represent-

71

ing a bi-dimensional stretching. In pipe flow, Sreenivasan [48] determined a corresponding value of

≈ 2/3. In forced homogeneous and isotropic turbulence that value was expected to be higher, due to no

geometrical confinement by the boundaries in the small scales regions, being a higher value confirmed

with this present work.

4.3.4 Preliminary Study Near Maximum Dissipation Reduction

In pipe flow it is commonly seen that, for a given Reynolds, no matter the polymer properties, there

is a maximum drag reduction limit. So, if for a given Reynolds one could vary the polymer proper-

ties, a search for the maximum drag reduction would provide a point on the maximum drag reduction

asymptote.

The concept of drag reduction is connected with wall shear stress relief when polymers are added

to a given solvent. In a non-dimensional basis, drag reduction is associated to the friction factor relief

for the same Reynolds number. Nevertheless, both concepts end up being equivalent if the parameters

used to non-dimensionalize the wall shear stress are the same both for the polymeric and solvent-alone

solutions.

To study this behavior in homogeneous isotropic turbulence, an analogy needs to be made first.

A logical one is to consider the dissipation reduction phenomenon. So, the counterpart of the wall

shear stress would be the dissipation rate, whilst that for the friction factor would be the dissipation

coefficient Cε. Furthermore, in isotropic turbulence it is often used the Taylor microscale-based Reynolds

number Reλ as a characteristic one. It is important to note, however, that the parameters used to non-

dimensionalize both Cε and Reλ vary under an addiction of polymers. So, due to this fact, it is important

to clarify that dissipation reduction and dissipation coefficient reduction are not necessarily equivalent.

In order to perform an experiment like that mentioned above, only the parameters β and τp are varied,

being the reference conditions kept constant. This allows one to fix a Rerefλ . Nevertheless, the statistics

are shown with the real Reλ computed.

As a preliminary study about how Cε behaves near maximum dissipation reduction conditions, some

simulations were done for a given Rerefλ = 91. Figure 4.15 shows the results obtained for DR and Cε

both as a function of (1− β) /β and Weref .

Figure 4.15(a) shows that approaching (1− β) /β = 1, DR decreases its derivative. In addition, as

Weref increases, that effect becomes stronger. It is not consider β < 0.5 due to the fact that it represents

the region where polymers overlap, being β = 0.5 of the order of the limit when polymers begin to overlap.

Figure 4.15(b) shows what happens when Cε is considered. It is clear that a minimum Cε is present

for the majority of the cases and that the overall minimum is approximately equal to each minimum. The

minimum values for the larges Weref are concentrated at β = 0.98.

Both figures show in the left a Newtonian-like condition for comparison. Both figures also show

that maximum dissipation reduction and minimum Cε for each Weref are approximately equal among

them for each dissipation statistic, suggesting saturation both for increasing (1− β) /β and Weref . The

smooth convergence to a possible global maximum DR and to a possible global minimum Cε may

72

(a) (b)

Figure 4.15: Dissipation statistics as a function of (1− β) /β at various reference Weissenberg numbersWeref , (a) represents the dissipation statisticDR and (b) represents the statistic 1/C

1/2ε . These samples

are near the maximum dissipation reduction. A sample near Newtonian conditions, (DR ≈ 0), is alsogiven.

indicate that a region independent from polymer properties may be present.

Figure 4.16 shows the results of Cε as a function of Reλ. That relation is shown in Prandtl-Karman

(a) (b)

Figure 4.16: Relation between dissipation coefficient Cε and Taylor microscale-based Reynolds number,(computed with solvent properties and statistics), Reλ at various reference Weissenberg numbersWeref

and β concentrations, (a) in Prandtl-Karman coordinates and (b) in a Cε −Reλ plot. These samples arenear the maximum dissipation reduction. In (a), the lines joining the samples follow an ascending orderof (1− β) /β, being the lowest, (the starting point), in the lower-right corner, (considering the set of pointswith Weref 6= 0). In (b) it should be interpreted accordingly. Table 4.8 may also be used to identify thepoints.

coordinates, C−1/2ε − C1/2

ε Reλ, and in a coordinate system of the type Cε −Reλ. Both represent also the

Newtonian curve, the Weref = 0 curve. The lines connecting the points are linking them in ascending

order of (1− β) /β being the lowest, (the starting point), in the lower-right corner of curves represented

in Prandtl-Karman coordinates, while the representation in Cε −Reλ coordinates should be interpreted

accordingly and table 4.8 might help. The maximum deviation from the Newtonian curve in both plots

seems to converge independently from the polymer properties. A further investigation should be per-

73

formed in order to confirm a maximum deviating Cε (Reλ) curve independent from polymer properties,

namely for different Rerefλ maintaining a meaningful variation of the (β; τp) set of polymer variables.

74

Chapter 5

Conclusions

In this chapter the achievements of this thesis will be summarized. In addition, some proposals about

eventual future work are also done.

5.1 Achievements

In previous chapters DNS of several cases were conducted regarding different subjects. In chapter

4 they can be seen details on them. First, Newtonian simulations were performed to study the charac-

teristics of IVS for higher Reynolds numbers. Then a comparative study between a purely viscous fluid,

simulated with the Carreau-Yasuda model, and a viscoelastic fluid, simulated with the FENE-P model,

for the same shear-thinning behavior, took place. Finally, a survey with respect to FENE-P alone was

done concerning turbulent dissipation features.

Newtonian simulations performed regarding the study of IVS brought numerical results for forced

HIT for Reλ up to 429, greater than those found in literature, see table 2.1. It allowed to confirm

scalability laws already presented such as 〈Rivs〉 ≈ 4η, 〈Rivs〉 ≈ 0.98RB , 〈Uivs〉 ≈ 0.85u′. Other statis-

tic, 0.23 < 〈ω0〉 /(ω′Re

1/2λ

)< 0.41, shows, as well, agreement with results from Jimenez [21]. The

circulation-based Reynolds number relates with Reλ as 10.1 < ReΓ/Re1/2λ < 13.7, while similar values

can be found in table 2.1. The major achievement refers to the scalability law for 〈Livs〉, which was

shown to be 〈Livs〉 ∼ η. It was verified that although 〈Livs〉 can attain values O (L11), it scales, in fact,

with η. It can also be seen that the convergence of this statistic occurs for a higher Reynolds than that

for the others.

The comparative survey between a purely viscous fluid and a viscoelastic one, for the same shear-

thinning behavior, was important to depict the rather different dynamics underlying each phenomenon.

Basically, in the Carreau-Yasuda case, by an assessment of the kinetic energy evolution equation at

statistically steady state conditions, Newtonian-like spectra were seen for scales greater than the small-

est. Indeed, it was expected to see an agreement with Kolomogorov’s Second Similarity Hypothesis, as

the local Reynold number Re` increases upwards the cascade, and a viscosity-independent dynamics

that becomes relevant. On the other hand, FENE-P dynamics showed that the elastic mechanism alters

75

deeply the spectra. It was concluded that a purely viscous shear-thinning model, like Carreau-Yasuda,

could not simulate the energy flux cascade mechanism of a viscoelastic fluid, like FENE-P. Spectra

assessments revealed that the FENE-P elastic mechanism, even for high Reλ, might be felt from the

inertial range up to the energy-containing range, becoming, this influence, specially stronger as Weref

increases. Comparing statistics like ε with ε[p] or ε∞ with ε[s] clearly denotes that the strain correlation

structure is largely different between these fluid cases.

Finally, numerical experiments with the FENE-P model were done to explore the onset of dissipation

reduction and the Cε behavior near maximum dissipation reduction conditions. For the former, it was

tested de Gennes’ theory, specifically the scalability law (1− β) /β ∼We1−15p/4, which it predicts at

onset conditions. The results for two Rerefλ indicate a probable We exponent 1− 15p/4 ≈ −11/4, giving

the stretching exponent p a value p ≈ 1. This p value, as expected, was found to be greater than that for

pipe flow. According to Sreenivasan [48], that value, (for pipe flow), was determined to be p ≈ 2/3. Thus,

the scaling found, (for HIT), is of the form (1− β) /β ∼We−11/4. For the latter case, which constitute

itself a preliminary study of Virk’s-like asymptote in forced HIT, showed, for one Rerefλ , approximately the

same limiting Cε value by β and Weref variations. As a smooth convergence to a possible limiting value

for Cε is present, for the Rerefλ tested, it may indicate that a region independent from polymer properties

may be present.

5.2 Future Work

For future work some ideas are purposed as follows.

1. Still regarding the IVS, a deeper study about their stability with increasing Reλ may be done to

understand how IVS remain stable, due to the fact that ReΓ also increases.

2. Further explore the possible presence of a Virk’s-like asymptote in forced HIT.

3. Explore the behavior of passive scalars, e.g., heat, in viscoelastic fluids.

4. Begin to look at turbulence with MHD dynamics, as it is largely different from classical turbulence,

specially in what it concerns to small scale statistical structure.

76

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