Direct Numerical Simulation of Turbulent Planar Jets with

Polymer Additives

Nuno Filipe Cabral Pimentel

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

November 2018

ii

”Nothing in this world can take the place of persistence. Talent will not; nothing is more common than

unsuccessful people with talent. Genius will not; unrewarded genius is almost a proverb. Education will

not; the world is full of educated failures. Persistence and determination alone are omnipotent.”

- Calvin Coolidge

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Acknowledgments

Firstly, I would like to start thanking Professor Carlos Silva for giving me the opportunity to work in this

area. Also, for his availability to explain and teach many interesting things in the area, along with his

advices during the thesis.

I would like to thank Mateus Guimaraes, for the data post-processing and obtaining the first results

with the the algorithm developed within this work.

I would like to give a special thanks to my IST colleagues and friends Hugo Abreu and Afonso Ghira

for introducing me to Professor Carlos Silva and for the meaningful discussions throughout the thesis

development, which were crucial to clarify many things.

I would also like to give a special thanks to Millennium BCP, namely Dr. Antonio Carreira, for providing

a crucial time off work required to develop this work and supporting me throughout this thesis.

I would like to thank my friends and colleagues Bruno Conceicao, Diogo Antunes, Ricardo Verıssimo

and Pedro Abracos for their continuous support and motivation.

Finally, I would like to thank my family for supporting me throughout all my IST course, for their

understanding and motivation. This work is devoted to them.

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Resumo

Neste trabalho foram efetuadas simulacoes numericas diretas (DNS) do jato espacial plano turbulento

com moleculas polimericas para estudar o mecanismo de interacao do polımero no avanco do es-

coamento do jato. Estas simulacoes foram conseguidas com a implemtancao do modelo numerico

de fluido visco-elastico, representado pelo modelo constitutivo reologico FENE-P. O modelo numerico

para a equacao de transporte do tensor de conformacao foi adaptado as condicoes de fronteira nao

periodicas do escoamento, nao estando presente na literatura referencias ao caso em estudo. Na

concretizacao deste, foi tido em conta o desempenho computacional da simulacao, para o qual se

implementou um mecanismo de celulas fantasma por forma a reduzir o tempo de computacao. O

algoritmo desenvolvido foi alvo de uma extensa verificacao numerica, de forma a garantir a resolucao

correta das equacoes governantes para o escoamento com moleculas de polımero diluıdas. O algoritmo

e testado em simulacoes numericas diretas com variacao das caracterısticas fısicas das moleculas do

polımero, nomeadamente a concentracao polimerica e tempo de relaxacao. Os resultados permitiram

observer um decrescimo da espessura do jato com a presenca de fluido visco-elastico, assim como

uma reducao da dissipacao da energia viscosa. De notar que o algoritmo implementado representa um

enorme progresso nas simulacoes numericas de escoamento viscoelastico turbulento de jatos espaci-

ais, fornecendo os primeiros resultados numericos em estudos do genero.

Palavras-chave: Turbulencia visco-elastica, DNS, FENE-P, Jato plano turbulento espacial

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Abstract

In this study it was performed Direct Numerical Simulations (DNS) of a spatial turbulent planar jet with

polymer additives to further understand the mechanism of polymer interaction on jet flows. These sim-

ulations were achieved with the implementation of the visco-elastic fluid numerical model, represented

by the rheological constitutive FENE-P model. The numerical model for the conformation tensor trans-

port equation was adapted for the non-periodic boundary conditions of the flow, for which no references

are present in the literature. The development of the numerical algorithm took into consideration the

computational performance of the simulations, in which it was implement a ghost cell mechanism to

decrease the computation time. The developed algorithm has been extensively verified to ensure the

correct resolution of the governing equations for turbulent jet flow with polymer additives. The numerical

model was tested in direct numerical simulation for different polymer molecules physical characteristics,

namely the polymeric concentration and the relaxation time. The observed results verified a decrease in

the jet width on the presence of a visco-elastic flow, together with a decrease of the viscous energy dis-

sipation rate. It should be noted that the implemented numerical model represents a major progress on

the numerical simulations of turbulent spatial jet flow with polymer additives, providing the first numerical

results for this kind of study.

Keywords: Visco-elastic turbulence, DNS, FENE-P, Spatial turbulent planar jet

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Contents

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

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

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

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

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 Turbulent Planar Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Turbulence in dilute polymer solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Experimental discoveries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 High performance computing and parallelization strategies . . . . . . . . . . . . . . . . . . 15

2.4.1 Parallelization Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Load Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.3 Granularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.4 Distributed vs. Shared memory parallelization . . . . . . . . . . . . . . . . . . . . . 17

2.4.5 Hybrid parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Governing equations and Numerical Methods 21

3.1 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 The FENE-P constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Properties of the conformation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5.1 Pseudo-spectral scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

xi

3.5.2 Compact finite differences discretization . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.3 Conformation Tensor Transport Equation . . . . . . . . . . . . . . . . . . . . . . . 30

3.6 Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6.1 Pressure-velocity coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6.2 Non-Linear Term in the Momentum Equation . . . . . . . . . . . . . . . . . . . . . 36

3.6.3 Conformation Tensor Transport Equation . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Stability condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.8 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.8.1 Inlet boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.8.2 Lateral boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.8.3 Outlet boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.8.4 Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.9 Code architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.9.1 Parallel architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.9.2 Parallel domain decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.9.3 FFT calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.9.4 I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Code Verification 47

4.1 Conformation Tensor Transport Equation Verification . . . . . . . . . . . . . . . . . . . . . 48

4.2 Velocity - Conformation Tensor Coupling Term Verification . . . . . . . . . . . . . . . . . . 58

5 Results 63

6 Conclusions 73

6.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography 75

A Stability condition proof 81

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List of Tables

3.1 Gauss-Legendre integration nodes and weights. From [60]. . . . . . . . . . . . . . . . . . 33

4.1 Simulation parameters for transport equation verification. . . . . . . . . . . . . . . . . . . 50

4.2 Coupling term simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1 Computational domain parameters for the simulations. Number of grid points (nx, ny, nz);

non-dimensional grid size (Lx/h, Ly/h, Lz/h). . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Physical parameters for the simulations. Weissenberg number Wi; slot width, initial max-

imum jets velocity, solvent viscosity based Reynolds number Re; Averaged Taylor turbu-

lence micro-scale centreline velocity solvent viscosity based Reynolds number on the x

streamwise direction Reλ; polymer relaxation time τp; ratio of the solvent to solvent plus

polymer viscosity β; polymer normalized maximum extensibility L; mesh resolution nor-

malized by the Kolmogorov small scale at x/h = 14. . . . . . . . . . . . . . . . . . . . . . 64

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List of Figures

2.1 Examples of turbulent flows in engineering applications and nature. . . . . . . . . . . . . . 3

2.2 Time history of the axial component of velocity on the centreline of a turbulent jet. From

[11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Energy spectrum at high Re under the Kolmogorov hypothesis for a Newtonian fluid. Flow

scales are expressed as the log of the wavenumber, k = 1/l. L is the flow characteristic

length, l0 is the characteristics size of the largest eddies and η is the Kolmogorov scale.

lEI is the start of the inertial sub-range and lDI is the start of the dissipation sub-range.

From [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Model spectrum for turbulence in polymer solutions indicating the three regions: (I) κ <

1/lp, Kolmogorov’s inertial cascade; (E) 1/lp < κ < 1/ηp, Elastic subrange where fractions

of turbulence kinetic energy are transferred to elastic energy, which is then dissipated by

viscous drag of relaxing polymers and internal friction between the monomers of a single

polymer; (V) κ > 1/ηp, Viscous dissipation range where turbulence kinetic energy is

dissipated by viscous forces, just as the Newtonian spectrum. From [17]. . . . . . . . . . 6

2.5 Turbulent planar jet. From [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6 Turbulent planar jet regions. From [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7 Turbulent mixing of jets of water (left) and PEO solution (right). From [25]. . . . . . . . . . 9

2.8 Turbulent boundary layer velocity profile. From [26]. . . . . . . . . . . . . . . . . . . . . . 10

2.9 Schematic illustration of the onset of drag reduction and the maximum drag reduction

asymptote. The dotted line represents the case with a fixed polymer concentration C

and increasing Re. The dashed line is the case for a fixed Re, where the onset of drag

reduction is first observed, and the polymer concentration C is increased. f refers to the

friction drag for pipe flows. From [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.10 Mean velocity profiles during drag reduction. From [23]. . . . . . . . . . . . . . . . . . . . 12

2.11 FENE-P dumbbell model with a spring between two beads. From [35]. . . . . . . . . . . . 13

2.12 One-dimensional schematic of a shock (thick, solid line). From [36]. . . . . . . . . . . . . 14

2.13 Evolution of efficiency with the number of processors. From [18]. . . . . . . . . . . . . . . 16

2.14 Illustration of OpenMP architecture. From [50]. . . . . . . . . . . . . . . . . . . . . . . . . 17

2.15 Illustration of MPI architecture. From [50]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.16 Illustration of Hybrid MPI+OpenMP architecture. From [50]. . . . . . . . . . . . . . . . . . 19

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3.1 View of the computational domain with reference frame and notation. From [16]. . . . . . 24

3.2 Mean velocity profile imposed at the inlet. H stands for the width of the inlet slot. From [18]. 39

3.3 View of the domain partitioning with reference frame and notation. . . . . . . . . . . . . . 43

3.4 Ghost cell configuration data exchange on the streamwise direction for a domain parti-

tioned in 3 normal slices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1 u velocity profile in the normal y direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Numerical solution for the Cij tensor considering a u velocity profile in the normal y direction. 51

4.3 Numerical (n) and analytical (a) solutions for the Cii tensor considering a u velocity profile

in the normal y direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Numerical (n) and analytical (a) solutions for the Cij , i 6= j tensor considering a u velocity

profile in the normal y direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 u velocity profile in the spanwise z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6 Cij numerical (n) and analytical (a) solutions comparison considering a u velocity profile

in the spanwise z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.7 w velocity profile in the normal y direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.8 Cij numerical (n) and analytical (a) solutions comparison considering a w velocity profile

in the normal y direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.9 v velocity profile in the spanwise z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.10 Cij numerical (n) and analytical (a) solutions comparison considering a v velocity profile

in the spanwise z direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.11 w velocity profile in the streamwise x direction. . . . . . . . . . . . . . . . . . . . . . . . . 56

4.12 Cij numerical (n) and analytical (a) solutions comparison considering a w velocity profile

in the streamwise x direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.13 v velocity profile in the streamwise x direction. . . . . . . . . . . . . . . . . . . . . . . . . 57

4.14 Cij numerical (n) and analytical (a) solutions comparison considering a v velocity profile

in the streamwise x direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.15 C12 and C13 profiles prescribed for the coupling term on the u velocity component. H is

the unit of length used in the code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.16 Coupling term for u velocity for numerical (red) and analytical (blue) solutions. . . . . . . . 60

4.17 C12 and C23 profiles prescribed for the coupling term on the v velocity component. h is

the unit of length used in the code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.18 Coupling term for u velocity for numerical (red) and analytical (blue) solutions. . . . . . . . 61

4.19 C13 and C23 profiles prescribed for the coupling term on the w velocity component. h is

the unit of length used in the code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.20 Coupling term for u velocity for numerical (red) and analytical (blue) solutions. . . . . . . . 61

5.1 Streamwise evolution of non-dimensional (a) jet’s half-width and (b) centreline velocity for

fluids with different relaxation time for cases A, J, K and L. From [69]. . . . . . . . . . . . . 66

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5.2 Normalized mean velocity: (a) streamwise and (b) normal direction components, (c-e)

velocity root mean squared components and (f) Reynolds shear stress profiles for fluids

with different relaxation time for cases A, J, K and L. From [69]. . . . . . . . . . . . . . . . 67

5.3 Mean (a-c) and root mean squared (d-f) components of the conformation tensor for fluids

with different relaxation time at x/h = 12 for cases J, K and L. From [69]. . . . . . . . . . . 68

5.4 Centreline root mean squared evolution of (a) streamwise, (b) normal and (c) spanwise

directions velocity components and (d-f) conformation tensor components for fluids with

different relaxation time for cases J, K and L. From [69]. . . . . . . . . . . . . . . . . . . . 69

5.5 Stream-wise evolution of centreline Weissenberg numbers for all visco-elastic cases.

From [69]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6 Stream-wise evolution of centreline turbulent kinetic energy dissipation of the solvent for

all cases. From [69]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.7 Stream-wise evolution of centreline mean conformation tensor components for cases J, K

and L. From [69]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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Nomenclature

API Application Programming Interface.

CFD Computational Fluid Dynamics.

CFL Courant-Friedrichs-Lewy condition.

DFT Discrete Fourier Transform.

DNS Direct Numerical Simulations.

FENE-P Finitely Extensible Non-linear Elastic Peterlin model.

FFT Fast Fourier Transform.

HIT Homogeneous Isotropic Turbulence.

I/O Input/Output.

KT Kurganov-Tadmor.

LASEF Laboratory of Fluid Simulation in Energy and Fluids.

MDR Maximum Drag Reduction.

MHR Magneto-hydrodynamics.

MPI Message Passing Interface.

NUMA Non-uniform memory access.

PEO Polyethylene Oxide.

ppm parts per million.

SPD Symmetric Positive Definite.

T/NT Turbulent/Non-turbulent interface.

Greek symbols

αp, βp Runge-Kutta Coefficients.

β Polymer Concentration.

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∆t Time step.

δ Flow width.

δij Kronecker’s delta.

ε Turbulent kinetic energy dissipation.

η Kolmogorov length scale.

λ Taylor microscale.

λi Eigenvalues of Cij .

µ Dynamic Viscosity.

ν Kinematic Viscosity.

ρ Fluid density.

τη Kolmogorov time scale.

τp Zimm relaxation time of the polymer.

ξi Gauss-Legendre integration nodes.

Roman symbols

C Conformation tensor.

H Convective flux tensor.

u Velocity vector.

Ai Gauss-Legendre integration weights.

Cij Components i, j of the Conformation tensor.

E(k) Kinetic energy density function.

f(Ckk) Peterlin function.

h Width of the inlet slot.

k Wavenumber.

L Maximum polymer extension.

l Characteristic length scale.

Lx Computational domain length in the x direction.

Ly Computational domain length in the y direction.

Lz Computational domain length in the z direction.

xx

nx Number of grid points in the x direction.

ny Number of grid points in the y direction.

nz Number of grid points in the z direction.

p Pressure.

ri Instantaneous orientation of a polymer dumbbell.

Re Reynolds Number.

Reλ Reynolds Number based on the Taylor scale.

Sij Strain Rate tensor.

t Time.

Tij Stress tensor.

u First component (streamwise) of the velocity vector.

ui Velocity vector.

U1L Mean local co-flow velocity.

U1 Co-flow velocity.

U2 Peak velocity at the inlet.

Uc Mean local centreline velocity.

v Second component (normal) of the velocity vector.

w Third component (spanwise) of the velocity vector.

x First spatial (streamwise) coordinate.

y Second spatial (normal) coordinate.

z Third spatial (spanwise) coordinate.

Subscripts

i, j, k Computational indexes.

x, y, z Cartesian components.

Superscripts

[p] Polymer.

[s] Solvent.

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Chapter 1

Introduction

The objective of the current chapter is to describe the motivation underlying the present work and to

provide a brief description of the outline of the document.

1.1 Motivation

Since Toms (1948) [1] reported turbulent drag reduction by the addition of long-chain polymers by up to

80 percent, this effect has been studied for both practical and theoretical purposes. Currently, industrial

applications regarding polymer drag reduction are mostly related to long-distance liquid transportation

pipeline systems, marine vehicles and heating/cooling system water-circulating devices.

This work will focus on the study of this phenomenon on turbulent jet flows, to further understand the

mechanism of polymer interaction on jet flows and advance our understanding of fluid turbulence and

visco-elastic flows. It is important to note that the framework of this study has potential applications on

the aerospace industry, namely on heat transfer reduction by injection of micro-jets into turbine blades

[2]. There is a potential application on combining visco-elastic fluids with the micro-jet structures, as

the heat transfer behaviour of visco-elastic fluids has been studied recently, namely focused on cooling

applications for turbine disks [3, 4].

Moreover, it is possible to make a formal analogy between the Finite Extensible Nonlinear Elastic -

Peterlin (FENE-P) set of equations for visco-elastic flows and magneto-hydrodynamics (MHD) equations,

as mentioned in [5, 6]. Thus, the numerical model considered here to simulate polymer solutions (FENE-

P) will have a strong resemblance and application to the methodology used in numerical simulations of

MHD flows. As for aerospace industry applications, in recent years plasma flows have rapidly become

an important set of new technologies that find application to hypersonic propulsion, being evident a

growing interest in plasma-based flow manipulation through MHD forces [7–9]. This step, together with

the combining technological developments in electromagnetics, aerodynamics, and chemical kinetics,

might lead to a breakthrough in propulsion systems for improving aerospace vehicle performance [10].

As the current work will comprise the resolution of Navier-Stokes equations coupled with polymer

models using Direct Numerical Simulations (DNS), it is of the uttermost importance the optimization of

1

the computational performance of the numerical model. Such is entailed by considering parallelization

techniques for high performance computing, with transversal applications to numerical simulations in the

industry.

1.2 Objectives

The main objectives for the work presented here are:

1. To implement and verify a computational method to simulate turbulent jets with polymer additives

(FENE-P model) through Direct Numerical Simulations (DNS) on an existing turbulent jet DNS

code.

2. To optimize the computational performance of the resulting computational model.

3. To analyse the effects of polymer additives on the development of turbulent jets.

1.3 Thesis Outline

The present document is organized into six chapters.

In the first chapter, it is described the underlying usefulness of the current work, highlighting potential

future applications, along with a statement of the main objectives to be accomplished.

In the second chapter, it is introduced the main subjects fundamental to the present work. The topics

of turbulence and turbulent planar jet are presented, along with a review on flows with diluted polymer

solutions. It is also presented a review on parallelization techniques for high performance computing.

In the third chapter, it is described the constitutive equations concerning the velocity field and the

polymer transport equation, along with the detailed definition of the numerical methods that were used

to implement the equations for the spatial discretization, temporal advancement and the boundary con-

ditions. It is provided detail regarding the parallelization strategy, namely on the enhancements that

were developed during this work.

In the fourth chapter, it is presented the code verification with analytical solutions, for example, veri-

fication was performed with the Couette flow. The verification exercise was extended to all components

from the conformation tensor transport equation and, also, the coupling term on the velocity field equa-

tions.

In the fifth chapter, it is presented the results obtained from DNS of turbulent planar jet with polymer

additives. Several quantities are analysed for different polymer physical characteristics. It is worth

mentioning that these results are the first numerical results of DNS visco-elastic turbulent planar jet.

The conclusions are presented in the sixth chapter, as well as proposals for future work.

2

Chapter 2

Background

This chapter defines the concept of turbulence and planar jet flows, along with some of its characteristics.

Furthermore, it is introduced an overview of numerical methods for the phenomenon of polymer-induced

drag reduction (FENE-P) and High Performance Computing, with emphasis on parallelization strategies.

2.1 What is turbulence?

As highlighted by [11], turbulent flows are prevalent in engineering applications and nature. Indeed,

flows such as waterfalls, rivers or flows around vehicles (e.g. airplanes, vehicles) are of turbulent nature.

Figure 2.1 illustrates examples of this phenomena.

(a) Space shuttle Atlantis launch in 2006. Fromwww.nasa.gov/centers/marshall/history.

(b) View of the Mayon volcano at January 24, 2018.From Romeo Ranoco/Reuters.

Figure 2.1: Examples of turbulent flows in engineering applications and nature.

3

Turbulence is a characteristic of the flow, being generally associated with flows that are highly un-

steady with chaotic variations of velocity and pressure in space and time. The transition from a laminar

to a turbulent flow is defined by Reynolds number [12], which is given by:

Re =ρUl

µ=Ul

ν(2.1)

where µ is the molecular viscosity, ρ is the density and ν is the kinematic viscosity. U and l are the

characteristic velocity and length scale of the flow, respectively. In most cases, when Re exceeds a

certain critical value, the flow transits from laminar to turbulent flow. The Reynolds number expresses

the ratio of inertial/inviscid forces and viscous forces and indicates the flow’s reaction to disturbances,

because perturbations can be damped by viscosity or amplified by inviscid forces.

The turbulence chaotic nature is noted by watching the velocity variation in time measured on the

centreline of a turbulent jet, as illustrated on Figure 2.2.

Figure 2.2: Time history of the axial component of velocity on the centreline of a turbulent jet. From [11].

Despite the random behaviour observed on velocity U1(t), observations show that the average ve-

locity < U1 > and its fluctuations present a stable statistically behaviour, which justifies the study of

turbulent flows by means of statistics.

The first concept of energy transfer through scales in turbulence was introduced by Richardson [13],

which stated that turbulence is produced at a large scale and is progressively broken down into smaller

and smaller scales by inviscid forces, until reaching a sufficiently small scale where viscosity becomes

dominant and the mechanical energy is dissipated into heat.

This conceptual picture was translated into a quantitative theory by Kolmogorov [14], who summa-

rized the following hypothesis to define the energy transfer process in turbulence:

• The local isotropy hypothesis: At sufficiently high Reynolds number, the small-scale turbulent

motions are statistically isotropic.

• The first similarity hypothesis: At sufficiently high Reynolds number, the statistics of small-scale

4

motions have a universal form dependent only on viscosity and dissipation rate. This region is

named universal equilibrium range.

• The second similarity hypothesis: At sufficiently high Reynolds number, the statistics of motions

at scales that are much smaller than the integral scale and much larger than the smallest scale

have a universal form dependent only by the dissipation rate. This region is named inertial sub-

range. The zone in which the viscous effects start being dominant is named dissipation sub-range.

The smallest scales of turbulence are called the Kolmogorov scales. By using the first similarity

hypothesis, and performing a dimensional analysis, length (η), time (τη) and velocity (uη) are determined

by the viscosity ν and dissipation rate ε and are given by:

η ≡(ν3

ε

)1/4

, τη ≡√ν

ε, uη ≡ (νε)

1/4 (2.2)

Obukhov [15] formulated Kolmogorov’s hypothesis in the spectral space, in which the turbulence

eddies characteristics scale l is described as a function of a wave number k = 2π/l.

Figure 2.3: Energy spectrum at high Re under the Kolmogorov hypothesis for a Newtonian fluid. Flowscales are expressed as the log of the wavenumber, k = 1/l. L is the flow characteristic length, l0 is thecharacteristics size of the largest eddies and η is the Kolmogorov scale. lEI is the start of the inertialsub-range and lDI is the start of the dissipation sub-range. From [16].

The kinetic turbulence energy distribution along the spectrum of wave numbers is expressed in Figure

2.3. Ek is the turbulence kinetic energy, L is the characteristics length of the flow, l0 is the characteristics

size of the largest eddies and η is the Kolmogorov scale. lEI is the start of the inertial sub-range and

lDI is the start of the dissipation sub-range.

Using Kolmogorov’s second similarity hypothesis, that at the inertial sub-range viscous effects are

negligible, becoming only function of the dissipation rate and the eddies length scale l, it follows that the

5

energy spectrum is proportional to ε2/3κ−5/3 for a Newtonian fluid and expressed by:

E(k) = Ckε2/3κ−5/3 (2.3)

in which k is such that lEI > l > lDI , and Ck is the Kolmogorov’s constant.

For the case of a visco-elastic fluid, the polymer interaction with the flow leads to a steepening of

the kinetic energy spectrum beyond a wavenumber lp, which is the Lumley length-scale. In [17], it is

presented a model spectrum for the visco-elastic fluid consisting of 3 major regions: (I) Inertial cascade,

(E) Elastic subrange and (V) Viscous dissipation. This model spectrum is illustrated on Figure 2.4.

It is important to notice that the model spectrum for the visco-elastic fluid on Figure 2.4 has been

verified with experimental results by [17].

Figure 2.4: Model spectrum for turbulence in polymer solutions indicating the three regions: (I) κ < 1/lp,Kolmogorov’s inertial cascade; (E) 1/lp < κ < 1/ηp, Elastic subrange where fractions of turbulencekinetic energy are transferred to elastic energy, which is then dissipated by viscous drag of relaxingpolymers and internal friction between the monomers of a single polymer; (V) κ > 1/ηp, Viscous dis-sipation range where turbulence kinetic energy is dissipated by viscous forces, just as the Newtonianspectrum. From [17].

The (I) Inertial cascade on Figure 2.4 consists on a Newtonian inertial cascade at low wavenumbers,

whose governing equations have been above-mentioned.

According to [17], the (E) Elastic subrange is a spectral region separated from the inertial cascade

by the Lumley scale lp, which is determined by the elastic properties of the fluid and the turbulence

dissipation rate. In this region, fractions of the turbulence kinetic energy arriving from the inertial sub-

range are converted into elastic energy by polymer stretching. Subsequent relaxation of the stretched

polymers dissipates a part of this elastic energy due to the viscous drag of the polymer molecules in the

solvent and interactions between monomers of a single polymer. The remaining elastic energy is trans-

formed back to turbulence kinetic energy. This phenomenon is called back reaction and, consequently,

the energy flux on this region is continuously reduced from higher to lower wavenumbers.

6

Experimental data [17] suggested that this region follows a power-law with a slope of−3, as illustrated

on Figure 2.4, where the polymer stresses overcome the viscous stresses. The energy spectrum in a

turbulent structure of wavenumber k decreases according to:

E(k) = Ckε2/30 l5/3p (lpκ)

−3 (2.4)

where ε0 is equal to the energy flux from the Newtonian inertial cascade.

The (V) Viscous subrange consists on a region dominated by viscous stresses which transform

turbulence kinetic energy into heat, similarly to the dissipation range in the Newtonian spectrum, in

which the new viscous dissipation scale is given by:

ηp ≡(ν3

εv

)1/4

(2.5)

where εv is the energy flux contributing to the viscous dissipation at the end of the elastic subrange.

2.2 Turbulent Planar Jets

The turbulent plane jet flow is a type of flow belonging to the free shear flow type, which are charac-

terised by developing far away from the interaction with boundaries, and that advances along a preferred

direction. An example of a two-dimensional plane jet flow is presented in Figure 2.5.

Figure 2.5: Turbulent planar jet. From [18].

This type of flow is characterised by being statistically two-dimensional, with a dominant direction of

mean flow across the streamwise x direction and a mean velocity in the spanwise z direction of zero.

It is important to note some features of this kind of flow, namely that the velocity in the streamwise x

direction is much greater than the velocity in the normal y direction (u >> v), whilst the normal gradient

∂/∂y is much larger than the streamwise gradient ∂/∂x (∂/∂y >> ∂/∂x), as stated in [16]. Moreover, it

is worth mentioning the relation of flow scales for this type of flow, namely the shear layer thickness (δ)

and the flow streamwise length (L), for which L >> δ.

As the remaining shear flows, the turbulent plane jet is characterised by presenting a sudden change

from a turbulent rotational region, to an irrotational one. This transition is observed at the turbulent/non-

turbulent interface (T/NT), in which occurs exchanges of mass, momentum and other scalars while the

jet develops in the streamwise direction, spreading out. This dispersion of the flow makes it swell,

7

transmitting momentum to the adjoining fluid in the form of momentum and vorticity, dragging mass into

it in a phenomenon known as turbulent entrainment [16].

The plane jet can be split across different regions, as illustrated on Figure 2.6:

• The potential core region: It corresponds to the flow between the shear layers near the jet inlet,

originated by the high velocity gradient between the flow incoming from the jet inlet and that of the

remaining environment.

• The transition region: It comprises all the area in which the flow experiences transition into a fully

developed turbulent state.

• The self-similarity region: Region in which the flow characteristics, such as velocity and other

scalar properties, can be collapsed seamlessly after being properly scaled following the Kol-

mogorov’s hypothesis.

Figure 2.6: Turbulent planar jet regions. From [16].

2.3 Turbulence in dilute polymer solutions

2.3.1 Experimental discoveries

Since the discovery of the polymer drag reduction by Toms [1], there have been numerous experiments

done and theories proposed to explain the mechanisms underlying the drag reduction by dilute polymers

on turbulent flows. Indeed, Nadolink and Haigh [19] included over 2500 entries on the bibliography

review over this phenomenon. Nonetheless, Jin [20] refers that the most cited literature on polymer drag

reduction in boundary layers are Lumley [21, 22], Virk [23] and de Gennes [24], which also provide the

most successful attempts to explain the mechanism of polymer drag reduction.

8

Lumley [21] defined drag reduction as the reduction of skin friction in turbulent flow below that of

the solvent alone, by the addition of trace percentages of polymers. This phenomenon was observed

to occur in thermodynamically dilute solutions of long, flexible, expanded high-molecular-weight linear

polymers, considering typical polymer-solvent systems, such as the polyethylene oxide (PEO or Polyox)

in water. This one has become the most common solution due to its inexpensiveness, easiness to handle

and effectiveness (33% reduction in skin friction over that of water alone by the addition of 18 parts per

million (ppm) of PEO of molecular weight of 0.76× 106), as stated by [21]. An illustrative example of the

effect of PEO solution on turbulent flows is presented on Figure 2.7.

Figure 2.7: Turbulent mixing of jets of water (left) and PEO solution (right). From [25].

Further to that, Lumley [22] said that in regions where polymer molecules are elongated, viscosity is

enhanced. Moreover, Lumley stated the at sufficiently high wall shear stresses, the fluctuating strain rate

causes polymer molecules to expand and, thus, increasing viscosity. Furthermore, Lumley concluded

that in turbulent flows elongated polymer molecules are found in the wall layer, but not in the viscous

sub-layer, as vorticity and strain flow rate are not correlated (molecules do not expand, since there is

only simple shear on the viscous sub-layer).

Therefore, increased viscosity will only occur in the wall layer, leading to dissipation of turbulence

9

Figure 2.8: Turbulent boundary layer velocity profile. From [26].

fluctuations and mitigation of turbulence stresses. Small eddies on the wall layer will be damped by

the increased viscosity, and the resulting lower Reynolds stress at the buffer layer will thicken the vis-

cous sub-layer. The large eddies will expand on the viscous sub-layer and lead to higher streamwise

fluctuating velocities in this region. On the maximum drag reduction (MDR) regime large eddies will be

dominant.

Virk [27] compiled many experimental data available at that time on pipe flows with polymer additives

and smooth surfaces. Virk summarized three regimes of pipe flows with polymer additives:

• Laminar regime: Under a laminar flow regime, no drag reduction is observed, and the skin friction

obeys Poiseuille’s law, because the overall viscosity is very similar to the solvent viscosity in the

dilute solution.

• Polymeric regime: The flow is turbulent and the drag reduction effect is observed. Furthermore,

Virk states that the onset of drag reduction occurs at a universal characteristic wall shear stress,

and that the amount of drag reduction is dependent on the polymer characteristics, such as poly-

mer molecular weight and polymer concentration.

• Maximum drag reduction (MDR): It was determined the existence of a universal drag reduction

saturation asymptote, i.e. independent of system and polymer properties. The maximum drag

reduction is observed in the velocity profile and in the friction flow rate domain.

The drag reduction phenomenon appears to lie between two universal asymptotes: the Newtonian

turbulent flow (the so-called Prandtl-Karman law) and the MDR (see Figure 2.9). The experimental data

compiled in [23] confirmed the agreement with the MDR asymptote and drag reduction onset shear

stress relation. Indeed, it was concluded that MDR occurs when the effects of polymers are felt over all

scales, causing the buffer layer thickness to extend across the entire boundary layer.

10

Figure 2.9: Schematic illustration of the onset of drag reduction and the maximum drag reduction asymp-tote. The dotted line represents the case with a fixed polymer concentration C and increasing Re. Thedashed line is the case for a fixed Re, where the onset of drag reduction is first observed, and thepolymer concentration C is increased. f refers to the friction drag for pipe flows. From [28].

Moreover, Virk [23] observed and stated that the mean velocity profile can be divided into three zones

(see Figure 2.10):

• Standard viscous sub-layer: Usual region from the wall outward.

• Elastic sub-layer: This layer exists between the ”Newtonian plug” region and the usual viscous

sub-layer. It is characteristic of drag reduction and its extent increases with increasing drag reduc-

tion, up to the maximum drag reduction. On the latter situation, this layer is spread through the

entire cross section.

• Newtonian log layer: Also named ”Newtonian plug”, this layer has the universal slope logarithmic

profile. On this layer the velocity profile is shifted upwards, but parallel to the Newtonian law of the

wall.

The turbulence structure information can be analogously categorized into the same three zones.

Years later, experiments were performed in which it was observed that drag reduction initiates as

polymer enters lower inertial wall layer from above or below, suggesting that the source of the drag

reduction lies within the lower inertial range [24, 29]. Therefore, De Gennes [24] proposed a model based

on the elastic properties of the polymer particles, instead of the viscosity effects stated by Lumley. The

elastic theory postulates that the elastic energy stored by the partially stretched polymers is an important

variable for drag reduction and suppression of turbulence, and the increase in the effective viscosity is

small and inconsequential.

Moreover, the onset of drag reduction occurs when the cumulative elastic energy stored by the par-

tially stretched polymers becomes comparable with the kinetic energy in the buffer layer. The small

scales will be damped, leading to a buffer layer increase and reduced drag. The polymer concentration

11

Figure 2.10: Mean velocity profiles during drag reduction. From [23].

is included in the onset criterion because the cumulative elastic energy of the polymers is a function of

the concentration [24].

Recently, experiments have been made with fully developed three-dimensional turbulence that do

not support convincingly elastic theory. Instead, an approach based on an ”energy flux theory” is pro-

posed, that states that the turbulence energy flux in the cascade process is gradually reduced by the

energy transfer into the elastic motion of polymers, which becomes dominant in the small scales. An

experimental work that supports this approach is presented by [30].

2.3.2 Numerical simulations

Over the past 20 years, Direct Numerical Simulations (DNS) have played an increasingly important

role in the investigation of turbulent drag reduction mechanisms by polymer additives, namely for wall-

bounded turbulent flows. Despite the limitedness of DNS accuracy, due to the model for polymer stress,

inability to resolve all polymer scales and potential numerical instability, DNS provides the advantage of

describing the orientation of the polymer micro-structure in addition to the velocity field and Reynolds

stresses. This is especially relevant over laboratory experiments, allowing additional meaningful and

comprehensive insight over drag reduction mechanisms.

Typically, numerical simulations consider polymer molecules as two beads connected by an elastic

spring. The polymer dynamics are then described by the evolution of the end-to-end vector connecting

the two beads (see Figure 2.11), using constitutive equations such as Oldroyd-B and FENE-P [31].

Recently, the FENE-P polymer model has been widely used in DNS to study turbulence polymer drag

reduction, in such works as [32], [33] and [34].

The polymer orientation is represented as a continuous second-order tensor field, the so-called con-

formation tensor. This conformation tensor is defined as the normalised second moment of the end-to-

12

Figure 2.11: FENE-P dumbbell model with a spring between two beads. From [35].

end vector between the two beads [36], described as:

Cij =〈rirj〉

13 〈r2〉eq

(2.6)

where ri is the instantaneous orientation of a polymer dumbbell, r2eq is the square of the equilibrium

separation distance, and the angle brackets imply an ensemble average over the configuration space of

the dumbbell.

The conformation tensor Cij is a symmetric and positive definite (SPD) matrix. This property chal-

lenges the DNS of turbulence coupled FENE-P, as it is required that Cij remains semi-positive. While

mathematically both constraints are satisfied by the governing equations, these properties can be lost

due to cumulative numerical errors. Early attempts at numerical simulation of visco-elastic turbulence

were plagued by Hadamard instabilities that resulted from the numerical loss of the SPD property [37].

Sureshkumar and Beris [38] overcame these instabilities by introducing a stress diffusion term into

the equation for the conformation tensor. Variations of this approach were used in several investigations.

In 1997, the first DNS of channel flow [32] was able to show the Drag Reduction phenomenon, although

with lower Reynolds numbers in the numerical simulations in comparison to the conditions under which

Drag Reduction is experimentally observed with dilute polymers.

Vaithianathan [39] exploited the SPD property of Cij to derive independent equations for the eigen-

vectors and eigenvalues of the conformation tensor. In this formulation, Cij must remain greater than

zero and the eigenvalues of Cij should comply with the finite extensibility of the polymer:

λ1 + λ2 + λ3 ≤ L2 (2.7)

where λi are the eigenvalues of Cij and L is the maximum polymer extension (non-dimensional). Even

though this implicit formulation guarantees that Equation 2.7 is satisfied, the compact finite-difference

method that was used by [40] did not guarantee the eigenvalues remain positive. Instead, realizability

was enforced by setting the negative eigenvalues to zero before constructing the conformation tensor,

ensuring numerical stability. However, the uncontrolled, spatially distributed adjustments of the eigenval-

ues destroyed overall conservation of the conformation tensor, and spatial averages of the conformation

tensor contained spurious contributions from the convective term.

13

Thus, the decomposition applied by [39] guaranteed stability (by providing easy access to the eigen-

values) but did not guarantee conservation. This issue can be traced back to early numerical approaches

to compressible flows, which often suffered from loss of conservation [41]. It is related with the hyper-

bolic nature of the equation for Cij in the Oldroyd-B, FENE-P and Giesekus models, which admits shocks

(discontinuities) in the polymer stress tensor [42]. Discontinuities in the polymer stress cannot be fully

resolved by a grid, and so the main responsibility of the numerical scheme is to correctly predict the

jump magnitude. Jumps in the conformation tensor should satisfy the overall conservation balance to

guarantee correct elastic wave propagation. This behaviour is illustrated on Figure 2.12.

Figure 2.12: One-dimensional schematic of a shock (thick, solid line). Black dots represent the gridpoints. Thick dashed line is an ideal representation of the shock on the grid. A spectral representation,without an artificial stress diffusivity, would look like the thin dashed line, with overshoots and under-shoots (Gibbs phenomenon). The dotted line indicates the effect of adding the stress diffusivity to thespectral representation. From [36].

The solid line indicates a jump in the polymer stress tensor across a discontinuity. The other curves

illustrate the equivalent numerical representation based on a finite-difference and spectral scheme. The

Gibbs phenomenon observed in the spectral representation can be attenuated by introducing an artificial

diffusivity. However, artificial diffusion can also reduce the magnitude of the jump. There are more

sophisticated approaches to filtering the spectral modes to (just) eliminate the ringing, but they are still

at a relatively early stage of development [43]. In contrast, specific finite-difference schemes have been

designed to maintain the magnitude of the jump and avoid excessive spreading of the discontinuity.

Early hyperbolic solvers were first-order in space, robust and reliable, yet often overly dissipative

[41]. More recent approaches have overcome these shortcomings while still preserving the simplicity

of implementation and robustness. The approach taken by [36, 44] consists on an algorithm based

on the method of Kurganov and Tadmor (KT). This second-order scheme guarantees that a positive

scalar will remain so at all points and it was generalized to guarantee that an SPD tensor also remains

SPD. Furthermore, the method dissipates less elastic energy than methods based on artificial diffusion,

resulting in stronger polymer–flow interactions. This approach has been successfully used in DNS of

Shear Flow, Decaying and Forced HIT [45–47], and, as such, will be used in this dissertation.

14

2.4 High performance computing and parallelization strategies

As stated in [48], there is an increasing need of computational resources to perform increasingly complex

DNS simulations. In the case of large computational simulations of fluid flow, and to take advantage of

these resources, it is necessary to develop codes that can run in parallel and are suited to the cluster

configurations that exist nowadays [49].

2.4.1 Parallelization Efficiency

In this subsection, it is presented the metrics used for the performance of the code, namely the speed-up

and the efficiency of the parallel code. While the speed-up provides a good measure of the performance,

it would be irrelevant if the number of processors was not known, while the efficiency provides a way of

comparison between the different configurations.

The speed-up is defined as the ratio between the sequential time ts and the parallel equivalent tp,

given by equation (2.8):

Speed− up =tstp

=σ(n) + ψ(n)

σ(n) + ψ(n)p + κ(n, p)

(2.8)

in which σ is the time in inherently serial tasks, ψ is the time spent in parallelizable tasks, p is the number

of processing units, n is a measure of the problem size (such as mesh points), and κ is the measure of

communication introduced by parallelization, which is a function of both the problem size and the number

of processors. The σ(n) + ψ(n) reflects the total sequential time of the code. It is convenient to define

this metric in terms of fractions of the code with reference to the total time. Thus, it is introduced the

parallel Fp and sequential Fp fractions of the code. Ideally the parallel fraction would be 100%, but it is

not possible since there are always some initialization procedures that cannot be performed in parallel.

The parallel and sequential fractions are related according to equation (2.9):

Fp =ψ(n)

σ(n) + ψ(n)= 1− Fs (2.9)

If one considers that κ is neglectable, it is obtained the Amdahl’s Law from (2.10). Amdahl’s Law provides

an upper limit to the performance of a parallel code and it shows that unless the sequential fraction is

very small, there will be no gain in increasing the number of processors.

Speed− up =1

1 + Fp

(1p − 1

) (2.10)

The efficiency of a parallel code (equation (2.11)) is given by dividing the speed-up by the number

of processes. It provides a comparable measure of performance between configurations that differ in

problem size and/or number of processors.

Efficiency =1

p+ Fp (1− p)(2.11)

In Figure 2.13 it is shown the evolution of efficiency with the number of processors. It becomes clear why

15

scalability is critical, as larger numbers of processors will yield less and less performance improvements.

Figure 2.13: Evolution of efficiency with the number of processors. From [18].

2.4.2 Load Balancing

Load balancing is an important concept to assess performance of parallelization code. It consists on

dividing a task among different processing units in a synchronized way, that is each unit takes the same

amount of time to complete its part, without having processes finishing earlier than the others and

waiting, resulting in idling resources. The slowest process will be the limiting factor when assessing load

balancing.

Another form of imbalance is related to memory imbalance between processors. On machines with

little memory per node, load imbalance can become burdensome.

2.4.3 Granularity

Code granularity refers to measure between computation and communication, once a process in parallel

programming is split across multiple processing units that work simultaneously. The following types of

granularity are defined in [49]:

• Coarser grain: The task is divided in large components, with greater computation work between

communication stages, which implies a better opportunity for performance increase. The load

balancing is more difficult to achieve, with possibly longer synchronization waiting periods.

• Finer grain: The task is divided into small components and less computation is required between

communication events. Despite being an advantage for load balancing, it will lead to higher com-

munication overheads, thus leading to potential worse performance.

16

2.4.4 Distributed vs. Shared memory parallelization

Shared memory parallelization

Shared memory parallelization refers to code being executed by various processors accessing a shared

memory. Thread-based models deal with each thread behaving as a processing unit that has no sepa-

rate memory of its own and shares its memory with the other threads, each thread doing its part of the

processing on the same shared memory, contributing to the desired result. This implies that all threads

or processes see and have access to the same memory. An illustration of this set-up is presented in

Figure 2.14.

Figure 2.14: Illustration of OpenMP architecture. From [50].

The standard application for shared memory configuration is OpenMP. It uses the Fork-Join model, in

which the master thread is responsible for synchronizing the run and distributing the code amongst other

threads. Then, the code is executed between threads and, after the parallelization step, the remaining

threads are joined with the master thread so the code proceeds in sequential mode.

According to [50], this configuration presents the following limitations:

• All processes or processors demand access to the same memory. Even though a software layer

that deals with simulating a shared memory can be used, the performance will be severed.

• Scalability is physically limited by the size of the shared memory node on which it is running.

• Some variables and locations in memory also must be duplicated to avoid race conditions, where

multiple threads try to update the same shared variable simultaneously. Care must be taken not to

overlap multiple processors changing the same place in memory simultaneously.

• Not suitable for dynamic problems (i.e. when the workload fluctuates rapidly during the execution).

Distributed memory parallelization

Contrary to shared memory parallelization, distributed memory relies on communication between pro-

cessing units, each one with its individual memory.

17

Message Passing Interface, or MPI, is the standard API used to parallelize programs that run on

distributed memory architectures. It provides different types of communications, such as point-to-point

and collective (e.g. gathering or broadcasting of data). The communications must be explicitly handled

by the user. Moreover, it provides other functionalities, such as parallel I/O and derived datatypes. An

illustration of this configuration is presented in Figure 2.15.

Figure 2.15: Illustration of MPI architecture. From [50].

It is important to note that the current work considers a parallelized code under MPI.

According to [50], this configuration presents the following limitations:

• Final speed-up limited by the purely sequential fraction of the code.

• Scalability is limited due to the additional costs related to the MPI library and load balancing man-

agement.

• MPI is a flat model, treating each MPI process as a separate physical processor and memory,

independently of physical architecture. This means that unnecessary communications might take

place between processes that physically have access to the same memory.

• Certain types of collective communications become more and more time-consuming as the number

of processes increases, namely MPIAlltoall, which is used in this work.

• As each processor gets its own portion to process before the task is under-way, the assumption of

equal load distribution is taken, which might not be the case as the processing progresses. This

load unbalance might be accounted for by the user, but it involves great programming effort as MPI

is not developed with this taken into consideration, and overheads of more frequent communication

for load distribution appear.

2.4.5 Hybrid parallelization

Hybrid parallel programming consists of mixing several parallel programming paradigms in order to ben-

efit from the advantages of the different approaches. In general, MPI is used for communication between

18

processes, and another paradigm, such as OpenMP, is used inside each process. An illustration of this

configuration is presented in Figure 2.16.

Figure 2.16: Illustration of Hybrid MPI+OpenMP architecture. From [50].

A hybrid programming configuration presents the following advantages in comparison to the dis-

tributed and shared memory [49]:

• Improved scalability through a reduction in both the number of MPI messages and the number of

processes involved in collective communications. This gain is specially verified if the non-hybrid

code uses communications of type MPI AlltoAll.

• More adequate to the architecture of modern supercomputers (e.g. interconnected shared-memory

nodes, NUMA machines, ...), whereas MPI used alone is a flat approach.

• Optimization of the total memory consumption, due to the OpenMP shared-memory approach;

less replicated data in the MPI processes. According to [49], benchmark of pure MPI to Hybrid

provided a memory gain between 80% to 480%.

• Fewer simultaneous accesses in I/O and a larger average record size, with potential significant

time savings on a massively parallel application.

• Better load balancing, by joining OpenMP load balancing features with MPI.

However, implementation of hybrid programming comes with higher level of complexity, with gains in

performance not being guaranteed according to the final application [50].

19

20

Chapter 3

Governing equations and Numerical

Methods

This chapter describes in detail the implementation of the Navier-Stokes equations that govern viscous

fluid flows, considering a coupling mechanism to represent the polymer additive-turbulence interaction,

along with the numerical methods considered in the main code that were used for the simulation of turbu-

lent planar jet with polymer additives. It is highlighted the discretization schemes considered throughout

jet simulations, as well as the parallelization techniques.

It is important to note that the starting point of the numeric work here presented is a DNS code for

turbulent jet described in Ricardo Reis’ PhD thesis [16] and Diogo Lopes’ PhD thesis [18]. Both thesis

were concerned about DNS of Newtonian fluids, presenting extensive verification and validation of the

DNS of turbulent jets. Furthermore, the core of the parallelization strategy was developed in the above-

mentioned works. Subsequently, the main features that were added to the DNS code are as follows:

• Introduction of the Conformation tensor transport equation;

• Coupling of the Navier-Stokes momentum equation with the Polymer Stress Tensor;

• Use of ghost cells data on streamwise slab configuration.

An overview regarding the discretization schemes considered on this work is now presented:

• The Navier-Stokes momentum equation is solved considering a pseudo-spectral scheme for spa-

tial discretization, together with fully explicit third order Runge-Kutta for temporal advancement;

• The Conformation tensor transport equation is solved according to the following structure:

– The convective term is discretized according to the Kurganov-Tadmor (KT);

– The stretching term is discretized using the finite difference method;

– The method used for temporal advancement is the explicit third order Runge-Kutta method.

21

3.1 Velocity field

On the velocity field governing equations, it is assumed that the fluid is incompressible and that it satisfies

the continuity and momentum equations with an additional term related with divergence of polymer

stress. Below it is presented the continuity and momentum, respectively:

∂ui∂xi

= 0 (3.1)

∂ui∂t

+ uk∂ui∂xk

= −1

ρ

∂p

∂xi+

1

ρ

∂Tij∂xj

(3.2)

where ui is the i component of the velocity vector, ρ is the constant fluid density, p is the local pressure

and Tij is the combination of the viscous and polymer stress tensors. The stress tensor can be ex-

pressed as a linear sum of contributions from the Newtonian stress (T [s]ij ) and the polymer stress (T [p]

ij ),

as shown below:

Tij = T[s]ij + T

[p]ij (3.3)

where the Newtonian stress tensor T [s]ij is given by:

T[s]ij = 2ν[s] ∂Sij

∂xj(3.4)

being ν[s] the kinematic viscosity of the fluid. The Sij is the Strain Rate Tensor defined as:

Sij =1

2

(∂ui∂xj

+∂uj∂xi

)(3.5)

The Polymer Stress Tensor T [p]ij is described in the next sub-section.

3.2 The FENE-P constitutive model

Among the conformation models used in turbulence polymer simulations, the Finitely Extensible Non-

Linear Elastic Peterlin Model (FENE-P) is the most widely used configuration [28, 45]. FENE-P model

considers the polymer solution as a flowing suspension sufficiently dilute that the polymer molecules

do not interact with each other and that each molecule is idealized as an elastic dumbbell composed

by two beads connected by a non-linear spring with a maximum length [31]. Under this configuration,

and considering a uniform polymer concentration field, the constitutive equation of the Polymer Stress

Tensor yields the following relationship:

T[p]ij =

ρν[p]

τp[f(Ckk)Cij − δij ] (3.6)

where Cij is the conformation tensor, ν[p] the zero shear-rate polymeric viscosity, δij is the Kronecker

delta, L is the maximum possible extension of polymers and τp is the Zimm relaxation time of the

22

polymer. The polymer viscosity ν[p] is included in the model by a non-dimensional parameter β, which

represents the ratio between the solvent and the zero-shear-rate viscosity:

β =ν[s]

ν[p] + ν[s](3.7)

The function f(Ckk) is the Peterlin function given by:

f(Ckk) ≡ L2 − 3

L2 − Ckk(3.8)

where Ckk = Cxx + Cyy + Czz is the trace of the conformation tensor, which represents the extension

length. This function ensures finite extensibility, as it gives rise to a non-linear spring force that diverges

as√Ckk → L, preventing the spring from extending beyond L [36].

To complement the continuity equation and the conservation of momentum equation, a transport

equation for the conformation tensor Cij is required. Under the FENE-P model, the conformation tensor

evolution equation is given by:

∂Cij∂t

+ uk∂Cij∂xk

=∂ui∂xk

Cjk +∂uj∂xk

Cik −1

ρ

T[p]ij

ν[p](3.9)

The former equation is solved simultaneously with the velocity field flow equations, ensuring polymer-

flow interaction as the polymer molecules are deformed by the velocity field and, in turn, the resulting

conformation tensor introduces a polymeric stress on the flow structure.

3.3 Properties of the conformation tensor

The effects of polymer stretching on the flow development may be understood as either an elastic effect

or a viscous effect. In the framework of the FENE-P model, if the problem is approached from the

perspective of the elastic theory of drag reduction, the transport equation for the elastic energy can be

studied to understand the energy transfer between the polymers and the flow [51, 52]. Alternatively, if

polymer interaction is approached from the perspective of the viscous theory, Benzi [53] highlighted the

different physical roles played by the different components of the conformation tensor Cij , highlighting

the important role of Cyy which appears in the momentum and kinetic energy equations as an effective

viscosity.

Furthermore, the conformation tensor Cij is a measure of the second-order moment of the end-

to-end distance vector of the polymer dumbbell. It can be written as (2.6) where the vector r is the

separation vector between the two beads of the dumbbell. From the definition, it follows that the confor-

mation tensor is a symmetric positive definite (SPD) matrix [39]. Hulsen [54] proved that during exact

time evolution the conformation tensor must remain positive definite if it were initially. Nonetheless, cu-

mulative numerical errors that arise from virtually all initial value problem algorithms can give rise to

negative eigenvalues, which in turn cause the unbounded growth of Hadamard instabilities that quickly

overwhelm the calculation [55].

23

Another important property of the conformation tensor is that the trace, which represents the square

of the separation distance, must always be less than the square of the maximum extension, i.e., r2 <=

L2. The model guarantees this property through the force term f(Ckk), which diverges in strength as

this limit is approached, as mentioned previously. Hence for flows of arbitrary strength, the restoring

force is always sufficient to maintain this constraint. However, numerical errors in the evaluation of Ckk

can lead to violations of this constraint. Extension past L2 causes the force to change sign, resulting in

the rapid divergence of the calculation [39].

3.4 Computational domain

The computational domain for the turbulent jet spatial simulation consists of a box with uniform grid and

with Lx,Ly and Lz dimensions in each of the x (streamwise), y (normal) and z (spanwise) directions

(see Figure 3.1). The y and z directions have periodic boundary conditions, whilst the streamwise x is a

non-periodic direction with inflow and outflow boundaries.

Figure 3.1: View of the computational domain with reference frame and notation. From [16].

3.5 Spatial discretization

This section details all the spatial discretization procedures. The code employs the following structure:

Momentum equation terms highly accurate pseudo-spectral schemes to solve derivative on the normal

and spanwise directions and uses a compact scheme on the streamwise direction. As for the Con-

formation Tensor Transport equation, it is employed the 2nd order scheme Kurganov-Tadmor and finite

difference method.

3.5.1 Pseudo-spectral scheme

Pseudo-spectral schemes are used in spatial discretization in the normal and spanwise direction, as

these schemes are limited to the treatment of periodic functions. These schemes allow the representa-

24

tion of periodic functions in the Fourier space (also called spectral) in terms of frequencies for temporal

functions, and in terms of wavenumbers for spatial functions. Due to their high-level of accuracy without

numerical dispersion (to the machine precision), pseudo-spectral schemes are extensively used in CFD

[56]. Given a periodic flow variable along the normal y and spanwise directions z in the physical space,

φ(x, y, z, t), an inverse 2D discrete Fourier transform can be used to expand it through (3.10),

φ(x, y, z, t) =

ny2 −1∑

j=−ny2

nz2 −1∑

k=−nz2

φ(x, ky, kz, t)eι(kyy+kzz) (3.10)

where j, k are the indexes of the points along the normal y and spanwise z direction, respectively, and

(ky, kz) are the Fourier wave numbers defined by,

ky =2π

Lyj

kz =2π

Lzk. (3.11)

where i =√−1 is the imaginary unit, Ly and Lz are the spatial dimensions along the y and z directions,

and ny and nz are the number of discrete mesh points along those same directions, respectively.

Each Fourier coefficient φ(x, y, z, t), can then be obtained using the (direct) 2D discrete Fourier

transform,

φ(x, ky, kz, t) =1

nynz

ny−1∑j=0

nz−1∑k=0

φ(x, y, z, t)e−ι(kyy+kzz) (3.12)

It is important to note that the spectral variable φ is defined as a function of streamwise distance x,

as this direction does not present a periodic behaviour on the planar jet. Having defined the general

Discrete Fourier Transform (DFT), it is important to note that one of the main features of this method

is the simplicity of most operations in the spectral space [57]. In fact, derivatives in the physical space

result from the simple multiplication in the Fourier space, like

∂φ

∂y= ιkyφ

∂φ

∂z= ιkzφ (3.13)

resulting in a computationally light operation, with the accuracy only suffering from the rounding errors

and the discretization for the Fourier Transform. In addition to the high-level accuracy, the Fourier Trans-

form became more computationally interesting in 1965 with the publication of the Fast Fourier Transform

(FFT) algorithm. This algorithm presents a computational cost proportional to N log2N , where N is the

number of discretization points used. This cost is quite low comparing to the straightforward computation

of one-dimension Fourier transform with a proportional computational cost of N2 [58].

25

3.5.2 Compact finite differences discretization

The streamwise direction does not present a periodic behaviour, with one boundary being the jet inlet

with a predefined velocity and noise, and the other boundary being the outlet which ensures the flow is

not perturbed. Hence, spectral schemes are excluded to discretize the streamwise direction. In order

to maintain high accuracy, the code employs a 6th order compact finite difference scheme, described

in various references such as [40], [16] and [59]. It has been shown that these schemes provide easily

high order accuracy and possess some good ”spectral characteristics” [59] [18]. The basic idea of this

scheme is the definition of a derivative in terms of its neighbouring derivatives, which means the scheme

is implicit and requires solving a linear system of equations.

Considering a uniform grid with spatial coordinates xi = (i − 1)∆x, with ∆x = Const. (i ∈ [1, nx],

xi ∈ [0, Lx]), and using Taylor expansions of the function fi = f(xi) = f(x) with first derivative f ′i =

f ′(xi) = dfdx (xi) and second derivative f ′′i = f ′′(xi) = d2f

dx2 (xi), one can defined the general formula as

p∑j=−p

αjf′i+j =

q∑k=−q

akfi+k +O(∆xn) (3.14)

with p, q ∈ N. The order of the approximation n depends on the restrictions imposed upon the numbers

αj and ak. For the 6th order compact scheme, p = 2 and q = 3 in (3.14). Thus, (3.15) is obtained for the

first derivative calculation.

βf ′i−2 + αf ′i−1 + f ′i + αf ′i+1 + βf ′i+2 = afi+1 − fi−1

2∆x+ b

fi+2 − fi−2

4∆x+ c

fi+3 − fi−3

6∆x+O(∆xn) (3.15)

For this case the restrictions on (α, β, a, b, c) are:

• a+ b+ c = 1 + 2α+ 2β if n ≥ 2.

• a+ 22b+ 32c = 2 3!2! (α+ 22β) if n ≥ 4.

• a+ 24b+ 34c = 2 5!4! (α+ 24β) if n ≥ 6.

• ...

The calculation of each streamwise derivative depends on each particular point location. All interior

points, i.e., points between i = 3, ..., nx − 2, are calculated with a 6th order accuracy central compact

formulation given by:

αf ′i−1 + f ′i + αf ′i+1 = afi+1 − fi−1

∆x+ b

fi+2 − fi−2

∆x(3.16)

where α = 1/3 and enforcing the following restrictions on a and b,

a = (α+ 2)/3 = 7/9

b = (4α− 1)/12 = 1/36(3.17)

For those points near the two boundaries along the streamwise direction, i.e., i = 2 and i = nx−1, a

classical 4th order Pade scheme was used, corresponding to setting α = 1/4 and,

26

a = (α+ 2)/3 = 3/4

b = (4α− 1)/12 = 0.(3.18)

To complete the calculation of the derivatives in the streamwise x direction, a non-centred 3rd order

implicit scheme is used with forward and backward differencing for the inlet (i = 1) and outlet boundary

(i = nx), respectively.

f ′1 + 2f ′2 =1

2∆x(−5f1 + 4f2 + f3) (3.19)

f ′nx+ 2f ′nx−1 =

1

2∆x(5fnx

− 4fnx−1 − fnx−2) (3.20)

The linear system of equations to calculate the first derivative in the streamwise direction is sum-

marised in (3.21) in matrix form. In practice, the calculation of a single derivative at given point

1 α1

α2 1 α2

α 1 α

. . . . . . . . .

α 1 α

. . . . . . . . .

α 1 α

αn−1 1 αn−1

αn 1

f ′1

f ′2

f ′3...

f ′i...

f ′nx−2

f ′nx−1

f ′nx

=

1

∆x

a1 b1 c1

−a2 0 a2

−b −a 0 a b

. . . . . . . . . . . . . . .

−b −a 0 a b

. . . . . . . . . . . . . . .

−b −a 0 a b

−an−1 0 an−1

cn bn an

f1

f2

f3

...

fi...

fnx−2

fnx−1

fnx

(3.21)

with the following coefficients, as discussed above,

α1 = αn = 2 , a1 = −an = −5

2, b1 = −bn = 2 , c1 = −cn =

1

2

α2 = αn−1 =1

4, a2 = an−1 =

3

4

27

α =1

3, a =

7

9, b =

1

36

The compact scheme for the second derivative is obtained in a similar manner as the first derivative.

An analogous equation to (3.22), where f ′i− > f ′′i , can be written as

βf ′′i−2 + αf ′′i−1 + f ′′i + αf ′′i+1 + βf ′′i+2 = afi+1 − 2fi + fi−1

∆x2+

bfi+2 − 2fi + fi−2

4∆x2+ c

fi+3 − 2fi + fi−3

9∆x2+O(∆xn)

(3.22)

and the restrictions on (α, β, a, b, c) become:

• a+ b+ c = 1 + 2α+ 2β if n ≥ 2.

• a+ 22b+ 32c = 2 4!2! (α+ 22β) if n ≥ 4.

• a+ 24b+ 34c = 2 6!4! (α+ 24β) if n ≥ 6.

• ...

Like for the first derivatives, the second derivative of all interior points (from i = 3 to i = nx − 2) were

calculated with a 6th order central compact formulation, given by

αf ′′i−1 + f ′′i + αf ′′i+1 = afi+1 − 2fi + fi−1

∆x2+ b

fi+2 − 2fi + fi−2

∆x2(3.23)

where α = 2/11 and,

a = 4(1− α)/3 = 12/11b = (4α− 1)/12 = 3/44 (3.24)

As before, the points next to boundary (at i = 2 and i = nx−1) are obtained with a classical 4th order

Pade scheme. For the second derivative this is done with α = 1/10 and,

a = 4(1− α)/3 = 12/10b = (4α− 1)/12 = 0 (3.25)

At the boundary points (at i = 1 and i = nx) it is considered a non-centred 3rd order implicit scheme

given by

f ′′1 + 11f ′′2 =1

2∆x2(13f1 − 27f2 + 15f3 − f4) (3.26)

f ′′nx+ 11f ′′nx−1 =

1

2∆x2(13fnx

− 27fnx−1 + 15fnx−2 − fnx−3) (3.27)

28

The final system of equations for the second derivative results in,

1 α1

α2 1 α2

α 1 α

. . . . . . . . .

α 1 α

. . . . . . . . .

α 1 α

αn−1 1 αn−1

αn 1

f ′′1

f ′′2

f ′′3...

f ′′i...

f ′′nx−2

f ′′nx−1

f ′′nx

=

1

∆x2

a1 b1 c1 d1

a2 −2a2 a2

b a −2(a+ b) a b

. . . . . . . . . . . . . . .

b a −2(a+ b) a b

. . . . . . . . . . . . . . .

b a −2(a+ b) a b

an−1 −2an−1 an−1

dn cn bn an

f1

f2

f3

...

fi...

fnx−2

fnx−1

fnx

(3.28)

with the coefficients,

α1 = αn = 11 , a1 = an = 13 , b1 = bn = −27 , c1 = cn = 15 , d1 = dn = −1

α2 = αn−1 =1

10, a2 = an−1 =

12

10

α =2

11, a =

12

11, b =

3

44

(3.21) and (3.28) are presented more compactly by

A1f′ =

1

∆xB1f (3.29)

A2f′′ =

1

∆x2B2f (3.30)

where A1, A2, B1 and B2 are (nx × nx) matrices and f , f ′ and f ′′ are (nx × 1) vectors.

Having defined the matrices A1, A2, B1 and B2, the calculation of the derivative vectors f ′ and f ′′

from vector f is the following:

1. Inversion of A1 and A2 matrices to obtain A1−1 and A2

−1

29

2. Multiplication of matrices B1 and B2 by the vector 1∆xf , resulting in two new vectors of size (nx×1)

(1

∆xB1f

),

(1

∆x2B2f

)(3.31)

3. Multiplication of matrices A1−1 and A2

−1 by the vectors 1∆xB1f and 1

∆x2B2f to obtain the first and

second derivatives, respectively:

A1−1

(1

∆xB1f

), A2

−1

(1

∆x2B2f

). (3.32)

It is important to note that this procedure is computationally lighter than computing the inversion of

M1 = A1−1B1 and M2 = A2

−1B2 matrices and then performing the multiplication of M1 and M2 by f .

As M1 and M2 are full matrices, this option would require several operations proportional to n2x, whilst

the above procedure only requires the inversion of matrices A1 and A2, resulting in a computational cost

proportional to nx.

3.5.3 Conformation Tensor Transport Equation

The spectral and high-order compact schemes are not suitable for solving the conformation tensor trans-

port equation, since they lose spectral convergence near the descontinuities [36]. Therefore, in spatial

domain, a second-order central difference scheme is adopted for the discretization, except for the con-

vection term in which it is considered the Kurganov-Tadmor scheme (KT).

∂Cij∂t

+ uk∂Cij∂xk︸ ︷︷ ︸

Convection

=∂ui∂xk

Cjk +∂uj∂xk

Cik︸ ︷︷ ︸Stretching

−1

ρ

T[p]ij

ν[p](3.33)

Convection Term: Kurganov-Tadmor (KT) method

The convection term is discretized with a second-order KT scheme, described in [36], to guarantee

the Symmetric and Positive Definite (SPD) property of the conformation tensor, C, at all times and all

points. Spatial derivatives are second-order accurate everywhere, except for the grid points losing SPD

property. Where it occurs the scheme automatically reverts to first-order accurate for that grid point to

maintain the SPD property is maintained. The equations for the six independent components of C are

coupled to the velocity field and must be solved simultaneously. This scheme is given by:

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, (3.34)

where the convective 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) (3.35)

30

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) (3.36)

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) (3.37)

The superscripts ‘+’ and ‘–’ on the right hand side in (3.35), (3.36) and (3.37) designate the values

of the conformation tensor at the interface obtained in the limit approaching the point of interest from

the right (+) or left (–) side. The conformation tensor C at the interface is constructed from the following

second-order linear approximations:

C±i+1/2,j,k = Ci+1/2±1/2,j,k ∓(

∆x

2

)(∂C

∂x

)i+1/2±1/2,j,k

(3.38)

C±i,j+1/2,k = Ci,j+1/2±1/2,k ∓(

∆y

2

)(∂C

∂y

)i,j+1/2±1/2,k

(3.39)

C±i,j,k+1/2 = Ci,j,k+1/2±1/2 ∓(

∆z

2

)(∂C

∂z

)i,j,k+1/2±1/2

(3.40)

This central difference approximation to the convective term not only allows to capture the potentially

sharp variations in the conformation field, but also by writing the convective flux with a difference for-

mula the conservation of the mean conformation tensor is automatically satisfied [36]. To complete the

expression, approximations for the spatial derivatives of the conformation tensor are required. These

quantities must be defined in a way that maintains SPD property for each C±. The potential candidates

for approximating the gradient 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

(3.41)

It is selected the derivative approximation that can yield SPD results for C+i−1/2 and C−i+1/2. When two

or more candidates satisfy the criterion, it is selected the one which maximizes the minimum eigenvalue

for these two tensors. When none of them meet this criterion, the derivative is set to zero, reducing to

first-order accurate. This slope limiting procedure will ensure that all C± are SPD.

The update of the conformation tensor C requires the area-averaged velocity at the edge of the

volume surrounding each grid point. However, a complication of using a pseudo spectral code in the

velocity update is that the area-averaged velocities at cell edges that are not known. As mentioned in

[36], a simple estimate would be to average the nearest neighbouring grid points, but this approach

31

would not satisfy:

ui+1/2,j,k − ui−1/2,j,k

∆x+vi,j+1/2,k − vi,j−1/2,k

∆y+wi,j,k+1/2 − wi,j,k−1/2

∆z= 0, (3.42)

which follows from the continuity equation over a control volume. Violation of (3.42) leads to numerical

errors in the volume average of C. The exact expression for the area-averaged velocity is given by:

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 (3.43)

vi,j+1/2,k =1

∆x∆z

∫ xi+∆x/2

xi−∆x/2

∫ zk+∆z/2

zk−∆z/2

u

(x, yi +

∆y

2, z

)dxdz (3.44)

wi,j,k+1/2 =1

∆x∆y

∫ xi+∆x/2

xi−∆x/2

∫ yj+∆y/2

yj−∆y/2

u

(x, y, zk +

∆z

2

)dxdy (3.45)

where ui+1/2,j,k, vi,j+1/2,k and wi,j,k+1/2 are the area-averaged velocities at the edge of the volume

surrounding each grid point. In [45] and [36], the velocities u(x, y, z), v(x, y, z) and w(x, y, z) are ob-

tained from the inverse transform of the Fourier coefficients u(kx, ky, kz), v(kx, ky, kz) and w(kx, ky, kz).

However, since the streamwise direction x is not periodic this method is not applicable. Thus, the area-

averaged velocities are computed considering the physical domain along the streamwise direction x and

in the spectral domain along the normal y and spanwise z directions. The resulting expressions for the

cell velocities are:

ui+1/2,j,k = FT−1

{u(x, ky, kz)

sin(ky∆y/2)

ky∆y/2

sin(kz∆z/2)

kz∆z/2

}(3.46)

vi,j+1/2,k =1

∆x

∫ xi+∆x/2

xi−∆x/2

FT−1

{v(x, ky, kz)e

iky∆y/2 sin(kz∆z/2)

kz∆z/2

}dx (3.47)

wi,j,k+1/2 =1

∆x

∫ xi+∆x/2

xi−∆x/2

FT−1

{w(x, ky, kz)e

ikz∆z/2 sin(ky∆y/2)

ky∆y/2

}dx (3.48)

where FT−1 is the inverse fast Fourier transform, u(x, ky, kz), v(x, ky, kz) and w(x, ky, kz) are the Fourier

coefficients at each grid point. For the streamwise direction it is required to calculate the velocity and

integrate along x in the physical space. Along the streamwise direction x, the velocity on the cell edge

is interpolated with 5th order Lagrange Polynomials and the integration along the streamwise direction

is performed considering Gauss-Legendre 3 point rule. The computation procedure to assess the area

averaged velocity is the following:

1. Interpolation of velocity along the x direction for the required points. When assessing the area

averaged velocity u(x, y, z), the streamwise velocity component is interpolated on xi+1/2. When

assessing the area averaged velocities v(x, y, z) and w(x, y, z), the required points to interpolate

are the Gauss-Legendre nodes for the 3 point rule. In both cases, the interpolation is considered

a centred 5th Lagrange Polynomial for interior points (from i = 3 to i = nx−3). On the remaining

points, the polynomial order is maintained and the grid points used to interpolate are shifted in

32

order to only consider information within the domain (for example, to interpolate the velocity at

i = 1 + 1/2 it is considered the points from i = 1 to i = 6). Moreover, on the velocity points that are

outside the domain boundaries (for example, edge i− 1/2 for i = 1 and edge i+ 1/2 for i = nx) it

is assumed that the edge u velocity value is equal to the i grid point u velocity value;

2. 2D discrete Fourier transformation of the velocity field;

3. Computation of the integral along the periodic directions y and z on the spectral space;

4. Computation of the cell edge velocity for the velocities v and w along the periodic directions on the

spectral space;

5. Inverse 2D discrete Fourier transform of the velocity field and computation of the velocity integral

along the streamwise direction x. It is important to note that this step is not required for the area

averaged u velocity. The integration on the physical space is performed with the Gauss-Legendre

3 point rule, except for the boundary points in which the Gauss-Legendre reverts to the 1 point rule.

The Gauss-Legendre 3 point formulation for the area averaged normal velocity v can be written

as:

v(i, j + 1/2, k) =1

2

[A1F

(x+

∆x

2ξ1

)+A2F

(x+

∆x

2ξ2

)+A3F

(x+

∆x

2ξ3

)](3.49)

F (x) = FT−1

{v(x, ky, kz)e

iky∆y/2 sin(kz∆z/2)

kz∆z/2

}(3.50)

where ξi and Ai are the Gauss nodes and weights, respectively, and are given by

3 point rule 1 point rule

ξi Ai ξi Ai

node

1 −√

15/5 5/9 0 2

2 0 8/9

3√

15/5 5/9

Table 3.1: Gauss-Legendre integration nodes and weights. From [60].

The area averaged spanwise velocity w is obtained analogously to v.

It should be noted that the formulation presented is applicable to the velocity on positive edges i+ 1/2,

j + 1/2 and k + 1/2. The area averaged velocities on negative edges i − 1/2, j − 1/2 and k − 1/2

are obtained by shifting one position backwards the positive edge velocity field in the respective velocity

direction. This shifting requires special treatment on the boundaries. For the normal v and spanwise

w velocities, the lower boundary for the negative edge cell velocities is equal to the upper boundary

positive edge cell velocities, thus ensuring a period behaviour. Regarding the streamwise velocity u, the

negative cell edge velocity at the inlet boundary is assumed to be the velocity value u at the inlet.

33

Stretching Term and Polymer Stress Coupling Term: Finite difference method

The code was developed for use with a finite difference approach in which the differential equations

are approximated at each point. For the stretching term and the polymer stress coupling term, the

first derivatives were computed with a central difference scheme second-order accurate everywhere in

space, except for the boundaries on the streamwise direction.

For the stretching term, this scheme results in:

∂ui∂xk

Cjk +∂uj∂xk

Cik =(ui)m+1 − (ui)m−1

2∆xkCjk +

(uj)m+1 − (uj)m−1

2∆xkCik (3.51)

where i, j and k are the computational indexes and m is an auxiliary variable that represents i, j and

k for the ∂/∂x, ∂/∂y and ∂/∂z derivatives, respectively. On the inlet and outlet boundaries the central

difference scheme is reverted to a forward and backward finite difference first-order accurate scheme,

respectively. The forward finite difference scheme is given by:

∂ui∂x

Cj1 +∂uj∂x

Ci1 =(ui)2 − (ui)1

∆xCj1 +

(uj)2 − (uj)1

∆xCi1 (3.52)

The backward finite difference scheme is given by:

∂ui∂x

Cj1 +∂uj∂x

Ci1 =(ui)nx

− (ui)nx−1

∆xCj1 +

(uj)nx− (uj)nx−1

∆xCi1 (3.53)

For the polymer stress coupling term on the momentum equations, this scheme results in:

∂T[p]ij

∂xj=ν[p]

τp

[(f(Ckk)Cij − δij)m+1 − (f(Ckk)Cij − δij)m−1

2∆xk

](3.54)

where i and j are the computational indexes and m is an auxiliary variable that represents i, j and k

for the ∂/∂x, ∂/∂y and ∂/∂z derivatives, respectively. Similarly to the stretching term, the coupling term

on the inlet and outlet boundaries the central difference scheme is reverted to a forward and backward

finite difference first-order accurate scheme, respectively. The forward finite difference scheme is given

by:∂T

[p]ij

∂xj=ν[p]

τp

[(f(Ckk)Cij − δij)2 − (f(Ckk)Cij − δij)1

∆x

](3.55)

The backward finite difference scheme is given by:

∂T[p]ij

∂xj=ν[p]

τp

[(f(Ckk)Cij − δij)nx

− (f(Ckk)Cij − δij)nx−1

∆x

](3.56)

34

3.6 Temporal discretization

The time advancement method of the momentum equation and for the conformation tensor transport

equation is an explicit 3rd order Runge-Kutta scheme of numerical integration. The equations to be

solved can be written as follows:

∂ui∂t

= N (ui) + L[s] (ui) + L[p] (Cij)−∇πp (3.57)

where N(ui) is the non-linear convective term

N(ui) = −uj∂ui∂xj

+∂

∂xi

ujuj2

= ui × ωi (3.58)

and L[s](ui) represents the viscous term due to the solvent

L[s](ui) = ν∂2ui∂xj∂xj

= ν∇2u (3.59)

and L[p](Cij) represents the viscous term due to the polymer

L[p] (Cij) = ν[p]∇ ·(f(Ckk)Cij − δij

τp

)(3.60)

and πp is the modified pressure given by

π =p

ρ+ujuj

2(3.61)

Having determined the terms in (3.59) to (3.61), it is employed the 3 step, 3rd order Runge-Kutta scheme,

which calculates each sub-step velocity up from the last two time sub-steps p− 1 and p− 2 using

uip − uip−1

∆t= αp

[N(ui

p−1) + L[s](uip−1) + L[p](Cij

p−1)]

+ βp

[N(ui

p−2) + L[s](uip−2) + L[s](Cij

p−2)]− ∂πp

∂xi

(3.62)

The coefficients αp and βp in (3.62) for a 3rd order accuracy with 3 time sub-steps p = 1, 2, 3 are defined

in (3.63) [61].

α1 = 8/15 β1 = 0

α2 = 5/12 β2 = −17/60

α3 = 3/4 β3 = −5/12

(3.63)

3.6.1 Pressure-velocity coupling

To solve (3.62), the pressure field π must be known at each time sub-step p. In order to know the

pressure term, each Runge-Kutta step is divided into two sub-steps using a method called fractional

step ([62, 63] through [16]). This method is used to solve the simultaneous calculation of the velocity and

pressure field. This is necessary to insure incompressibility of the velocity field at each time sub-step and

35

it implies that a Poisson equation must be solved at each sub-step of the Runge-Kutta scheme. Firstly,

the terms N(ui), L[s](ui) and L[p](Cij) are computed, and an intermediate velocity uip∗ is calculated as

uip∗ − uip−1

∆t= αp

[N(ui

p−1) + L[s](uip−1) + L[p](Cij

p−1)]

+ βp

[N(ui

p−2) + L[s](uip−2) + L[p](Cij

p−2)] (3.64)

Secondly, the pressure field is calculated by imposing the continuity equation. Subtracting (3.64) from

(3.62), one obtainsuip∗ − uip−1

∆t= −∂π

p

∂xi(3.65)

By taking the divergence of (3.65), the Poisson equation (3.66) is obtained.

∇ · uip −∇ · uip∗

∆t= −∇2πp (3.66)

At each Runge-Kutta step the continuity equation must be verified,

∇ · uip = 0 (3.67)

When this is applied to (3.66), one obtains:

−∇ · uip∗

∆t= −∇2πp (3.68)

Thus, solving the Poisson equation (3.68) will result in the modified pressure, which, when used in (3.65),

yields the velocity at the end of each Runge-Kutta step p.

3.6.2 Non-Linear Term in the Momentum Equation

The terms of velocity momentum equation are discretised in the Fourier space for the spanwise and

normal directions (y, z) and 6th order compact differences for the derivatives in the streamwise direction

(x) [16]. The velocity at each sub-step p is determined by the following equations:

uip∗ − uip−1

∆t= αp

[Nx

p−1+ ν(

d2

dx− k2)ui

p−1 + Lx[p]

(Cijp−1)

]+ βp

[Nx

p−2+ ν(

d2

dx− k2)ui

p−2 + Lx[p]

(Cijp−2)

]vip∗ − vip−1

∆t= αp

[Ny

p−1+ ν(

d2

dx− k2)vi

p−1 + Ly[p]

(Cijp−1)

]+ βp

[Ny

p−2+ ν(

d2

dx− k2)vi

p−2 + Ly[p]

(Cijp−2)

]wip∗ − wip−1

∆t= αp

[Nz

p−1+ ν(

d2

dx− k2)wi

p−1 + Lz[p]

(Cijp−1)

]+ βp

[Nz

p−2+ ν(

d2

dx− k2)wi

p−2 + Lz[p]

(Cijp−2)

]

(3.69)

36

where d/dx represents the operator of compact second order differentiation and each component of the

non-linear term is given by:

Nxp = vpωpz − wpωpy

Nyp = wpωpx − upωpz

Nzp = upωpy − vpωpx

(3.70)

The vorticity components in (3.70) are given by:

ωpx = ikywp − ikz vp

ωpy = ikzup − dwp

dx

ωpz = dvp

dx − ikyup

(3.71)

In (3.70), k2 is the norm of the wavenumber vector k that appears from the calculation of a second

derivative in the spectral space.

k2 = k2y + k2

z (3.72)

After solving (3.69), the correction pressure field πp is obtained solving the Poisson problem. This

computation is performed on the Fourier space and can be written as:

(d2

dx2− k2

)πp =

1

∆t

(dup∗

dx+ iky v

p∗ + ikzwp∗)

(3.73)

Having calculated the pressure field, the final velocity field uip at the end of each Runge-Kutta step is

obtained by

up = up∗ −∆t(dπp

dx

)vp = vp∗ −∆t (ikyπ

p)

wp = wp∗ −∆t (ikzπp)

(3.74)

3.6.3 Conformation Tensor Transport Equation

To know each new velocity at each new sub-step, it is necessary to know the conformation tensor at the

previous sub-step(s). Equation (3.9) can be written in the following form:

∂Cij∂t

= N ′ (ui, Cij) +M (ui, Cij) + L[p]′ (Cij) (3.75)

where N ′ (ui, Cij) is the convection term

N ′ (ui, Cij) = −uk∂Cij∂xk

(3.76)

and M (ui, Cij) is the stretching term

M (ui, Cij) = Cjk∂ui∂xk

+ Cik∂uj∂xk

(3.77)

37

and L[p]′ (Cij) is the viscous term

L[p]′ (Cij) = −f(Ckk)Cij − δijτp

(3.78)

The same process that was applied before to the Navier-Stokes momentum equation in (3.62), is

done here. Thus, the following is obtained:

Cijk − Cijk−1

∆t= αk

[N ′(uik−1, Cij

k−1)

+M(uik−1, Cij

k−1)

+ L[p]′(Cij

k−1)]

+βk

[N ′(uik−2, Cij

k−2)

+M(uik−2, Cij

k−2)

+ L[p]′(Cij

k−2)] (3.79)

where αk and βk are the coefficients of (3.63).

It is important to notice that the Runge-Kutta step of equation (3.79) is not the same as that of the

equation (3.69). In equation (3.79), Cijk−2 ≡ Cijn−2 and Cij

k ≡ Cijn−1, whereas in equation (3.69),

Cijk−2 ≡ Cijn−1.

3.7 Stability condition

The time step used in the numerical resolution of momentum and conformation tensor transport equa-

tions is bounded by stability restrictions. In convection dominated problems (such as the one considered

in this work), the main parameter that governs the stability limits imposed on the size of the mesh ele-

ments and the time step is the Courant-Friedrich-Levy condition or Courant number (CFL) expressed by

(3.80) [58]:

CFL =|ui|∆t

∆x(3.80)

In the simulations, the Courant number is found by assessing (3.80) at each iteration for each flow

direction. The resulting ∆t can be written as:

∆t = CFL×min

{∆x

|u|max,

∆y

π|v|max,

∆z

π|w|max

}(3.81)

where in (3.81) |u|max, |v|max, |w|max are the maximum velocity in each respective direction in the veloc-

ity field. The π factor in the y and z direction arises due to the spectral nature of the spatial discretisation

in those directions [64]. In the current work, a visco-elastic fluid is considered with a conformation tensor

having a second order discretisation scheme. Vaithianathan [36] demonstrated that to ensure the con-

formation tensor Cij remains positive as the simulation develops, the following Courant condition must

be satisfied:

CFL <1

6≈ 0.1667 (3.82)

which represents a more severe stability condition than the one considered for Newtonian fluids CFL <

0.6 [18]. For this reason, the simulations consider a maximum CFL number of 0.16. The derivation of

the maximum CFL number is presented in appendix A.

38

3.8 Boundary and Initial Conditions

In this section it is described the different boundary conditions deployed in each location of the domain

for the spatial plane jet.

3.8.1 Inlet boundary

The inlet conditions are imposed by forcing a velocity profile and a conformation tensor profile at each

time step for the streamwise component of the velocity (U ) and for the polymer conformation tensor

(Cij), respectively.

The velocity profile imposed at the inlet at each timestep is given by:

u(x0, t) = Umed(x0) + Unoise (3.83)

where x0 = (x = 0, y, z) is the inlet plane, Umed is the mean inlet profile and Unoise is a superimposed

random numerical noise. For the planar jet, the mean velocities in the normal y and spanwise z directions

are set to zero at the inlet. In the streamwise direction x the prescribed velocity profile is based in a

hyperbolic tangent profile used in [65] and [16]. The mean inlet velocity is expressed in (3.84).

Umed(x0) =U2 + U1

2+U2 − U1

2tanh

[h

4θ0

(1− 2|y|

h

)](3.84)

In (3.84) θ0 is the momentum thickness of the initial shear layer, h is the jet inlet slot width and y is the

distance from the centre of the jet, i.e., y = 0. U1 is the co-flow velocity and U2 is the centreline velocity.

The co-flow is required to allow the growth of the jet’s shear layer, otherwise hampered by the lack of

natural entrainment from the periodic lateral boundaries. However, U1 cannot be too high, as it might

change the instability characteristic of the initial mean velocity profile [16]. An example of the mean

velocity profile is shown in Figure 3.2, exhibiting inlet velocity from y/h = −0.5 to y/h = 0.5.

Figure 3.2: Mean velocity profile imposed at the inlet. H stands for the width of the inlet slot. From [18].

39

The numerical noise, Unoise, is superposed to the mean profile in (3.84), at each timestep t by

imposing a three-component fluctuating velocity field (u, v and w), to exhibit the statistical characteristics

of isotropic turbulence [65]. The velocity noise profile is given by

Unoise = ampUbase~u′ (3.85)

where amp is an amplification factor typically ranging from [0, 0.15] and ~u′ = (u′, v′, w′) comes from a

random number generator chosen to comply with the following energy spectrum

E(k) ∝ kse− s2 (k/k0)2 (3.86)

where s is the noise spectrum slope in the large scales region and, k0 the wave number location of

every maximum. These parameters are chosen to provide an energy input dominant at the small scales.

The numerical noise is only imposed in the shear-layer region of the mean streamwise velocity profile

through a proper convolution function.

An inlet boundary is also considered for the polymer conformation tensor. The inlet profile is based

on the assumption of steady flow, fully developed in the streamwise direction x. Considering the inlet ve-

locity profile without co-flow (see (3.84)), it is possible to notice that there are no normal v and spanwise

w velocities. Taking the previous remarks into account, one obtains an analytical profile for the polymer

conformation tensor, given by the real root of the following cubic equation:

C223 +

L2

2τp2(∂U∂y

)2C22 −L2

2τp2(∂U∂y

)2 = 0 (3.87)

Further detail regarding the derivation of this solution, as well as the value for each component of the

conformation tensor, is presented in section 4.1.

3.8.2 Lateral boundaries

As explained in section 3.4, the normal y and spanwise z directions of the domain are discretized with

spectral methods, which required the boundaries to be periodic. This periodicity of the lateral boundaries

is stated for all three components of the velocity field, the pressure and the polymer conformation tensor.

These can be written as:~u(x, y, z, t) = ~u(x, y + Ly, z, t)

~u(x, y, z, t) = ~u(x, y, z + Lz, t)

p(x, y, z, t) = p(x, y + Ly, z, t)

p(x, y, z, t) = p(x, y, z + Lz, t)

Cij(x, y, z, t) = Cij(x, y + Ly, z, t)

Cij(x, y, z, t) = Cij(x, y, z + Lz, t)

(3.88)

The boundaries must be set sufficiently far away from the turbulent region, otherwise any perturbation

that leaves the domain will influence the opposite side of the computational domain and the simulation

40

will diverge [66]. Moreover, the distance of the lateral boundaries is also dictated by the need to avoid

spurious reflections which would upset the flow. It must be noted that the above-mentioned issue only

affects the normal direction y under the planar jet configuration, as the spanwise direction z is naturally

periodic.

3.8.3 Outlet boundary

The outlet boundary is the most sensitive boundary condition of the planar jet. The fluid structures that

leave the domain must exit unperturbed by the outflow condition and ensuring that reflected waves do not

go back into the domain. Therefore, the code employs a non-reflective boundary condition used by [67],

where all the terms of the Navier-Stokes equations - convective and viscous - are explicitly advanced.

This method begins with the x momentum equation, in which the x component of the diffusive term

Re−1 ∂2u∂x2 and the pressure gradient ∂p

∂x and included in an unknown variable Cu(y, z). Thus, the final

equation is:∂u

∂t= −Cu(y, z)

∂u

∂x− v ∂u

∂y− w∂u

∂z+

1

Re

(∂u2

∂y2+∂u2

∂z2

)(3.89)

The Cu(y, z) term is calculated at i = nx − 1 to determine the value of streamwise velocity u at each

Runge-Kutta sub-timestep p with(3.90):

up−1 − up−2

∆t= −Cu(y, z)

∂u

∂x− v ∂u

∂y− w∂u

∂z+

1

Re

(∂u2

∂y2+∂u2

∂z2

)(3.90)

Using (3.90) the streamwise derivative is calculated with a 1st order backward differences scheme,

whilst the normal and spanwise directions are determined with a pseudo-spectral scheme. Next, the

same equation is used with the Runge-Kutta time scheme to obtain the streamwise velocity at the outlet

plane i = nx. After this, the code solves the Poisson equation, as described in section 3.8.4, to obtain

the corrected normal v and spanwise w velocities at i = nx with the pressure gradient that result from

the solution of the Poisson equation. This step is shown in (3.91).

v(y, z) = v∗y(y, z)−∆t ∂p∂y

w(y, z) = w∗z(y, z)−∆t∂p∂z

(3.91)

where v∗y,z and w∗y,z are the velocity components for the normal and spanwise directions, respectively,

obtained from the Navier-Stokes equations without the pressure gradient term. This scheme was shown

by Reis [16] to ensure a smooth exit of the coherent turbulent structures, with negligible perturbations

from the presence of the outflow boundary condition. It must be noted that no special treatment of the

boundary outlet condition is required on the polymer conformation tensor, as the outflow condition will

be reflected through the velocity field on the polymer conformation transport equation.

3.8.4 Poisson equation

In the case of the Poisson equation, the lateral boundaries for the spanwise and normal direction are

periodic, due to the pseudo-spectral scheme. For the inlet and outlet boundaries, the pressure boundary

41

conditions are of Neumann type. This requires values of the pressure gradient outside of the computa-

tional domain, for x = −∆x and x = Lx + ∆x]. The values on these points are taken as being equal to

the corresponding boundary points:∂p

∂x x=−∆x=

∂p

∂x x=0(3.92)

∂p

∂x x=Lx+∆x=

∂p

∂x x=Lx

(3.93)

The streamwise pressure gradients at the inlet are obtained from the imposed velocity fields at the inlet.

At the outlet it is obtained from the intermediate velocity at each Runge-Kutta step before the correction.

These steps are illustrated on (3.94) and (3.95), respectively.

−∆t∂p

∂x i=1= ui=1 − u∗i=1 (3.94)

−∆t∂p

∂x i=nx

= ui=nx− u∗i=nx

(3.95)

For discretisation purposes, (3.92), (3.95) corresponds to

f ′i=0 = f ′i=1 (3.96)

f ′i=nx+1= f ′i=nx

(3.97)

where 0 and nx + 1 are fictitious nodes placed outside the computational domain. Consequently, the set

of expressions in (3.98) is obtained, which allows for the inversion of the Poisson equation.

fi=1 = 87fi=2 − 1

7fi=3 − 67∆xf ′i=1

fi=0 = fi=1 −∆xf ′i=1

fi=nx= 8

7fi=nx−1 − 17fi=nx−2 − 6

7∆xf ′i=nx

fi=nx+1 = fi=nx−∆xf ′i=nx

(3.98)

3.9 Code architecture

As mentioned previously in section 3, the starting point of the numeric work here presented is a DNS

code for turbulent jet described in Ricardo Reis’ PhD thesis [16] and Diogo Lopes’ PhD thesis [18], having

also defined the core of the parallelization strategy was developed in the above-mentioned works. In this

section it is described the parallel code architecture, such as the data dependency, read/write operations

and domain data decomposition.

3.9.1 Parallel architecture

The parallelization strategy was designed with the aim of minimizing communication between proces-

sors, whilst being able to meet the requirements of the problem. The current numerical work is paral-

lelized with MPI and is described in detail in [16]. The main operations to be performed by the numerical

42

scheme for each Runge-Kutta step are the calculation of the non-linear and the viscous terms, to obtain

the velocity field, followed by the resolution of the Poisson equation to obtain the pressure field and the

corrected velocity.

3.9.2 Parallel domain decomposition

The computational domain is implemented on a mixed decomposition, with the co-existance of stream-

wise slabs and normal slices (see Figure 3.3). The reasoning underlying this configuration is presented

in [16].

Figure 3.3: View of the domain partitioning with reference frame and notation. From [18].

The data field configuration is swapped according to the operation taking place, according to the

following structure:

• The direct and inverse Fourier transforms are computed in independent yz planes, thus being

intrinsically connected to the normal slice configuration;

• The compact scheme for the streamwise derivatives is computed on the slab configuration, as it

requires a complete line of data on the streamwise direction not available on normal slice configu-

ration;

• The streamwise derivatives for the conformation tensor transport equation do not require complete

lines of data on the streamwise x direction. Thus, it was opted to perform the calculation on

the normal slice configuration with the aid of ghost cells, avoiding the need to perform intensive

communications between processors to swap data configuration.

The data exchange between the two data configurations uses the function MPI AlltoAll and is per-

formed in every Runge-Kutta sub-step.

Data exchange with ghost cells

Ghost cells (also referred as halo cells on the literature [50]) here consist in replicated data from neigh-

bouring partitions that is passed between processors. The replicated data is only performed on the

streamwise direction, as the remaining directions are periodic. This additional chunk of data in each

43

normal slice allows the computation of the streamwise derivative, avoiding the use of a MPI AlltoAll

communication transfer. This process requires data exchange at each Runge-Kutta sub-step. For a

given centred domain partitioned on normal slices, two streamwise data planes are communicated on

each side of the slices, except on the ones that are located on the streamwise boundaries. An illustrative

example of ghost cell application is presented in Figure 3.4.

(a) Normal slice configuration without ghost cell data on the xy plane.

(b) Normal slice configuration with ghost cell data on the xy plane.

Figure 3.4: Ghost cell configuration data exchange on the streamwise direction for a domain partitionedin 3 normal slices.

3.9.3 FFT calculation

The Fast Fourier Transforms are calculated with the Fastest Fourier Transform in the West (FFTW)

library [68]. FFTW is a high-performance library, released under the GNU General Public License1,

with portable features, maintaining the program performance on most computer architectures without

the need of modification. Due to the nature of FFT algorithms, the Fourier transform computation speed

is maximized when the prime factorization of the number of grid points in the spectral directions is

comprised by low prime numbers [18].

3.9.4 I/O

DNS programs have input and output (I/O) intensive procedures, being one of the slowest operations

found in computing systems [16]. Moreover, the amount of space required is quite demanding, increas-

ing three times with the mesh size. For example, the reference simulation performed in [18] comprised a

1See https://www.fftw.org

44

600 million point mesh, resulting in 2.3GB per data file. Several I/O methods were considered, which are

described in detail in [16] and [18]. The preferred I/O method is the MPI-IO, which allows the processors

to write and read their part of a common file. In addition, as there is a provision to dynamically adjust the

scope of action for each processor when interacting with the file, this method allows simulation restarts

using a different number of processors.

45

46

Chapter 4

Code Verification

To ensure full reliability on the results generated by the spatial jet DNS code with FENE-P, extensive code

verification tests with known results were performed. The code verification exercise was conceptualized

with the purpose of assessing all terms related to the visco-elastic characteristics of the simulation that

were added to the existing DNS code. Thus, the exercise was organized according to the following

structure:

1. Conformation Tensor Transport Equation

• It was considered a 2D Couette flow for verification purposes. With this particular flow, an

analytical solution was computed for the polymer conformation transport equation;

• The code was modified in order to allow the interaction of the numerical scheme with this flow.

Namely, the initial velocity profile was frozen for each Runge-Kutta step, which means it was

kept constant throughout the simulation;

• Being a 2D flow scheme, the verification analysis was performed for all 2D planes to trigger

and verify all the equation components.

2. Velocity - Conformation Tensor Coupling term

• It was performed the verification of the visco-elastic coupling term on the momentum equation

by triggering it with a known Cij conformation tensor;

• By having a known imposed Cij , an analytical solution was found and compared to the result

from the numerical scheme.

It should be noted that the verification exercise of the overall 3D numerical simulation code does

not comprise a comparison to other simulation results, as no comparable simulation set-ups, i.e., con-

sidering non-periodic boundary conditions on a spatial jet with polymer additives, were found on the

literature.

47

4.1 Conformation Tensor Transport Equation Verification

The Conformation Tensor Transport Equation was verified by performing an exercise with a frozen in

time 2D Couette flow, thus allowing the comparison with a steady analytical solution. The velocity profile

was kept constant throughout the simulation by imposing it in each Runge-Kutta step.

Being a 2D flow scheme and as the streamwise direction does not present a periodic behaviour in

the spatial jet flow (e.g. discretization is not performed with a pseudo-spectral scheme in this direction),

the verification analysis was performed for all 2D planes so that all the transport equation components

are triggered.

In this section it is presented in detail the analytical solution for the transport equation, assuming a

2D steady flow for the u velocity component fully-developed in the x direction. The derivation consid-

ering these assumptions is also presented on [57]. It should be noted that the derivation is performed

analogously for the remaining velocity profiles considered in this verification exercise.

The velocity profile for the u velocity component in function of y normal direction is presented in

Figure 4.1.

Figure 4.1: u velocity profile in the normal y direction.

From Figure 4.1 it is noticeable that the velocity profile is a mirrored Couette profile, due to the nu-

merical scheme and the need to ensure periodic conditions. Therefore, the discontinuity points observed

on the profile (y/h = −5, 0, 5 for Figure 4.1) are not comparable to the analytical solution.

The profile of the conformation tensor was set to an identity matrix of size 3 at the beginning of the

simulation.

The conformation tensor transport equation is presented previously in (3.9). Considering the Polymer

Stress Tensor relationship from (3.6), the transport equation can be written as follows:

∂Cij∂t

+ uk∂Cij∂xk

=∂ui∂xk

Cjk +∂uj∂xk

Cik −1

τp[f(Ckk)Cij − δij ] (4.1)

As mentioned previously on Section 3.3, the conformation tensor Cij is an SPD matrix by defini-

48

tion. For this reason, Cij = Cji and, therefore, the system of linear equations only encompasses six

independent variables, corresponding to the independent components of the conformation tensor.

As it is considered a steady and fully-developed flow in the streamwise x direction, the following

assumptions are enforced:∂

∂t= 0,

∂

∂x= 0 (4.2)

Given the two-dimensional flow (velocity profile u(y)) and the conservation of mass, the following is

obtained:

v = 0, w = 0,∂

∂z= 0 (4.3)

Taking into consideration the flow assumptions from Equations (4.2) and (4.3), the conformation

tensor transport equation can be rewritten in this case as:

uk∂Cij∂xk

=∂ui∂xk

Cjk +∂uj∂xk

Cik −1

τp[f(Ckk)Cij − δij ] (4.4)

Having equation (4.4), each entry of the conformation tensor Cij must be assessed. The resulting

equations for each entry are presented below.

For component C11:

2∂u

∂yC12 −

1

τp[f(Ckk)C11 − 1] = 0 (4.5)

For component C12:∂u

∂yC22 −

1

τp[(Ckk)C12] = 0 (4.6)

For component C13:∂u

∂yC23 −

1

τp[(Ckk)C13] = 0 (4.7)

For component C22:

f(Ckk)C22 − 1 = 0 (4.8)

For component C23:1

τp[(Ckk)C23] = 0 (4.9)

For component C33:

f(Ckk)C33 − 1 = 0 (4.10)

From equation (4.9) it follows that C23 = 0, as the diagonal component of the conformation tensor

Cij are positive. Having this result, equation (4.7) can be simplified, leading to C13 = 0.

Moreover, comparing equations (4.8) and (4.10), it is noticeable that the following relationship must

be held: C22 = C33.

Rearranging equation (4.8), and considering that the Peterlin function is defined as (3.8), the following

is obtained:L2 − 3

L2 − (C11 + 2C22)C22 = 1 (4.11)

49

from which it is obtained

C11 = C22

(1− L2

)+ L2 (4.12)

From equation (4.5), it is possible to organize it as function of C22, if one considers the relationship

between variables C11, C12 and C22 from equations (4.6) and (4.12). By doing so, the following equation

is achieved:

2

(∂u

∂y

)2

τ2pC

222 =

1

C22

(C22

(1− L2

)+ L2

)− 1 (4.13)

Rearranging (4.13) for C22, it is obtained a cubic polynomial in function of C22:

C322 + C22

L2

2(∂u∂y

)2

τ2p

− L2

2(∂u∂y

)2

τ2p

= 0 (4.14)

The real root of the cubic polynomial (4.14) is the following:

C22 = 3

√√√√√ L2

4(∂u∂y

)2

τ2p

+

√√√√ L6

216(∂u∂y

)6

τ6p

+L4

16(∂u∂y

)4

τ4p

+

3

√√√√√ L2

4(∂u∂y

)2

τ2p

−√√√√ L6

216(∂u∂y

)6

τ6p

+L4

16(∂u∂y

)4

τ4p

(4.15)

The verification exercise for the conformation tensor equation considers the following simulation pa-

rameters:

General properties Polymer properties

Box height h 2π Zero shear-rate viscosity ν[p] 0.002

Box dimensions 0.9h× h× 0.118h Maximum molecular extensibility L 10

Mesh 128× 128× 128 Relaxation time (s) τp 0.1

Maximum velocity umax 1 Polymer concentration β 0.8

Table 4.1: Simulation parameters for transport equation verification.

Considering parameters from Table 4.1 and the velocity profile from 4.1, the following velocity u

derivatives are obtained:

∂u

∂y=

1π , if yh < 0

− 1π , if yh ≥ 0

(4.16)

Taking into consideration the previous remarks, the following analytical solution for the conformation

tensor Cij is obtained:

50

Cij ≈

1.002 0.0318 0

0.0318 1.000 0

0 0 1.000

for yh < 0

1.002 −0.0318 0

−0.0318 1.000 0

0 0 1.000

for yh ≥ 0

(4.17)

The numerical solution obtained from this simulation is presented in Figure 4.2.

Figure 4.2: Numerical solution for the Cij tensor considering a u velocity profile in the normal y direction.

Having both analytical and numerical results, it is performed the comparison between both. This

verification is presented in Figures 4.3 and 4.4 for the diagonal and non-diagonal conformation tensor

components, respectively. It is important to note that for each figure the dashed lines correspond to the

numerical solution, whilst the solid lines refer to the analytical solution.

From Figures 4.3 and 4.4 it is possible to observe that the numerical and analytical solutions present

the same results, thus ensuring the verification of the triggered equation components in this scenario.

51

Figure 4.3: Numerical (n) and analytical (a) solutions for the Cii tensor considering a u velocity profile inthe normal y direction.

Figure 4.4: Numerical (n) and analytical (a) solutions for the Cij , i 6= j tensor considering a u velocityprofile in the normal y direction.

52

Now it is presented the verification results for the remaining 2D planes.

For a u velocity component in function of z spanwise direction:

Figure 4.5: u velocity profile in the spanwise z direction.

As it is considered a steady and fully-developed flow in the streamwise x direction, the following

assumptions are enforced:∂

∂t= 0,

∂

∂x= 0 (4.18)

Given the two-dimensional flow (velocity profile u(z)) and from the continuity equation, the following

is obtained:

v = 0, w = 0,∂

∂y= 0 (4.19)

(a) Cii tensor components (b) Cij , i 6= j tensor components

Figure 4.6: Cij numerical (n) and analytical (a) solutions comparison considering a u velocity profile inthe spanwise z direction.

53

For a w velocity component in function of y normal direction:

Figure 4.7: w velocity profile in the normal y direction.

As it is considered a steady and fully-developed flow in the spanwise z direction, the following as-

sumptions are enforced:∂

∂t= 0,

∂

∂z= 0 (4.20)

Given the two-dimensional flow (velocity profile w(y)) and from the continuity equation, the following

is obtained:

u = 0, v = 0,∂

∂x= 0 (4.21)

(a) Cii tensor components (b) Cij , i 6= j tensor components

Figure 4.8: Cij numerical (n) and analytical (a) solutions comparison considering a w velocity profile inthe normal y direction.

54

For a v velocity component in function of z spanwise direction:

Figure 4.9: v velocity profile in the spanwise z direction.

As it is considered a steady and fully-developed flow in the normal y direction, the following assump-

tions are enforced:∂

∂t= 0,

∂

∂y= 0 (4.22)

Given the two-dimensional flow (velocity profile v(z)) and from the continuity equation, the following

is obtained:

u = 0, w = 0,∂

∂x= 0 (4.23)

(a) Cii tensor components (b) Cij , i 6= j tensor components

Figure 4.10: Cij numerical (n) and analytical (a) solutions comparison considering a v velocity profile inthe spanwise z direction.

55

For a w velocity component in function of x streamwise direction:

Figure 4.11: w velocity profile in the streamwise x direction.

As it is considered a steady and fully-developed flow in the spanwise z direction, the following as-

sumptions are enforced:∂

∂t= 0,

∂

∂z= 0 (4.24)

Given the two-dimensional flow (velocity profile w(x)) and from the continuity equation, the following

is obtained:

u = 0, v = 0,∂

∂y= 0 (4.25)

(a) Cii tensor components (b) Cij , i 6= j tensor components

Figure 4.12: Cij numerical (n) and analytical (a) solutions comparison considering a w velocity profile inthe streamwise x direction.

56

For a v velocity component in function of x streamwise direction:

Figure 4.13: v velocity profile in the streamwise x direction.

As it is considered a steady and fully-developed flow in the spanwise y direction, the following as-

sumptions are enforced:∂

∂t= 0,

∂

∂y= 0 (4.26)

Given the two-dimensional flow (velocity profile v(x)) and from the continuity equation, the following

is obtained:

u = 0, w = 0,∂

∂z= 0 (4.27)

(a) Cii tensor components (b) Cij , i 6= j tensor components

Figure 4.14: Cij numerical (n) and analytical (a) solutions comparison considering a v velocity profile inthe streamwise x direction.

It is possible to observe that the numerical and analytical solutions present the same values for

all steps in the verification exercise of the conformation tensor transport equation. Thus, the code is

deemed as verified.

57

4.2 Velocity - Conformation Tensor Coupling Term Verification

As mentioned before, the velocity - conformation tensor coupling term was verified by imposing a known

Cij field and comparing the obtained result from the numerical scheme with an analytical solution. It

should be noted that this verification assumes a frozen in time Cij field to match the analytical solution.

The velocity - conformation tensor coupling term, expressed in (3.2) and (3.6), can be written as:

∂T[p]ij

∂xj=ν[p]

τp

∂ [f(Ckk)Cij − δij ]∂xj

(4.28)

On (4.28) the terms ν[p], τp and δij are constants. Moreover, the coupling term must be verified for each

of the velocity components. Therefore, for the purpose of obtaining an analytical solution for the coupling

term, it was assumed that the main diagonal values of the Cij tensor are set to one, as this setting would

nullify the coupling term’s derivative on the same direction of the respective velocity component. The

non-diagonal values of the Cij tensor were imposed and adapted to each velocity component coupling

term.

The analytical solution is derived with detail in this section for the u velocity component. It should be

noted that the coupling term for the remaining velocity components was derived analogously.

The coupling term for the u velocity component can be expanded to the following form:

∂T[p]ij

∂x=ν[p]

τp

(∂ [f(Ckk)C11 − δ11]

∂x+∂ [f(Ckk)C12 − δ12]

∂y+∂ [f(Ckk)C13 − δ13]

∂z

)(4.29)

From (4.29) it is noticeable that the ∂x term is zero, as the Peterlin function f(Ckk) and C11 are

equal to one. For the terms C12 and C13, it was assumed a linear profile along the y and z directions,

respectively.

Considering the previous remarks, equation (4.29) can be rewritten as follows:

∂T[p]ij

∂x=ν[p]

τp

(∂C12

∂y+∂C13

∂z

)(4.30)

Therefore, C12 and C13 profiles were computed as a function with independent variable equal to the

respective partial derivative on equation (4.30). The considered profiles are shown in 4.15.

Profiles shown on 4.15 can be written as:

C12 =

zhπ5 + π, if yh < 0

− zhπ5 + π, if yh ≥ 0

(4.31)

C13 =z

h

π

5(4.32)

For this verification exercise, the following simulation parameters were considered. It should be noted

that these parameters are valid for the coupling term verification exercise in all velocity components.

Thus, for the verification test, given the properties from Table 4.2 and profiles from Figure 4.15, the

58

(a) C12 profile (b) C13 profile

Figure 4.15: C12 and C13 profiles prescribed for the coupling term on the u velocity component. H is theunit of length used in the code.

General properties Polymer properties

Box height h 2π Zero shear-rate viscosity ν[p] 0.002

Box dimensions 10× 10× 10 Maximum molecular extensibility L 100

Mesh 128× 128× 128 Relaxation time (s) τp 0.1

Polymer concentration β 0.8

Table 4.2: Coupling term simulation parameters.

analytical solution for tensor Cij is the following:

∂T[p]ij

∂x'

0.0398, if yh < 0

0, if yh ≥ 0

(4.33)

Having computed the analytical solution, it is performed the comparison with the numerical solution.

The result for the coupling term on the u velocity is presented in Figure 4.16.

From Figure 4.16, it is possible to see that the dashes lines regarding the numerical (red colour) and

analytical (blue colour) solutions follow the same trend and present the same values.

As mentioned before, the verification exercise was performed analogously for the remaining velocity

components. Now it is presented the verification results for the coupling term in the v velocity component

in Figures 4.17 and 4.18.

As observed on the coupling term for the u velocity, the dashes lines regarding the numerical (red

colour) and analytical (blue colour) solutions follow the same trend and present the same values on the

coupling term for the v velocity.

Similarly, to the v velocity component, it is shown the verification results for the coupling term in the

w velocity component in Figures 4.19 and 4.20.

As observed on the coupling term for the u velocity, the dashes lines regarding the numerical (red

colour) and analytical (blue colour) solutions follow the same trend and present the same values on the

coupling term for the w velocity.

59

Figure 4.16: Coupling term for u velocity for numerical (red) and analytical (blue) solutions.

(a) C12 profile (b) C23 profile

Figure 4.17: C12 and C23 profiles prescribed for the coupling term on the v velocity component. h is theunit of length used in the code.

Having obtained the results for the coupling term verification exercise, it is possible to conclude that

the code is verified given the precision of the results between analytical and numerical solutions.

60

Figure 4.18: Coupling term for u velocity for numerical (red) and analytical (blue) solutions.

(a) C13 profile (b) C23 profile

Figure 4.19: C13 and C23 profiles prescribed for the coupling term on the w velocity component. h is theunit of length used in the code.

Figure 4.20: Coupling term for u velocity for numerical (red) and analytical (blue) solutions.

61

62

Chapter 5

Results

This chapter is devoted to the analysis of new results obtained with the code developed within this thesis,

namely the study of the influence of polymer additives on the turbulent planar jet. It should be noted that

the results were produced as part of an ongoing investigation with Mateus C. Guimaraes [69].

The addition of polymer additives to a Newtonian solvent has proven to strongly influence the flow

physics. However, the flow physics of the visco-elastic scenario still lack understanding in comparison

to the Newtonian case, mainly due to its complexity and lack of data at an experimental and numerical

level. Some studies have been performed considering homogeneous isotropic turbulence [35, 47, 57],

but flows with a non-zero mean shear, such as planar jets, have not yet been investigated numerically

or experimentally.

It is important to note that the validation of the Newtonian code with other literature references is not

presented in this section. There has been an extensive validation process presented by [16, 18] for the

Newtonian turbulent planar jet.

Having the code been verified (see Chapter 4 for further detail), studies were carried out using DNS

of FENE-P turbulent plane jets considering the simulation parameters shown in Tables 5.1 and 5.2.

The physical length of the computational box in the x streamwise direction was chosen to allow for

the simulation of a fully developed turbulent jet. Concerning the normal direction y, the computational

domain must be large enough to accommodate the jet without interacting with the normal boundaries.

According to [18], the mesh size should be of the order of the Kolmogorov small scale, to be able to

capture the turbulent phenomenon. Moreover, since turbulence is isotropic at the small scales, the

mesh size is required to be approximately equal in all direction. In addition to this, the number of points

in the y normal and z spanwise directions are required to be decomposable into powers of 2 for better

performance of the FFTW library. Also, due to the parallelization strategy the number of points in the x

streamwise and y spanwise directions must be even.

63

nx ny nz Lx/h Ly/h Lz/h

512 512 128 18 18 4

Table 5.1: Computational domain parameters for the simulations. Number of grid points (nx, ny, nz);non-dimensional grid size (Lx/h, Ly/h, Lz/h).

It should be noted that parameters from Table 5.1 are equal to all cases presented on Table 5.2.

Moreover, the grid is uniform in the three spatial directions and that the total number of mesh points

amounts to a total of just over 33 million.

As for the physical simulation parameters, results in this section considered the cases presented in

Table 5.2.

Case Wi Re Reλ τp(s) β L ∆x/η14h

A 0 3500 100 0 1 0 4.0

B 0.33 3500 110 0.025 0.8 100 4.0

C 0.65 3500 110 0.05 0.8 100 4.0

D 1.19 3500 110 0.10 0.8 100 3.9

E 1.66 3500 140 0.20 0.8 100 3.4

F 0.20 3500 120 0.025 0.6 100 3.8

G 0.27 3500 160 0.20 0.6 100 3.3

H 1.35 3500 160 0.025 0.4 100 3.3

I 1.55 3500 180 0.20 0.4 100 3.1

J 2 3500 200 0.40 0.8 100 2.8

K 2.6 3500 220 0.60 0.8 100 2.5

L 3.3 3500 250 0.80 0.8 100 2.3

Table 5.2: Physical parameters for the simulations. Weissenberg number Wi; slot width, initial maximumjets velocity, solvent viscosity based Reynolds number Re; Averaged Taylor turbulence micro-scale cen-treline velocity solvent viscosity based Reynolds number on the x streamwise direction Reλ; polymerrelaxation time τp; ratio of the solvent to solvent plus polymer viscosity β; polymer normalized maximumextensibility L; mesh resolution normalized by the Kolmogorov small scale at x/h = 14.

The Wi represents the Weissenberg (non-dimensional) number, which is given by the ratio between

the polymer relaxation time and Kolmogorov time scale:

Wi =τpτs

(5.1)

where τs represents the Kolmogorov time scale (τs = (ν[s]/ε[s])(1/2)).

The Reynolds number based on the Taylor microscale is defined as:

Reλ =u′λ

ν[s](5.2)

where u′ is the reference perturbation velocity and λ is the Taylor scale. The Taylor scale λ is given by

64

the viscous dissipation rate assuming local isotropy at the jet centreline.

ε[s] = 15ν[s]

⟨(∂u′

∂x

)2⟩

= 15ν[s] u′2

λ2⇔ λ =

√15ν[s]u′

2

ε[s](5.3)

As the Taylor scale is a function of the x streamwise direction, the value of Reλ presented in Table

5.2 refers to an averaged value on z spanwise direction and time along the centreline for y = 0.

In order to simulate the smallest scales that are present in the flow, the grid size must be of the order

of the Kolmogorov scale η, defined in terms of the solvent turbulent viscous energy dissipation rate ε[s]

and the solvent viscosity ν[s], as shown in equation (5.4).

η =

((ν[s])3

ε[s]

)1/4

(5.4)

According to [18], several authors state that the ratio of the element size ∆x to the Kolmogorov scale

η should have a value of the order of 2.0 to 3.0. In [11], it is stated that ∆x/η = 2.1 is a good reference

value for DNS simulations of isotropic turbulence.

∆x/η presents an evolution across the x streamwise direction on the centreline, with the highest

value located at the beginning of the self-similar region and decreasing from then on. It was opted to

select to analyse this quantity at x/h = 14, as this region refers to a fully developed turbulent region

of the jet, subject to the current study. By having mesh resolution normalized by the Kolmogorov small

scales within the values stated in the literature, it was considered that the mesh size was adequate for

the study of turbulence in the fully developed region of the jet, namely for cases J to L which are the

focus of this work.

The simulations are started with variable time step and constant Courant number (CFL = 1/6).

Once self-similarity is attained, the time step is kept constant at a lower level than the minimum reached

during the first part of the simulation. At this stage, statistical quantities are computed by accumulating

data over 150.000 iterations (3 weeks computing run).

The half-width or mean flow thickness δ is defined as follows according to [69]:

δ =

∫ ∞0

u− U2L

Uc − U2Ldy (5.5)

where Uc is the centreline velocity and U2L is the local co-flow velocity.

The streamwise velocity is non-dimensionalized according to the following expression:

(U1 − U2

〈Uc〉 − U2

)2

(5.6)

where U1 is the inlet centreline velocity and U2 is the inlet co-flow velocity.

It is important to notice that all variables presented in this section are an average on time and on the

z spanwise direction.

65

0 5 10 15 20

x/h

0.5

1

1.5

2δ/h

(a)

δ = 0.110xδ = 0.083xτp = 0.0τp = 0.4τp = 0.6τp = 0.8

0 5 10 15 20

x/h

1

1.5

2

2.5

3

3.5

[(Uc−U2)/(U

1−U2)]−2

(b)

0.210(x/h − 1)0.158(x/h − 1)τp = 0.0τp = 0.4τp = 0.6τp = 0.8

Figure 5.1: Streamwise evolution of non-dimensional (a) jet’s half-width and (b) centreline velocity forfluids with different relaxation time for cases A, J, K and L. The Newtonian case refers to τp = 0. From[69].

From Figure 5.1, it is possible to identify the self-similar region by fitting straight lines to the linear

region of each curve. A good linear fit was found between x/h = 6 to x/h = 17 for the Newtonian case,

whilst for the visco-elastic case with τp = 0.8 a good linear fit was observed spanning from x/h = 10 to

x/h = 17. These windows identify the self-similar region. Despite the good linear fit observed on Figure

5.1 (b), results from case τp = 0.8 display the need of additional convergence.

Moreover, it is possible to observe that for higher polymer relaxation time there is a different flow

behaviour between the Newtonian case and the FENE-P solution. From 5.1 (a) it is noticeable a lower jet

width on the visco-elastic cases in comparison to the Newtonian case. Similarly, the centreline velocity

shows a larger value on the visco-elastic cases than on the Newtonian case. Both these remarks are

indicative of a lower energy dissipation rate for the FENE-P solutions due to the interaction of the polymer

particles on the flow physics. This result follows the expected trend, as polymer particles are expected

to absorb elastic energy at a higher rate from the flow, reducing the overall energy dissipation of the flow

and, consequently, inducing drag reduction.

On Figure 5.2 it is shown the normalized mean velocities by the local Uc centreline velocity for a fully

turbulent streamwise cross-section (x/h = 12) for cases A, J, K and L. It is also shown the root mean

squared turbulent velocity perturbation components and the Reynolds shear stress profiles for different

relaxation time.

It is possible to observe that the streamwise u velocity profiles collapse for different relaxation time.

This behaviour is like the one observed on a Newtonian case for different streamwise cross-sections, for

which the streamwise u velocity profiles collapse for different self-similar stations locations [18]. Con-

cerning the normal y velocity profile, it is noted that a higher polymer relaxation time leads to, typically,

a lower average normal velocity magnitude. It is important to notice that the spanwise w average ve-

locity is not presented, since it presents an average zero value for a planar jet [18]. As for the velocity

perturbation root mean squared components, it is observed an overall decrease of all components with

increasing polymer relaxation time. Moreover, it is seen that all perturbation velocities converge to zero

with higher distances from the centreline, as the flow is moving towards zones not perturbed by the jet.

66

Concerning the Reynolds shear stresses, the examination of Figure 5.2 indicates that for higher polymer

relaxation time the shear stress reduces, namely for cases K and L. This observation reinforces the

previous remarks, indicating a smaller energy dissipation rate for the polymer solution case due to the

energy-absorption effect of the polymers.

0 0.5 1 1.5 2 2.50

0.5

1

(u−U2L)/(U

c−U2L)

(a)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2-0.06

-0.04

-0.02

0

0.02

v/(Uc−U2L) (b)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2 30

0.1

0.2

√

u′2/(Uc−U2L)

(c)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2 30

0.1

0.2√

v′2/(Uc−U2L)

(d)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2 3

y/δ

0

0.1

0.2

√

w′2/(Uc−U2L)

(e)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2 3

y/δ

0

0.01

0.02

0.03

u′v′/(Uc−U2L)2 (f)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

Figure 5.2: Normalized mean velocity: (a) streamwise and (b) normal direction components, (c-e) ve-locity root mean squared components and (f) Reynolds shear stress profiles for fluids with differentrelaxation time for cases A, J, K and L. From [69].

67

The conformation tensor Cij are plotted for the same streamwise cross-section (x/h = 12) in Fig-

ure 5.3. It is worth mentioning that the averaged conformation tensor components C13 and C23 are not

presented in Figure 5.3, as due to the nature of the turbulent flow is it possible to prove that the aver-

age value of these components is zero. This was observed on the simulation results. Concerning the

remaining conformation tensor components, it is seen that with increasing polymer relaxation time, the

magnitude of the averaged Cij increases. Moreover, one notices that the average values converges to

the polymer equilibrium state, that is value 1 for the diagonal tensor terms and 0 for the remaining tensor

terms, on the non-perturbed region of the flow, as expected. As the elastic energy stored by a stretched

polymer is proportional to the trace of the conformation tensor [70], the observed higher average values

of the conformation tensor components with increasing polymer relaxation time indicate that the polymer

molecules have an increasing absorbed elastic energy.

0 1 2 3

0

20

40

Cij

(a)ij = 11

ij = 22

ij = 33

ij = 12

τp = 0.4

0 1 2 3

-50

0

50

100

150

Cij

(b)ij = 11

ij = 22

ij = 33

ij = 12

τp = 0.6

0 1 2 3

y/δ

0

100

200

Cij

(c)ij = 11

ij = 22

ij = 33

ij = 12

τp = 0.8

0 1 2 30

50

100√

c′2 ij

(d)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.4

0 1 2 30

50

100

150

√

c′2 ij

(e)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.6

0 1 2 3

y/δ

0

100

200

√

c′2 ij

(f)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.8

Figure 5.3: Mean (a-c) and root mean squared (d-f) components of the conformation tensor for fluidswith different relaxation time at x/h = 12 for cases J, K and L. From [69].

68

On Figure 5.4 it is presented the velocity perturbation components on the streamwise direction nor-

malized by the local centreline velocity, along with the root mean squared conformation tensor compo-

nents.

It is observed that with increasing polymer relaxation time, the perturbation velocity profile (typically

starting around x/h = 3) for all velocity components presents a smoother transition behaviour with a

lower perturbation velocity magnitude.

As for the root mean squared conformation tensor components, it is noticeable their values are

increasing with higher polymer relaxation time, with foremost relevant changes on terms C11 and C12.

Also, it is clear the presence of an abrupt increase on the root mean squared conformation tensor

components, which is indicative of a regime modification in the flow.

0 5 10 15 200

0.1

0.2

√

u′2/(Uc−U2L)

(a)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

0 5 10 15 200

0.1

0.2

√

v′2/(Uc−U2L)

(b)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

0 5 10 15 20

0

0.05

0.1

0.15

√

w′2/(Uc−U2L)

(c)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

-5 0 5 10 150

20

40

60

80

√

c′2 ij

(d)

ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.4

-5 0 5 10 15

x/h

0

50

100

√

c′2 ij

(e)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.6

-5 0 5 10 15

x/h

0

50

100

150

√

c′2 ij

(f)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.8

Figure 5.4: Centreline root mean squared evolution of (a) streamwise, (b) normal and (c) spanwise direc-tions velocity components and (d-f) conformation tensor components for fluids with different relaxationtime for cases J, K and L. From [69]

69

The Weissenberg number compares the elastic forces to the viscous forces and an analysis for the

visco-elastic simulations is presented in Figure 5.5. It is observed that for higher polymer relaxation

time with the same β (especially clear on cases with β = 0.8), Wi number increases. Simultaneously,

the same cases when viewed from a turbulent kinetic energy dissipation perspective (see Figure 5.6)

present a lower viscous energy dissipation, which in turn would contribute to a lower Wi number. Thus,

this indicates that the polymer elastic forces on the flow are far more significant than the solvent viscous

forces.

If one considers cases with an equal τp, it is noticeable that higher β leads to a higher Wi number.

This is observed, for example, in cases with τp = 0.2s. Nonetheless, the variation on the ratio between

elastic and viscous forces on the flow is clearly smaller with the influence of β than with the influence of

τp.

0 5 10 15 20 25

x/h

0

1

2

3

Wi=

τ p/√

ν[s] /ε[s] τp = 0.025, β = 0.8

τp = 0.025, β = 0.6τp = 0.025, β = 0.4τp = 0.050, β = 0.8τp = 0.100, β = 0.8τp = 0.200, β = 0.8τp = 0.200, β = 0.6τp = 0.200, β = 0.4τp = 0.400, β = 0.8τp = 0.600, β = 0.8τp = 0.800, β = 0.8

Figure 5.5: Stream-wise evolution of centreline Weissenberg numbers for all visco-elastic cases. From[69].

As for the viscous energy dissipation rate presented in Figure 5.6, it is noted that the presence of

polymer additives on the flow leads to a lower viscous energy dissipation rate. Moreover, for the same τp

a lower β leads to lower viscous energy dissipation. Similarly, for the same β a higher τp leads to lower

viscous energy dissipation.

0 5 10 15 20 25 30

x/h

0

2

4

6

8

10

ε[s]h/U

3 1

Newtonian fluidτp = 0.025, β = 0.8τp = 0.025, β = 0.6τp = 0.025, β = 0.4τp = 0.050, β = 0.8τp = 0.100, β = 0.8τp = 0.200, β = 0.8τp = 0.200, β = 0.6τp = 0.200, β = 0.4τp = 0.400, β = 0.8τp = 0.600, β = 0.8τp = 0.800, β = 0.8

Figure 5.6: Stream-wise evolution of centreline turbulent kinetic energy dissipation of the solvent for allcases. From [69].

70

In Figure 5.7 it is presented the average conformation tensor trace components along the x stream-

wise direction.

It is noticeable that a higher τp induces higher conformation trace components. This behaviour is

also verified for the root mean squared components presented in Figure 5.4. Together with the observed

lower jet width and overall lower turbulent energy dissipation rate for higher polymer relaxation time, it is

indicative that the polymer stress due to larger polymer elongations (Cij) compensates the effect of the

increased relaxation time in reducing the polymer dissipation.

0 5 10 15 20 25

x/h

0

50

100

Cij

C11, τp = 0.4C22, τp = 0.4C33, τp = 0.4C11, τp = 0.6C22, τp = 0.6C33, τp = 0.6C11, τp = 0.8C22, τp = 0.8C33, τp = 0.8

Figure 5.7: Stream-wise evolution of centreline mean conformation tensor components for cases J, Kand L. From [69].

Future work will be developed on this area and ongoing investigation will be presented in [71].

71

72

Chapter 6

Conclusions

6.1 Achievements

In the present work it was implemented the Finite Extensible Nonlinear Elastic continuous model closed

with the Peterlin approximation (FENE-P) within an existing DNS tool, resulting in a DNS tool for turbulent

spatial jet with polymer additives. It is important to stress that DNS or experimental analyses of turbulent

spatial jet have not been found in the literature. As such, this work provides the first time a turbulent

spatial jet is analysed with a FENE-P visco-elastic configuration, leading to a new ground breaking

numerical tool with potential for new study areas. It is also worth mentioning that there is already

being performed investigation on a more thorough level regarding the influence of polymer additives on

turbulent planar jets with polymer additives in LASEF, which will result in conference proceeding and

technical articles [71].

During the development of the current work, extensive care was taken concerning the numerical

implementation to ensure impact minimization of the implementation of FENE-P on the overall DNS code

performance. Such was achieved, at a certain level, with the inclusion of the ghost cells mechanism and

minimization of the call of MPI commands, which allowed a relevant performance enhancement due

to the need of less communication time. In fact, iteration times with the inclusion of FENE-P on the

spatial jet code resulted in an increase of 7-fold, whilst for comparable homogeneous isotropic turbulent

FENE-P simulations a 10-fold increase was observed.

Further to the previous remarks, it was performed an extensive verification of the implemented code

with satisfactory result. This step is rather important because no comparable simulations were found in

the literature regarding FENE-P. Having the code verified, the results obtained were able to provide a

first meaningful insight regarding turbulent spatial jet simulations with polymer additives.

As for the results of DNS analyses of turbulent planer jets with polymer additives, it was possible to

obtain meaningful insight on the physics of the turbulent flow.

On the turbulent regime, it was analysed the evolution of the jet’s characteristics, the conformation

tensor components evolution and the turbulent energy dissipation rate. Regarding the jet’s characteris-

tics, focus was given on the jet’s width, velocity components evolution and Reynolds shear stress profile.

73

For all the abovementioned, the difference between the Newtonian fluid solution and FENE-P solution

is clear. The jet width in the FENE-P solution is lower than the one observed for the Newtonian case,

which indicates that the viscous energy dissipation is smaller for the polymer solution case. Similarly,

the centreline velocity shows a higher value for polymer solution than for the Newtonian case, confirm-

ing the previous remark. As for the remaining components, results show that velocity perturbations on

turbulent regime are lower in the visco-elastic solution and that the Reynolds shear stress profiles de-

creases on the FENE-P solution. Both previous remarks confirm a smaller energy dissipation rate for

the polymer solution case due to the energy-absorption effect of the polymers, which is indicative of the

drag reduction phenomenon.

As for the conformation tensor components evolution, it is observed an increase of these components

with higher polymer relaxation time. This is followed by a lower jet width and overall lower turbulent

energy dissipation rate for higher polymer relaxation time. Such indicates that the polymer stress due to

larger polymer elongations (Cij) compensates the effect of the increased relaxation time in reducing the

polymer dissipation. On works such as [47] related to homogeneous isotropic turbulence, it has been

shown that for increasingly large Wi there is a decrease in the fraction of the power dissipated by the

polymers. This scenario was not observed on the turbulent planar jet with polymer additives.

6.2 Future Work

In order to continue the work done in the current thesis, the following topics could be addressed:

• To analyse with more detail the interaction of polymer particles on the turbulent spatial jet. Addi-

tional analyses could include a better understanding the energy dissipation of the polymer or study

the polymer characteristics to achieve the upper limit of drag reduction. As mentioned previously,

there is already being performed investigation on a more thorough level regarding the influence of

polymer additives on turbulent planar jets with polymer additives in LASEF.

• Study the implementation of a hybrid parallelization approach, as the problem under studying is

highly scalable and would benefit from an hybrid MPI/OpenMP approach, namely on communica-

tion time and I/O. It should be noted that a first step has been performed for this scenario in [72],

with promising results for the Newtonian turbulent jet case.

74

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80

Appendix A

Stability condition proof

In this appendix, it is shown the proof for the Courant condition that must be satisfied in order to ensure

that the conformation tensor Cij remains positive definite. The proof is presented in [36] and is based

on the fact that a convex sum of positive definite matrices is itself positive definite. For example, matrix

C∗ defined as

C∗ =

N∑l=1

slCl (A.1)

where the coefficients sl ≥ 0 satisfy:N∑l=1

sl = 1 (A.2)

is positive definite if the matrices Cl are positive definite.

The Courant condition for numerical stability is obtained by considering the convection term of equa-

tion (3.33).

The numerical approximation of the convective term in many schemes strives to satisfy the maximum

principle of the underlying partial differential equation, leading to a convex representation. This is a

convenient representation for a bounded scalar (tensor), as it immediately implies the bounds will be

obeyed. For first-order methods, the sl weights depend only on the advecting velocity field and are

independent of the scalar (tensor) values themselves. However, for a second-order update, the weights

are functions of the velocity and the scalar (tensor) values at the node of interest and the neighbouring

nodes. Therefore, ensuring the weights remain convex therefore depends upon how the scalar (tensor)

values at the cell edges are approximated.

Considering the convection term alone, the following relationship can be derived from equations

(3.35), (3.36) and (3.37):

Ci,j,k =1

6

(C−i+1/2,j,k + C+

i−1/2,j,k + C−i,j+1/2,k + C+i,j−1/2,k + C−i,j,k+1/2 + C+

i,j,k−1/2

)(A.3)

81

Applying the previous relationship to equation (3.33), it is obtained:

C[n+1]i,j,k = s1C

+i+1/2,j,k + s2C

−i+1/2,j,k + s3C

+i−1/2,j,k + s4C

−i−1/2,j,k + s5C

+i,j+1/2,k + s6C

−i,j+1/2,k+

s7C+i,j−1/2,k + s8C

−i,j−1/2,k + s9C

+i,j,k+1/2 + s10C

−i,j,k+1/2 + s11C

+i,j,k−1/2 + s12C

−i,j,k−1/2

(A.4)

where the sl weights are given by:

s1 ≡(−ui+1/2,j,k + |ui+1/2,j,k|

)( ∆t

2∆x

), s2 ≡

1

6+(−ui+1/2,j,k − |ui+1/2,j,k|

)( ∆t

2∆x

),

s3 ≡1

6+(ui−1/2,j,k − |ui−1/2,j,k|

)( ∆t

2∆x

), s4 ≡

(ui−1/2,j,k + |ui−1/2,j,k|

)( ∆t

2∆x

),

s5 ≡(−vi,j+1/2,k + |vi,j+1/2,k|

)( ∆t

2∆y

), s6 ≡

1

6+(−vi,j+1/2,k − |vi,j+1/2,k|

)( ∆t

2∆y

),

s7 ≡1

6+(vi,j−1/2,k − |vi,j−1/2,k|

)( ∆t

2∆y

), s8 ≡

(vi,j−1/2,k + |vi,j−1/2,k|

)( ∆t

2∆y

),

s9 ≡(−wi,j,k+1/2 + |wi,j,k+1/2|

)( ∆t

2∆z

), s10 ≡

1

6+(−wi,j,k+1/2 − |wi,j,k+1/2|

)( ∆t

2∆z

),

s11 ≡1

6+(wi,j,k−1/2 − |wi,j,k−1/2|

)( ∆t

2∆z

), s12 ≡

(wi,j,k−1/2 + |wi,j,k−1/2|

)( ∆t

2∆z

)

(A.5)

The condition that the weights sl ≥ 0 is ensured if the following Courant number is satisfied:

CFL ≡ max(|u|∆x

,|v|∆y

,|w|∆z

)∆t <

1

6(A.6)

Summing the sl weights, one obtains:

∑l

sl = 1− ∆t

2

[ui+1/2,j,k − ui−1/2,j,k

∆x+vi,j+1/2,k − vi,j−1/2,k

∆y+wi,j,k+1/2 − wi,j,k−1/2

∆z

](A.7)

From equation (A.7) it is possible to notice that the term in brackets is equal to the term presented

in equation (3.42), which has been shown to be equal to zero. In combination with equation (A.1), it is

ensured that the eigenvalues of C[n+1]i,j,k remain positive and that its sum remain below L2.

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