Diplomarbeit - UPM · Diplomarbeit (Proyecto Fin de Carrera) An inverse RANS simulation of a...

Diplomarbeit(Proyecto Fin de Carrera)

An inverse RANS simulationof a turbulent channel flowat moderate Reynolds numbers

Student: Florian TuerkeSupervisors: Prof. Dr. Javier Jimenez

Prof. Dr.-Ing. Frank Thiele

Universidad Politécnica de Madrid

Escuela Técnica Superior deIngenieros AeronáuticosDepartamento de Ingeniería Termodinámicay MotorpropulsionProf. Dr. Javier Jiménez

Technische Universität Berlin

MadridMay 24, 2011

Fakultaet V Verkehrs- und MaschinensystemeInstitut für Strömungsmechanik undTechnische AkustikFachgebiet Computational FluidDynamics and AeroacousticsProf. Dr.-Ing. Frank Thiele

Eidesstattliche Erklärung

Hiermit erklare ich an Eides statt, die vorliegende Arbeit selbststandig und nur

unter Verwendung der angegebenen Literatur und Quellen erstellt zu haben.

Florian Tuerke

Madrid, May 24, 2011

Acknowledgements

I would like to thank a number of people for their assistance, discussions, ideas

and interest, as well as their support throughout the research project, including

the following: Professor Dr. Javier Jimenez, Dr. Frank Thiele, Dr. Octavian

Frederich, Adrian Lozano Duran, Guillem Borrell, Dr. Ricardo Garcia Mayoral,

Dr. Ayse Gul Gungor, Juan A. Sillero, Pablo Garcia Ramos and Professor Dr.

Sergio Hoyas. This work was also made possible by the generous collaboration with

the Marenostrum Supercomputing Center in Barcelona, who lent their processing

computers and storage facilities.

Also many thanks, to the Erasmus foundation and the TU Berlin for their financial

support and for having offered me the possibility to study abroad, as well as to my

parents for their financial support.

For inspiration and support I want to thank Hari, Maria y Costanza.

Contents

1 Introduction 18

1.1 The nature of turbulence . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2 Motivation for current work . . . . . . . . . . . . . . . . . . . . . . 22

2 General Description of Turbulence 23

2.1 The Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 The equations of fluid motion . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 The continuity equation . . . . . . . . . . . . . . . . . . . . 25

2.2.2 The momentum equation . . . . . . . . . . . . . . . . . . . . 25

2.3 Statistical description of turbulent flow . . . . . . . . . . . . . . . . 26

2.3.1 Reynolds decomposition . . . . . . . . . . . . . . . . . . . . 26

2.3.2 The mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.3 Statistics for the Channel Experiment . . . . . . . . . . . . . 28

2.4 Scales of turbulent motion . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.1 The turbulent kinetic energy spectrum . . . . . . . . . . . . 30

2.4.2 The energy cascade . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.3 The Kolmogorov hypotheses . . . . . . . . . . . . . . . . . . 33

3 Wall-Bounded Turbulent Flow 36

3.1 Models for the near wall region . . . . . . . . . . . . . . . . . . . . 38

3.1.1 The viscous sublayer . . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 The log-layer . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Dynamics of wall bounded flow . . . . . . . . . . . . . . . . . . . . 40

3.2.1 The viscous sublayer . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 The logarithmic region . . . . . . . . . . . . . . . . . . . . . 42

6

Diplomarbeit Contents

4 The Numerical Method 44

4.1 Direct Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . 44

4.2 The numerical procedure . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Derivation of the governing equations . . . . . . . . . . . . . 46

4.2.2 Initial and Boundary Conditions . . . . . . . . . . . . . . . . 48

4.2.3 Spectral Method . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.4 Spacial Resolution . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.5 Time Resolution . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.6 Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 The Numerical Experiment 54

5.1 Computational Domain and Numerical Issues . . . . . . . . . . . . 54

5.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2.1 Fixing the mean velocity profile . . . . . . . . . . . . . . . . 57

5.2.2 Natural and unnatural profiles . . . . . . . . . . . . . . . . . 57

5.2.3 Influence on the Reynolds number . . . . . . . . . . . . . . . 63

5.3 Blending mean profiles . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3.1 Blending technique . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.2 Variation of blending loctation . . . . . . . . . . . . . . . . . 65

6 Results 69

6.1 Statistics for β-cases . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2 Spectral results of β-cases . . . . . . . . . . . . . . . . . . . . . . . 75

6.3 Results of blended cases . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4 Intersection Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.5 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.6 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.6.1 Forcing Term . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.6.2 Reynolds stress budget . . . . . . . . . . . . . . . . . . . . . 97

6.7 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 100

6.7.1 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7

Diplomarbeit Contents

6.7.3 Sensibility study of the linear model . . . . . . . . . . . . . 103

6.8 Coherent Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.9 Release of fixed mean profile . . . . . . . . . . . . . . . . . . . . . . 113

7 Discussion and Conclusions 116

8

List of Figures

1.1 Space Shuttle launch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2 Turbulent flow of a cigarette . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Total Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Diviatoric Reynolds stress . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 1D streamwise energy spectra of channel flow at y+ = 15 . . . . . . . . . . . 30

2.4 2D engery spectra at y+ = 15 . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.1 Computational Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3 Mean profile and total stress for various channel sizes . . . . . . . . . . . . 57

5.4 Fit of Cess formula to “Torroja” profile . . . . . . . . . . . . . . . . . . . 60

5.5 Variation of A and κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.6 Variation of mean velocity profile . . . . . . . . . . . . . . . . . . . . . . 62

5.7 Zoomed-in view of figure on the left . . . . . . . . . . . . . . . . . . . . . 62

5.8 Mean velocity profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.9 Zoomed in view of left graph . . . . . . . . . . . . . . . . . . . . . . . . 62

5.10 Blending of two mean profiles . . . . . . . . . . . . . . . . . . . . . . . . 65

5.11 β = −0.5 blendings and their first derivative for Reτ = 550 . . . . . . . . . . 66

5.12 β = −1.0 blendings and their first derivative for Reτ = 950 . . . . . . . . . . 67

6.1 Statistics for fixed mean velocity profile at Reτ = 550 . . . . . . . . . . . . 70

6.2 Statistics for β = −0.5 and β = −1.0 profiles . . . . . . . . . . . . . . . . 71

6.3 Comparison of Reynolds stress for Reτ = 550 and Reτ = 950 cases . . . . . . 72

9

Diplomarbeit List of Figures

6.4 Isotropy coefficients and structure coefficients . . . . . . . . . . . . . . . . 73

6.5 2D streamwise velocity spectra . . . . . . . . . . . . . . . . . . . . . . . 75

6.6 2D streamwise vorticity spectra . . . . . . . . . . . . . . . . . . . . . . . 75

6.7 Pre-multiplied 1D spectra of TKE . . . . . . . . . . . . . . . . . . . . . 77

6.8 Energy in zero modes of streamwise direction . . . . . . . . . . . . . . . . 78

6.9 Energy in zero modes of wall normal direction . . . . . . . . . . . . . . . . 78

6.10 Energy in zero modes of spanwise direction . . . . . . . . . . . . . . . . . 79

6.11 Statistics of β = −0.5 blendings . . . . . . . . . . . . . . . . . . . . . . . 80


6.13 Structure coefficient and fluctuations . . . . . . . . . . . . . . . . . . . . 81

6.14 Comparison of Reynolds stress for Reτ = 550 and Reτ = 950 cases . . . . . . 82

6.15 Energy in streamwise zero modes for blendings . . . . . . . . . . . . . . . 83

6.16 2D spectral results of the streamwise velocity . . . . . . . . . . . . . . . . 84

6.17 2D spectral results of the ωz vorticity component . . . . . . . . . . . . . . 85

6.18 Intersection of Reynolds stresses . . . . . . . . . . . . . . . . . . . . . . 87

6.19 Mean profile and mean shear of the “intersection” analysis . . . . . . . . . . 87

6.20 Total stress of the “intersection” analysis . . . . . . . . . . . . . . . . . . 88

6.21 Total stress√

τtotal for blending cases . . . . . . . . . . . . . . . . . . . . 90

6.22 Normalized streamwise velocity fluctuations . . . . . . . . . . . . . . . . . 91

6.23 Normalized wall normal velocity fluctuations . . . . . . . . . . . . . . . . . 91

6.24 Normalized spanwise velocity fluctuations . . . . . . . . . . . . . . . . . . 91

6.25 Normalized streamwise velocity fluctuations . . . . . . . . . . . . . . . . . 92

6.26 Normalized wall normal velocity fluctuations . . . . . . . . . . . . . . . . . 92

6.27 Normalized spanwise velocity fluctuations . . . . . . . . . . . . . . . . . . 92

6.28 Comparison of 950 and 550 cases . . . . . . . . . . . . . . . . . . . . . . 93

6.29 Total stresses and extra energy . . . . . . . . . . . . . . . . . . . . . . . 97



6.32 Spectrum of Eigenvalues for a given pertubation . . . . . . . . . . . . . . . 101

6.33 Maximum linear transient amplification of perturbations . . . . . . . . . . . 102

6.34 Comparison calculation technique of linear stability analysis . . . . . . . . . 104

10

Diplomarbeit List of Figures

6.35 Stadard Deviation of D for various cases . . . . . . . . . . . . . . . . . . 107

6.36 Visualisation of clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.37 Histograms of clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.38 Joint p.d.f. of attached and detached clusters at Reτ = 550 . . . . . . . . . . 110

6.39 Joint p.d.f. of attached clusters at Reτ = 950 . . . . . . . . . . . . . . . . 112

6.40 Statistics for fixed mean velocity profile at Reτ = 550 . . . . . . . . . . . . 113

6.41 2D streamwise velocity spectra . . . . . . . . . . . . . . . . . . . . . . . 114

6.42 Energy during release of mean profile . . . . . . . . . . . . . . . . . . . . 115

11

List of Tables

5.1 Summary of box sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Flow quantities of 550 channel . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Flow quantities of 950 channel . . . . . . . . . . . . . . . . . . . . . . . 64

5.4 Summary of blending cases for 550 channel . . . . . . . . . . . . . . . . . 67

5.5 Summary of blending cases for 950 channel . . . . . . . . . . . . . . . . . 68

6.1 Summary of cases for transient linear stability analysis . . . . . . . . . . . . 104

6.2 Summary of fitted parameter for transient linear stability analysis . . . . . . 104

12

Nomenclature

Arabic Symbols

1D One Dimensional

2D Two Dimensional

f Instantaneous value of forcing term

u Instantaneous velocity component in x-direction

v Instantaneous velocity component in y-direction

w Instantaneous velocity component in z-direction

A Cess parameter

a Acceleration

b Channel width

C Cascade power law constant

D Discriminant of velocity gradient

D′ Standard deviation of discriminant of velocity gradient field

dt Time differential

E Energy

13

Diplomarbeit

ETT Eddy turnover time

F Mean value of forcing term

f Fluctuation of forcing term

FFT Fast Fourier Transform

FFT Fast fourier transform

H Convection term

h Channel half hight

h+ Channel half hight in wall units

Iu Streamwise isotropy coefficient

Iw Spanwise isotropy coefficient

K Turbulent kinetic energy

k Turbulent kinetic energy

Ka Karman constant

L Channel length

l Turbulent lenght scale

l0 Lenght scale of large eddies

Lǫ Integral length scale

m Mass

N Number of grid points

n Collocation points

NG Number of grid points

Nx Number of points in x-direction

14

Diplomarbeit

Ny Number of points in y-direction

Nz Number of points in z-direction

NOP Number of arithmetic operations

P Production

p Pressure

p.d.f. Probability density function

Pi Bezier points

Q Invariant of velocity gradient

R Invariant of velocity gradient

R.H.S. Right hand side

Re Reynolds number

Reτ Reynolds number base don the friction velocity

RMS Root mean square

T Wall time

t Time

Tm Chebyshev polynomial

Tn Chebyshev polynomial

TKE Turbulent kinetic energy

U Mean of Velocity

u Velocity fluctuation in x-direction

u′ R.M.S. of streamwise velocity fluctuation

u+ streamwise velocity component in wall units

15

Diplomarbeit

u0 Velocity scale of large eddies

uη Kolomogorov velocity scale

uτ Friction velocity

Ub Bulk Velocity

Ufalse Unnatural streamwise mean velocity profile

Utrue Natural streamwise mean velocity profile

v Velocity fluctuation in y-direction

v′ R.M.S. of wall-normal velocity fluctuation

Vclus Volume of cluster

w Velocity fluctuation in w-direction

w′ R.M.S. of spanwise velocity fluctuation

x Streamwise coordinate

Y With channal half hight h normalized wall normal coordinate

y Wall normal coordinate

y+ Wall normal distance in wall units

z Spanwise coordinate

Greek Symbols

β Profile mixing variable

∆t Time increment

∆x Spatial increment

∆ Increment

16

Diplomarbeit

δv Viscious length scale

ǫ Dissipation

η Kolomogorov length scale

η+ Kolmogorov length scale in wall units

κ Wave Number and Cess parameter

κ0 Wave number of zero modes

λ Wavelength

µ Dynamic viscosity

ν Kinematic viscosity

νtot Eddy viscosity

ω Vorticity

ρ Density

τ Shear stress

τ0 Time scale of large eddies

τη Kolomogorov time scale

τw Wall shear stress

θ Chebyshev variable

17

1 Introduction

The world around us is turbulent. In nature, turbulence is the norm, not the

exception - from the smoke of a cigartte, to stormy winds, to tumultuous flood

waters, to rivers and water falls, turbulence is everywhere and has captured the

attention of many researchers in the past century. In engineering applications

turbulent flows are omnipresent and of great interest to companies whose products

are influenced by or operate in fluids in motion. The flow of the air over an aircraft

wing, blood flow in arteries, oil transport in pipelines, lava flow after a volcano

eruption, atmospheric and oceanic currents or stellar nebula as well as the mixing of

fuel and air in combustion chambers of gas turbines, we are surrounded by, and make

use of turbulence in our daily life. Understanding the nature of turbulence allows

us to influence, control it and take advantage of it by for example enhancing mixing

in processes which would not be feasible without its presnece. Thus saving money

and resources through minimizing losses due to friction, imperfect combustion and

other forms of energy dissipation. According to [6] about half the energy spent

worldwide to move fluids around or to move vehicles through fluids, is dissipated

by turbulence in the immediate vicinity of the wall.

Turbulence or turbulent flow is characterized by chaos. A set of equations, namely

the Navier- Stokes Equations, give a complete description of the turbulent flow.

Though, their strength to describe every detail of the flow becomes their burden,

since they result in fairly complex behavior and analytical solutions to even the

simplest turbulent flow problems do therefore not exist. The flow variables as a

function of space and time can only be obtained numerically. The most accurate,

though also by far the most expensive technique (resolving all time and lengths scales

18

Diplomarbeit 1 Introduction

relevant to turbulence) to simulate a flow numerically, is called Direct Numerical

Simulation, or in short DNS. It is used in the present work to compute the flow

in a channel to study the influence of a fixed mean profile on the quantities such

as velocity fluctuations and structures. This way the mechanisms of how energy is

drawn from the mean flow and fed into smaller scales is studied.

Even though DNS is the most exact tool to calculate turbulent flow, it is not

applicable for engineering computations, due to its computational cost. The en-

gineering computation relies on simpler methods such as the computationally cheap

Rynolds-averaged Navier-Stokes (RANS) simulations or Large Eddy simluations

(LES), which are intermediate in complexity between RANS and DNS. Describing

the flow variables statistically leads to the notorious closure problem, which can

only be overcome by modelling the terms, that cannot be calculated directly. The

search for improved models through a better understanding of physical phenomena

is the main objective of modern day turbulence research.

In the early days of turbulence research those models were solely based on ex-

perimental data from channel experiments. Though, with the large increase in

computational power during the last two decades, DNS has become a strong and

impressive research tool. The power of DNS not only lies within obtaining the flow

variables in a high resolved threedimensional domain, but also in the capability of

studying unphysical flow phenomena and therefore testing, validating and improving

current understandings of turbulence and its underlying mechanisms. One such

unphysical phenomena is to fix the velocity profile of the mean flow while letting

the fluctuations evolve freely. In the present work, this feature was implemented in a

fully spectral, incompressible DNS code, to evaluate its influence on flow quantities

and turbulent structures in a fully developed channel flow at Reynolds numbers

based on the friction velocity of Reτ = 550 and Reτ = 950, respectively.

The work is structured as follows. After an introductory characterization of tur-

bulence and the motivation for the present work, the mathematical equations that

19


Figure 1.1: Turbulent flow during

the launch of Space Shut-

tle Atlantis

Figure 1.2: Turbulent motion in the

smoke of a cigarette

govern the flow as well as the statistical tools that are used to analyze it are pre-

sented in chapter 2. Also the importance of the multi-scale character of turbulence

and the energy spectrum are shortly discussed in the light of the probably most

important contributor to turbulence research Andrei Kolmogorov. In chapter 3 the

characteristics of wall-bounded flow are shortly reviewed, while chapter 4 introduces

the numerical method used in the present work. It follows chapter 5 where the

numerical experiment is outlined. The results and their discussion are presented in

chapter 6, while chapter 7 concludes.

1.1 The nature of turbulence

A turbulent flow is characterized by disorder in space and time, which leads to

chaos and thus naturally to instationary behavior. In contrast to a laminar flow

which is stable to small pertubations, a turbulent flow is unstable by nature and

small perturbation will amplify. Though, well organized structures can be observed

in different length and time scales. The multiscale character of turbulent flow is

one of the most important features, since it leads to the very problem of numerical

simulation of turbulence. Energy is fed into large scales, which are determined by

the geometry of the flow and is passed down, in what is called the energy cascade,

to smaller scales, which are considered isotropic and therefore independent of the

flow geometry, where it is dissipated. Since energy disspation plays a crucial part in

20


a turbulent flow, the spatial discritization has to be very fine (approach for DNS)

or a model has to be used for the small scales (approach for LES or RANS).

Turbulent flows are subject of heavy mixing, which greatly increases the transport

of matter, momentum and heat compared to laminar flow. Compared to the

turbulent diffusion, except for very close to the wall, the molecular diffusion can

be considered insignificant. From observation one will agree that turbulent flows

are rotative, which implies that vorticity (curl of velocity field) plays a major

role. Since vorticity behaves very differently in three dimensions than it does

in two dimensions, a turbulent flow has to be considered three dimensional. As

shown in [4] vorticity in two dimensions cannot be amplified, whereas for high

Reynolds number flows, in three dimensions vorticity is proportional to the angular

momentum of the fluid. Since the pressure gradient, which is the only real force in

an incompressible inviscid fluid, is irrotational and unable to influence the angular

momentum, vorticity represents a conserved quantity. Vortices are therefore good

candidates for the equivalent of objects that can be individually followed as the fluid

moves around.

Furthermore, turbulent flow is random and unpredictable, in the sense that a

small uncertainty at a given time will amplify in the manner that a deterministic

prediction of its evolution is impossible. Statistical tools as described in section 2.3

must be used to make the flow mathematically quantifiable.

Figure 1.1 shows exemplarily the nature of turbulence. The seemingly chaotic

exhaust during the launch of the Space Shuttle Atlantis exhibits turbulent motions

at different length scales. In figure 1.2 the flow of the cigarette smoke enters the

picture in a laminar motion (lower left corner) and transitions into chaotic motion:

Turbulence.

21


1.2 Motivation for current work

The motivation to fix the mean profile of the streamwise velocity component of

a turbulent channel flow originally arose from the desire to reduce the computing

time to obtain a converged solution of the turbulent flow field. Especially the

large scale structures, which for example for a channel flow are well know, take

expensive computing time to reach a converged state, while the smaller scales, due

to smaller time scales with which they are associated, reach the converged state

faster. Once the smaller structures had adapted to the larger structures, everything

could be released to compute the actual flow. Unfortunately this procedure did

not work but resulted in unexpected growth in the Reynolds stresses. It was

decided to investigate this phenomenon systematically and in greater detail, which

is the subject of the present work. A fixed mean profile, which can be interpreted

as an inverse RANS simulation since everything except for the mean profile is

calculated, was implemented in a fully spectral, incompressible DNS code, to study

the interaction between the mean flow and the fluctuations of turbulent motion.

That once the energy resides in the fluctuations, it is clear that it gradually moves

down through the energy cascade to smaller scales, where it is eventually dissipated.

However the mechanism, with which the fluctuations draw energy from the mean

profile and to what extend the Reynolds stresses and the mean velocity gradient

interact to produce turbulence is still subject of current investigation and not yet

very well understood. The hope of the present work is to find some further evidence

on how this interaction might work.

22

2 General Description of Turbulence

2.1 The Reynolds Number

The non-dimensional parameter, called the Reynolds number, was introduced by

Reynolds in 1883. It characterizes the relative importance of inertial forces over

viscous forces in the flow.

Re =inertia forces

viscous forces=

ρudu/dx

µd2u/dx2

Applying the scaling dV/dx = V/h, where h is the channel hight and u the instan-

taneous fluid velocity, equation 2.3 becomes

Re =ρuu/h

µu/h2=

ρuh

µ=

uh

ν

where ν is the kinematic viscosity and µ is the dynamic viscosity with ν = µρ. This

general definition of the Reynolds number given in equation 2.1 becomes

Re =Ub

ν(2.1)

for a channel flow, where Ub is the bulk velocity defined as

Ub =1

h

∫ h

0

u (2.2)

and h = 1. In the present work the bulk velocity was normalized to obtain Ub =

0.899 and held constant for all simulations.

The“Reynolds number”used in the code is for reasons of convenience simply defined

as the inverse of the kinematic viscosity

23

Diplomarbeit 2 General Description of Turbulence

Re =1

ν(2.3)

and therefore slightly higher then the acutal Reynolds number based on the bulk

velocity.

A flow is considered laminar for Re < 1, 350 and fully turbulent for Re > 1800

as stated in [1]. Since the flow for the current work reaches Reynolds numbers of

Re > 10000, the channel can be considered fully turbulent.

Several other definitions of Reynolds numbers, using different velocities, can be

defined. For the turbulent channel flow the friction velocity

uτ =

√

τw

ρ(2.4)

with the wall shear stress τw defined as

τw = ρν

(

dU

dy

)

y=0

(2.5)

where U denotes the mean velocity, is commonly used to define the friction Reynolds

number

Reτ =uτh

ν(2.6)

The friction Reynolds number of the simulations in the current work is held constant

at Reτ = 550 and Reτ = 950, respectively.

2.2 The equations of fluid motion

Applying the Navier-Stokes equations, the fluid is assumed to behave as a continous

medium. The so called continuum hypothesis holds for turbulent flow since the

smallest length and time scales encountered in turbulence are still several oders

of magnitude larger than the molecular scales. In this chapter the incompressible

(ρ = const = 1) equations of basic fluid dynamics are presented. The velocity is

24


assumed to be sufficiently low to omit the influence of compressibility. Therefore

the continuity equation and the momentum equations completely describe the flow

field.

2.2.1 The continuity equation

The conservation of mass is given by

∂ρ

∂t+ ∇ · (ρu) = 0 (2.7)

Where u is the fluid velocity and for constant density flow this yields

∇ · u = 0 (2.8)

which means that the flow is divergence-free or solenoidal.

2.2.2 The momentum equation

The conservation of momentum is based on Newtons’ second law: F = m · a. It

relates acceleration of fluid particles to the surface and body forces experienced by

the fluid. Neglecting gravity and for now any kind of body forces (later a body

force will be added by fixing the mean profile), the only remaining force is the stress

tensor τij, which in defined, assuming a Newtonian fluid, as

τij = −pδij + µ

(

∂ui

∂xj

+∂uj

∂xj

)

(2.9)

where µ is the dynamic viscosity and p the pressure. With equation 2.8 the shear

stress τij is comprised of the sum of the isoptropic contribution −pδij and the

diviatoric contribution µ(

∂ui

∂xj+

∂uj

∂xi

)

.

According to the momentum equation, forces cause the fluid to accelerate and it

follows

25


ρDui

Dt=

∂τij

∂xj

ρ∂ui

∂t+ ρuj

∂ui

∂xj

=∂τij

∂xj

(2.10)

or with 2.9

∂ui

∂t+ uj

∂ui

∂xj

= − ∂p

∂xi

+ ν∂2ui

∂x2i

(2.11)

where ν = µ/ρ is the kinematic viscosity. This set of equations is called the Navier-

Stokes equations and together with the continuity equation 2.8 it governs the flow

of a fluid, no matter laminar or turbulent.

2.3 Statistical description of turbulent flow

Since the turbulent velocity field of a fluid flow is random, statistical methods have

to be used to describe it. Even though the underlying Navier-Stokes equations

are a deterministic set of equations, turbulent flows display a strong sensitivity to

unavoidable perturbations in initial conditions, boundary conditions and material

properties and thus result in the random nature of turbulence. In order to quantify

turbulent flow, statistical methods are necessary. Furthermore the use of statistics

decrease the amount of data of a simulation, being considered, drastically. Imple-

menting the computation of statistics in the code reduces the size of output files

and thus make them easier and more economic to handle and to post process.

In this chapter an overview of the statistical tools and notation, used in the present

work is given.

2.3.1 Reynolds decomposition

Describing turublent velocity fields, a velocity component u is commonly split up

into a mean value U plus a fluctuation u.

u = U + u (2.12)

26


This is called the Reynolds decomposition. Plugging the Reynolds decomposition

into the equation for mass conservation 2.8 yields

∇ · U = 0 (2.13)

and

∇ · u = 0 (2.14)

which means that both, the mean of the velocity and its fluctuation are solenoidal.

The actual problem of turbulence modelling arises from the non-linear term in the

momentum equation. The Reynolds decomposition applied to the conservation of

momentum 2.11 yields the equation for the mean flow

DUi

Dt= − ∂p

∂xj

+ ν∇2Ui −∂uiuj

∂xj

(2.15)

The only (but crucial) difference to the Navier-Stokes equations given in 2.11 are

the covariances of the velocity fluctuations 〈uiuj〉, which are called the Reynolds

stresses. The tensor 〈uiuj〉 is commonly referred to as the Reynolds stress tensor.

Without the presence of this tensor the equations of u and U would be the same.

Therefore, the different behaviour in turbulent motion is attributable to the ap-

pearance of the Rynolds stresses. Since the smallest scales of motions of turbulent

fluctuations are very small and even decrease with an increasing Reynolds number,

the requirements for the resolution are very high. Thus, the only approach that can

resolve the Reynolds stresses correctly is the direct numerical simulation (DNS).

For all other modelling techniques no closed solution of the Navier-Stokes equations

is feasable. The Reynolds stress tensor has to be modeled, which results in the

notorious “closure problem”. Since in the present work DNS is used, no further

comments on the modelling of the Reynolds stresses will be given.

2.3.2 The mean

The solution of one realisation of the flow field yields the instantaneous velocity u.

For the current situtation where the boundary conditions are independent of time

27


the ensemble average (or mean) U of the velocity u for N independent realisations

of the flow field is calculated by

U =1

N

i∑

1

ui(x) (2.16)

where ui(x) denotes the instantaneous velocity component of the ith realisation.

The value of a mean quantity is marked by 〈·〉, except for the velocity components,

which are written in capital letters, if it is referred to the mean. In addition to

averaging over N realizations, the flow variables are averaged over homogeneous

directions (streamwise (z) and spanwise (x) directions), to improve the statistics,

since by definition flow variables are invariant under any translation in homogenous

direction. The fluctuation u is obtained by subtracting equation 2.12 from equation

2.16.

It is important to notice that the mean of a fluctuation is zero,

〈u〉 = 0 (2.17)

while the mean of a fluctuation multiplied with a fluctuation (or itself for that

matter) is not equal zero

〈uu〉 6= 0 (2.18)

This yields the famous closure problem of turbulence. Furthermore, the mean of

the mean is obviously equal to the mean

〈U〉 = U (2.19)

Those rules are used throughout the present work without further notice.

2.3.3 Statistics for the Channel Experiment

In order to obtain correct statistics a converged state of the flow has to be reached

(fully developed channel flow). A converged state was defined as a near linear profile

28


0 0.2 0.4 0.6 0.8 10

0.5

1

τ xy+

y/h

Figure 2.1: Total Stress

0 100 200 300 400 5000.02

0.025

0.03

0.035

0.04

0.045

<vw

>

T

Figure 2.2: Diviatoric Reynolds

stress

of the total stress as depicted in figure 2.1. In the present channel this takes about

5 eddy-turn-overs-times (ETT). The eddy-turn-over-time is calculated by

ETT =T · uτ

h(2.20)

where T is the wall-time of the simulation, uτ is the friction velocity and h the

channel half hight. For the current simulations this means to discard about the first

40% of the computed data to be on the safe side. Besides the linear profile of the

total stress, another good indicator whether the converged state has been reached is

the Reynolds stress 〈vw〉 which has to be constant and equal or close to zero for the

current flow configuration. Figure 2.2 depics the course of 〈vw〉 over the wall-time

T of a simulation. After fairly strong inital fluctuations 〈vw〉 settles for a value

close to zero. With uτ = 0.0488, h = 1 and ETT = 5, T is calculated to T = 100,

which if compared with figure 2.2 seems to be a fairly well converged state without

discarting an excessive amount of data.

After a steady state condition was reached, the equations were further integrated

forward in time, to obtain statistics. Statistics and spectra were calculated in the

code and written into a binary file that was then post processed using Matlab.

2.4 Scales of turbulent motion

In turbulence; a wide spectrum of length and time scales are observed, reaching in

size from the width of the flow h to very small length scales, which decrease even fur-

29


100

101

1020

0.5

1

1.5y+ = 15

κ

κ E

(κ)

Full ChannelTorroja fixed

Figure 2.3: 1D streamwise energy

spectra of channel flow at

y+ = 15

102

103

104

102

103

λ+x

λ+ z

Full ChannelTorroja fixed

Figure 2.4: 2D engery spectra at

y+ = 15. Contours at 0.4

and 0.7 of the maximum

of the unfixed case

ther with an increasing Reynolds number. The multi-scale character of turbulence

is one of the most important features, that distinguish it from laminar flow. In this

section the physical processes occuring at different scales of motion are introduced,

which are crucial to the understanding of turbulence and the mechanisms, discussed

later in the present work.

2.4.1 The turbulent kinetic energy spectrum

The one dimensional energy spectrum in streamwise direction, depicted in figure

2.3, shows how the turbulent kinetic energy is distributed over eddies of various

sizes in the respective direction.

Since the calculation in the code is carried out in Fourier space, the turbulent

kinetic energy is computed directly from the Fourier modes. The energy of the

Fourier modes for the streamwise coordinate is given as

Exx(κx) =1

2〈ux(κ)u∗

x(κ)〉 (2.21)

where the ”*“ denotes the complex conjugate of the Fourier transformed velocity

component ux(κ). The turbulent kinetic energy in streamwise direction is then

calculated to

30


k =∑

κ

Exx =1

2〈uxux〉 (2.22)

The advantage of expressing the kinetic energy in terms of the Fourier modes is that

it provides an easy way to quantify the energy at different scales of motion, which

are related to turbulent structures of different sizes. The length scale l (also called

the wavelength λ) is related to the wavenumber κ by κ = 2πl.

Since the energy spectrum is symmetric with respect to κ = 0, the one dimensional

streamwise energy spectrum is given by

⟨

u2x

⟩

=

∫

∞

0

Exx(κx)dκx (2.23)

which in practice (finite number of modes) means the summation of the energy over

all modes. It is common practice to plot the energy spectrum in semi-logarithmic

coordinates. To restore the integral property in that case, a pre-mulitplied energy

spectrum is used. It is given by

⟨

u2x

⟩

=

∫

∞

0

κxExx(κx)d(logκx) (2.24)

The area underneath the pre-multiplied spectrum thus corresponds to the energy

contained in the respective scale.

In the present work, a channel flow, with homegenous streamwise (x) and span-

wise (z) directions is being considered. It is useful to define a pre-multiplied two

dimensional energy spectrum for constant wall normal distances y, such that

E(κxκz) =

∫

∞

0

κxκzE(κx)d(logκx)d(logκz) (2.25)

E(κxκz) =

∫

∞

0

κxκzE(κx)dκx

κx

dκz

κz

E(κxκz) =

∫

∞

0

E(κx)dκxdκz

The resulting surface was cut at levels 0.4 and 0.7 of the maximum of the full channel

spectrum and is depicted in figure 2.4. The plots thus illustrates the streamwise

and spanwise sizes of structures, that reside at a given wall-normal distance y.

31


An approximation of the one dimensional energy spectrum in the inertial range is

given by Kolmogorov’s famous cascade power law [18]

Exx(κx) = Cǫ2/3κ−5/3 (2.26)

where C is an emperical constant. Since fairly low Reynolds numbers are used in

the present work the inertial range is not very well pronounced and this law would

only hold for a short intercept of the spectrum.

2.4.2 The energy cascade

As introduced by Richardson in 1922 the turbulent kinetic energy k is distributed

over the entire range of scales of turbulent motions. He thought of turbulence to

be composed of eddies of various sizes. Eddies are determined by the length scale

l and the time scale τ(l) = lu(l)

where u(l) is a characteristic velocity. Large eddies

can obviously contain smaller eddies with smaller time scales.

Eddies are defined as a turbulent motion within this region determined by l. The

largest eddies are of the size l0 and are determined by the geometric forcing of the

flow (e.g. channel hight h). The characteristic velocity for large eddies is in the

order of the bulk velocity Ub. The Reynolds number for the large scales is therefore

high and viscous effects are negligible.

Acording to Richardson, the kinetic energy enters the cascade at the large scales.

Those large eddies are unstable and break up, passing their energy down to some-

what smaller eddies. The smaller eddies experience the same break-up process and

energy is passed further down by inviscid processes to ever smaller scales. This is

called the energy cascade process. It is continued until the the Reynolds number

is sufficiently small (viscous forces dominate) to disspated the energy by viscous

mechanisms.

Figure 2.3 shows the pre-multiplied spectrum of the turbulent kinetic energy for the

flow analyzed in the present work. The large scales (low wave numbers) contain

most of the energy while the dissipation resides at small scales (not shown), where

the local Reynolds number is sufficiently small and therefore viscosity is active. The

size of the inertial range increases with the Reynolds number. For the Reτ = 550

32


cases the Reynolds number is too low and the inertial range almost vanishes.

The important conclusion from the energy cascade process is, that the place for

dissipation is at the smallest scales and therefore at the end of a sequence of invicid

processes. Thus, the energy transfer is given by the first process in the sequence,

which is the energy transfer from the largest eddies to somewhat smaller eddies.

As stated above the largest eddies of the size l0 contain energy of the order u20 and

therefore a times scale τ0 = l0u0

. It follows that the rate of transfer of energy (energy

flux from larger to smaller scales) is given by

T =u2

0

τ0

=u3

0

l0(2.27)

This indicates that the rate of disspation ǫ likewise scales asu30

l0and is therefore

independent of the viscosity and subsequently indepented of the Reynolds number.

2.4.3 The Kolmogorov hypotheses

Even though Richardson answered some fundamental questions about the processes,

occuring in turbulent motion, the answer, of how small are the smallest scales and

what do they depend on, remained unanswered. The size of the smallest scales in

which dissipation takes on action is of utmost interest, since it likewise determines

the grid spacing of the numerical descritization in order to resolve those scales and

thus capture all physical phenomena of the flow. In 1941 Kolmogorov answered

those questions in what is known today as the Kolmogorov hypotheses.

The first hypothesis is called ”Kolmogorov’s hypothesis of local (small scale) isotropy“

and states that for sufficiently high Reynolds numbers, the small scales are statis-

tically isotropic. The large eddies are anisotropic and depend on the forcing of

the flow. In the break-up processes of the energy cascade the smaller eddies loose

their memory of its initial directional orientation and can therefore be considered

isotropic (invariant to arbitrary rotation and reflexion of the coordinate system).

The scales in which this hypothesis holds is called the ”universial equilibrium range“.

33


The second hypothesis is called ”Kolmogorov’s first similarity hypothesis“ and it

answers the question what parameters does the universial equilibrium range depend

on. The time scales in this range are small so that small eddies can adapt quickly

to the dynamic equilibrium with the energy transfer T from large eddies. As stated

in 2.4.2 the energy transfer rate equals approximately the disspiation rate T ≈ ǫ.

The hypotheses states that the small scales are determined by ν and ǫ. With those

two paramters Kolmogorov formed the following length, time and velocity scales,

which are called Kolmogorov scales.

η =

(

ν3

ǫ

)1/4

(2.28)

τη = (ǫν)1/4

uη =(ν

ǫ

)1/2

The Reynolds number based on the Kolmogorov scales yields ηuτ

ν= 1 and confirms

that those scales are responsible for viscous dissipation and therfore characterize

the smallest eddies.

One important conclusion from this theory is that, in order to resolve the entire

energy spectrum, the grid spacing has to be chosen in the order of the Kolmogorov

scales. The ratios of the smallest to the largest scales are given by

η

l0∼ Re−3/4 (2.29)

uη

u0

∼ Re−1/4

τη

τ0

∼ Re−1/2

For increasing Reynolds numbers the smallest scales and therefore the grid spacing

decrease. As explained in sectino 4.1, this is the reason why DNS is nowadays only

feasable for moderate Reynolds number flows.

Kolmogorovs third hypothesis, called the ”Kolmogorvo’s second similarity hypoth-

esis“, tackles the range of scales between the Kolmogorov scales and the energy

34


containg large scales. Essentially it states that in this range (called the inertial

range) the statistics of motions are independent of the viscocity ν, since the Reynolds

number is still sufficiently high. No energy is therefore dissipated during the invicid

cascade process, but only passed down to the dissipation range.

35

3 Wall-Bounded Turbulent Flow

The subject of the present work is the turbulent flow in a channel. Therefore, a

short overview of the most characteristic features and basic theory of a channel flow

is given in this chapter. The flow is considered to be incompressible (ρ = const.).

L

flow

2h

bx,u

z,wy,v

Figure 3.1: Channel

A fully developped channel of the hight 2h (depicted in figure 3.7) is considered.

The channel consists of two boundary layers that have grown together, however a

channel boundary layer is different from a “regular” boundary layer in the sense

that there is no entrainment region in the channel boundary layer. Furhtermore a

“regular”boundary layer grows in streamwise direction and can therefore in contrast

to the channel boundary layer, not be considered homogeneous in x-direction.

The flow in a channel is predominantly in streamwise (x) direction and the velocity

varies mainly in wall-normal direction. The bottom wall is located at y = 0 and

the top wall is located at y = 2h. The width b and length L are considered to be

large compared to h. Thus, the flow is considered statistically independent of x

and z (statistically stationary in x and z) and therfore essentially one dimensional.

36

Diplomarbeit 3 Wall-Bounded Turbulent Flow

Furthermore the flow is symmetric about the horizontal plane y = h which is used

to furhter improve the statistics.

From the continuity equation follows with the spanwise mean velocity W = 0 that

also the wall-normal mean velocity V equals zero. Two important results from the

lateral and axial momentum equations are, that the axial pressure gradient

∂P

∂x=

dpw

dx(3.1)

is uniform along the streamwise direction and that it equals the wall-normal shear

stress gradient

dτ

∂y=

dpw

dx(3.2)

The channel flow is driven by a constant negative pressure gradient ∂P∂x

. The solution

of 3.2 results in the total shear stress with τw = τ(0)

τ(y) = τw

(

1 − y

h

)

= u2τ

(

1 − y

h

)

(3.3)

which is independet of any fluid properties. The total shear stress is the sum of the

Reynold shear stress and the viscous stress

τ(y) = −〈uv〉 + νdU

dy(3.4)

At the wall all Reynold stresses are zero and the wall shear stress only consists

of the viscous contribution. Viscosity therefore plays a crucial role near the wall,

while away from the wall the viscous stress is negligible compared to the Reynold

shear stresses. Therefore, important paramenters for the characterization of near

wall flow are the wall shear stress τw and the kinematic viscosity ν.

The friction velocity uτ is commonly used as a near wall velocity scale and is defined

as

uτ =√

τw (3.5)

and the viscous length scale is defined as

37


δv =ν

uτ

(3.6)

Quantities normalized with uτ and δv are said to be expressed in ”wall units“ and

are denoted by a ”+“ superscript. The distance from the wall measured in wall units

is

y+ =y

δv

=uτy

ν(3.7)

which can also be understood as a local Reynolds number for the size of the

structures at that hight. Low (and therfore near wall) y+ goes along with a relative

importance of viscous processes. The mean velocity expressed in wall units is given

by

U+ =U

uτ

(3.8)

3.1 Models for the near wall region

The importance of the near wall region in a turbulent channel flow becomes apparent

from the fact, that it is only near the wall where the local Reynolds number is low

enough to allow for viscous friction. The boundary layer is commonly devided into

distinct regions, which are defined by their wall-normal distance in wall units. From

the wall outwards, they are called:

The viscous sublayer (y+ < 5), the buffer layer (5 < y+ < 30), the logarithmic

(log) layer (y+ > 30) and the outer layer (y+ > 150). The first three layers are the

most characteristic features of wall-bounded turbulence [6] and constitute the main

difference between wall bounded turbulent flows and other types of turbulence.

For the detailed derivation of the models for the different regions, shortly described

in the following paragraphs, it is referred to [4] or [1]. Here only the results will be

presented.

38


3.1.1 The viscous sublayer

In the viscous sublayer, where viscosity is dominant and the set of scaling parameters

therefore are the kinematic viscocity ν and the friction velocity uτ , it can be shown

that the mean velocity profile follows a linear relation

U+ = y+ (3.9)

Most large eddies are excluded by the presence of the wall. As shown in [6] the

energy and the dissipation are at similar scales. The viscous sublayer is relatively

easy to simulate numerically, since the local Reynolds numbers are low. On the

other hand it is very difficult to study experimentally, since it is usually very thin

in laboratory flows.

3.1.2 The log-layer

The log-layer is easier to study experimentally but due to its higher local Reynholds

number more expensive to compute numerically.

The famous loglayer (log) law, introduced by von Karman in 1930 is a high Reynolds

number phenomenon. According to [6], its existence requires at least 0.2Reτ > 150,

which is only given for the Reτ = 950 simulations of the present work, but not for

the Reτ = 550 cases. The mean velocity profile for the logarithmic layer is defined

as

U+ =1

Kaln(y+) + A (3.10)

where A is a constant which depends on the details of the near wall and is commonly

set to A = 5 for smooth walls but changes with the roughness of the wall. Ka is

called the von Karman constant and usually takes a value of about Ka = 0.4.

For wall distances larger than y+ = 50 direct effects of viscosity are negligible and

inertial effects dominate the flow physics.

The relative importance of those two layers become apparent from the fact that

within those two layers (y+ < 150), about 83% of the near wall velocity drop takes

39


place.

3.2 Dynamics of wall bounded flow

Besides the statistical description of the flow, the dynamical structures found in

turbulent flow give further insights into the mechnisms that govern turbulence.

This section focuses on the qualitative current understanding of dynamical struc-

tures found in wall bounded turbulent flow. The structures found in wall-bounded

turbulent flow are significantly different from other turbulent flows since they are

forced by the impermeability of the wall. Very long and relatively wide structures

that correlate across the whole flow thickness [10] are found in the outer layer of

turbulent wall flows. Those structures even reach into the viscous sublayer and

appear as spectral handels in the 2D spectral density plots. The box size of present

simulation, however, is too small for those structures and it is referred to simulations

of larger boxes presented for example in [11].

3.2.1 The viscous sublayer

In the near wall viscous layer the flow is relatively smooth, since because of the

low local Reynolds number, viscosity plays a major role. Eddies in this region are

within the dissipative range. Though, due to the very high mean velocity gradient

and therefore high production, the viscous layer acts like a net source of turbulent

kinetic energy (TKE), rather than a sink. The TKE production peaks customarily

in the viscous layer (around y+ ≈ 15) and is then being transported into the outer

flow regions where the production is low, due to a shallow mean velocity gradient.

As described in [4], [6] and [10], two types of structures dominate the dynamics in

the viscous layer: streamwise velocity streaks and quasi-streamwise vortices. These

structures have a well defined length scale, namely the viscosity, which allows them

to be described as individual objects. Streaks are irregular arrays of long sinuous

alternating jets, which are superimposed on the mean shear. They are about 50+

40


wide and high, and show a streamwise seperation of roughly z+ ≈ 100. Low velocity

streaks, found in the viscous sublayer (below y+ ≈ 30), are longer (up to x+ ≈ 1000)

than high velocity streaks (x+ ≈ 250), found in the buffer layer (y+ > 40). The

vortices are slightly tilded away from the wall and stay in the near wall region only

for short distances of x+ ≈ 100 before they move on into the buffer- and log-layer.

That implies, that several vortices are associated with each streak.

The dynamics near the wall are commonly thought of as a closed cycle: The vortices

cause the streaks by deforming the mean velocity gradient, thus moving high speed

fluid towards the wall and low speed fluid away from the wall. The vortices in turn

are thought to be the results of the instability of the streaks and eventual burst,

thus closing the cycle. Furthermore, from [15] it was learned, that the near wall

region is an essentially autonomous feature of the wall regions, generating turbulent

fluctuations independently of the core region. Larger structures coming from the

outer flow hardly interfere with the viscous region, since the near-wall dynamics are

strong enough to be always dominant. This indicates, that the interactions of the

streaks and the mean velocity profile by which energy is drawn from the latter one,

to feed the fluctuations, is a predominantly local process. This hypothesis will be

revisited in the course of the current proyect.

The feed-back mechanism, proposed in [12] and readily mentioned in the past

paragraph, suggests that locally weak structures, with too little Reynolds stresses,

result in a local acceleration of the mean velocity profile, which in turn leads to

local enhancement of the velocity gradient and thus to the strengthening of the local

fluctuations. Furhermore it suggests that any interaction leading to the adjustment

of the intensities of the structures at different wall distances take place between

structures of similar sizes, without necessarily passing through the mean flow. This

feedback mechanism will be challenged in the course of the current work, when the

effect of a fixed mean profile on the development of the intensities is discussed.

To furhter distinguish structures that move away or to the wall, respectively, the

so called “Quadrant” analysis is used. It devides each point of the u-v-plane into

41


quadrants. Since most of the average tangential stress is contained in the second and

forth quadrant, the resulting structures are called ejections (Q2 with u < 0, v > 0)

and sweeps (Q2 with u > 0, v < 0), respectively. Ejections cluster in groups and

are associated with individual vortices. Sweeps and ejections do not stay in the

buffer layer, but extend all the way into the log-region were, they are associated

with vortex clusters. They move fast moving fluid to the wall (sweeps) and slow

moving fluid away from the wall (ejections), thus contributing to the heavy mixing

and momentum exchange that is associated with turbulence.

3.2.2 The logarithmic region

The second region that, due to its increased local Reynolds number, only became

numerically accessible in the past decade, is the logarithmic (log) layer. For the

Reτ = 550 simulations of the present work the log-layer does not even exist, since the

upper boundary (y/h = 0.2 or y+ = 110) lies below the lower boundary (y+ = 150).

For the Reτ = 950 simulations, however, the log-layer has a small range of y+ = 40.

Simulations of higher Reynolds number such as in [11] are necessary to understand

the logarithmic region. While due to the importance of viscosity, the structures are

quite smooth near the wall, above this layer the structures have high internal local

Reynolds number of y+ >> 1 and are most likely turbulent itself. They therefore

cannot be described as single scale objects but have to be treated statistically, since

they are itself part of a turbulent cascade process. Therefore the term “eddies”

rather than vortices is used to describe them, because vorticity are usually thought

of as objects of the size of the viscous Kolmogorov length scale.

Streaks from the viscous layer have essentially disappeared above y+ = 100 and

vorticity has become isotropic, with all three components around 40η. Large struc-

tures, however, are highly anisotropic alongated mostly in streamwise direction.

Structures centered at a wall distance y are, due to their different behavior, clasified

into two categories: attached and detached eddies, depending on whether they are

42


rooted in the near wall reagion or not. Detached eddies consist of small, roughly

isotropic vortex packets that behave more or less like in free shear flow and take

part in the Kolmogorov energy cascade processes. They experience the presence of

the wall only indirectly through the shear of the mean profile.

The tall attached eddies however, are larger than y and therefore anisotropic. They

are linked to velocity structures, that are more intense than their background. Due

to the impermeability condition of the wall, which damps the wall-normal velocity

component, they do not contain tangential Reynolds stresses. As describe in greater

detail in [10] their roots must therefore be irrotational and the pressure gradient is

the only force that acts on them.

Attached eddies can be devided into “active” isotropic eddies of the size of y and

“inactive” structures of sizes much larger than y. Attached “active” isotropic eddies

are part of the classical isotropic cascade process. Every structure in the log-layer,

however, that is larger than y, is anisotropic and therefore not part of the cascade

process. Thus these inactive structures obtained their name from the fact that they

reside above the classical isotropic Kolmogorov cascade without taking part in it.

However, due to their anisotropy, they carry Reynolds stresses and also contain

most of the fluctuating turbulent energy.

In [13] a feedback mechanism, similar to the one in the near wall region, was

suggested, in which clusters are repeatedly started by wakes that were left by still

larger clusters in front of them.

43

4 The Numerical Method

Direct Numerical Simulation (DNS) was used in the present work to simulate the

turbulent flow in a channel. A short overview and some background information on

DNS, followed by the explanation of the numerical method is given in the following

chapter.

4.1 Direct Numerical Simulation

DNS has been the driving force behind the revival of turbulence research in the past

view decades [6], after numerical simulations of turbulent flows became possible in

the late 1980’s and early 1990’s due to increasing computer power. DNS provides

an unprecedented level of detail on the flow and especially for near wall regions,

where experimental measurements are difficult to carry out, it has established itself

as an indispensable research tool.

DNS solves the Navier-Stokes equations by resolving the entire spectrum of length

scales of a given flow and given boundary conditions. The resolution of the full

spectrum is needed, since, as described in chapter 2, the kinetic energy and Reynolds

stresses are associated with length scales much larger than those responsible for

energy dissipation. DNS can be seen as the numerical equivalent to experiments.

However, while experiments can be thought of as an imperfect measurement of a

true system, DNS simulations would be a perfect measurement of an approximation.

The turbulent flow field is unsteady, as in a real flow and only smooth for length

scales smaller than 10η. This means that in order to resolve those small structures

the grid of DNS has to be very fine and powerful computers are needed. The smallest

44

Diplomarbeit 4 The Numerical Method

structures decrease with increasing Reynolds number and are proportional to Re3/4.

In a three dimensional domain that yields NG ∼ Re9/4, where NG are the number

of grid points needed to resolve the smallest structures. Two orders of magnitude

have to be added to account for the time resolution and thus the total number of

arithmetic operations that need to be computed to obtain meaningful statistics are

NOP ∼ Re11/4. In other words, an increase of the Reynolds number by the factor of

10, yields an increase of the factor of 500 in the number of arithmetic operations.

Even though the underlying Navier-Stokes equations have been known for over a

century, because of the requirements stated above, DNSs of turbulent flows were

unfeasible until the late 1980s when computers with sufficient capabilities became

available.

Conceptually DNS is the simplest approach, since no model is used and the entire

flow field is resolved. In that sense DNS simulations have several advantages

over experiments. Once a flow has been simulated all the data is available in a

three-dimensional domain and thus post-processing allows even to compute views

and terms which are difficult to obtain by experiments. Furthermore, imaginary

”unphysical“ flow phenomena can be simulated, by mposing boundary conditions,

that differ from the natural ones, to check for processes and validate hypothesis,

that could not be obtained from experiments. This makes DNS an excellent research

tool which will expand its influence with growing hardware capabilities.

As discussed in section 2.4, the smallest structures in the turbulent flow field and

therefore the grid spacing, decrease with increasing Reynolds number. That means,

that due to a lack of sufficient computing power, only moderate Reynold numbers

can be simulated. Though, as long as the physics (separation of energy containing

scales and dissipation scales) of the flow can be represented accurately, valuable

data and insights can be obtained from the simulations available today. It must be

stressed, that the objective of DNS is not to reproduce real life flows, but rather to

use it as an academic research tool, allowing the study of flow physics and thus the

45


development of improved turbulence models, which then can be used in commercial

flow solvers.

4.2 The numerical procedure

4.2.1 Derivation of the governing equations

In order to implement the equations of fluid motions, as given in 2.2, in the code

they have to be modified slightly. By doing so, continuity is imposed implicitely

and does not have to be accounted for seperately. The equations of conservation

of momentum and mass are taken from sections 2.2.2 and 2.2.1, respectively. For

reasons of simplicity and clarity the notation is chosen such that ∂x denotes the

operator ∂∂x

, etc. The convection term is denoted by Hj = ui∂uj

∂xi. Following [2], the

governing equations for the fluid can be written as

∂tuj = ∂xjp − Hj +

1

Re∇2uj (4.1)

and

∇ · u = 0 (4.2)

In order to eliminate the pressure gradient, the curl of the momentum equation 4.1

is taken and it follows

∂t (∇× uj) = ∇× Hj +1

Re∇2 (∇× uj) (4.3)

or written out for all three spatial directions

∂t [∂yuw − ∂zv] = (∂yH3 − ∂zH2) +1

Re∇2 (∂yw − ∂zv) (4.4)

∂t [∂zuu − ∂xw] = (∂zH1 − ∂xH3) +1

Re∇2 (∂zu − ∂xw) (4.5)

∂t [∂xuv − ∂yu] = (∂xH2 − ∂yH1) +1

Re∇2 (∂xv − ∂yu) (4.6)

where equation 4.5 is the equation for the normal component of vorticity ω

46


∂tω = (∂zH1 − ∂xH3) +1

Re∇2ω (4.7)

Equation 4.4 is multiplied by the operator ∂z and equation 4.6 is multiplied by the

operator ∂x and subsequently equation 4.6 is subtracted from equation 4.4. This

yields

∂t

[

∂2xv + ∂2

z v − ∂x∂yu − ∂z∂yw]

= R.H.S. (4.8)

where the R.H.S. (right hand side) is given by

R.H.S. =(

∂2xH2 + ∂2

zH2 − ∂x∂yH1 − ∂z∂yH3

)

+1

Re∇2

[

∂2xv + ∂2

z v − ∂x∂yu − ∂z∂yw]

(4.9)

The conservation of mass

∇ · u = ∂xu + ∂yv + ∂zw = 0 (4.10)

is used to eliminate the terms −∂x∂yu − ∂z∂yw in equations 4.8 and 4.9 and is

therefore implicitly imposed. With equation 4.10 it follows

∂yv = −∂xu − ∂zw

∂2y v = −∂x∂yu − ∂zw (4.11)

and thus equations 4.8 and 4.9 can be written as

∂t

[

∂2xv + ∂2

z v + ∂2y v

]

= −∂y [∂xH1 + ∂zH3]+H2

[

∂2x + ∂2

z

]

+1

Re∇2

[

∂2xv + ∂2

z v + ∂2y v

]

(4.12)

or

∂t∇v = −∂y [∂xH1 + ∂zH3] + H2

[

∂2x + ∂2

z

]

+1

Re∇4v (4.13)

The code solves for the Laplacian of the wall-normal velocity component and the

normal component of the vorticity using equation 4.13 and 4.7, respectively.

47


The definition of the vorticity ω = ∂zu − ∂xw and the continuity equation, given

in 4.10, are used to compute the stream- and spanwise velocity components. For

the computation, carried out using a spectral method (Fourier series in x and z

and Chebyshev ploynomial in y), all the derivatives become multiplications which

results in a favorable algebraic equation.

The price, paid for the elimination of the pressure and implicitly incorporating

the continuity equation into the Navier-Stokes equations, is the resulting 4th order

differential equation, which requires more grid points (higher computational cost)

to yield the same accuracy as a 2nd order equation. By substitution, the 4th order

equation is therefore split up into two second-order equations, to solve them more

efficiently.

For the convective terms a third order Runge-Kutta scheme is used to advance in

time. The Backward Euler Method (implicit) is used for the time advancement

of the viscous part. The Chebyshev-tau method is used to solve the discretized

equations as explained in further detail in [2].

For a constant density flow there is no connection between the pressure and the

density of the fluid and the pressure gradient is uniquely determined by the current

velocity field, independent of the flow’s history. Thus, the procedure stated above

does not require the calculation of the pressure. It was only calculated during

post processing, to obtain turbulence statistics, such as for example the budget of

the Reynolds stress, which involves pressure. It was computed from the normal

momentum equation with the wall pressure determined from the combination of

streamwise and spanwise momentum equations.

4.2.2 Initial and Boundary Conditions

Since the channel is assumed to be periodic in streamwise direction, the initial

conditions do not play a crucial role and were taken from previous analysis of

the same channel. As long as the initial condition roughly represent the large

48


scale fluctuations and mean velocity profile, the channel will adapt itself. The

intermittency in stream- and spanwise direction also omits the problem of finding

adequate inflow and outflow conditions and boundary conditions, respectively. The

outflow on one side is simply recycled as the inflow on the other side, thus creating

an infinite channel length and width, respectively. Although the box size has to be

chosen sufficently large in order to represent correctly the large scale structures in

the flow.

no-slip boundary condition were implemented in wall-normal directions (equation

4.13) at y = ±1, such that

v(±1) =∂v

∂y(±1) = 0 (4.14)

4.2.3 Spectral Method

In computational fluid dynamics, finite difference or spectral methods are used

to discretize the equations of fluid motions and calculate a numerical solution of

them. Finite difference methods approximate the solution locally and thus result

in sparsly filled matrices, which can be solved using specialized methods, exploiting

the diagonal predominance. Spectral methods, instead, approximate the solution

globally with a series sum of orthogonal basis functions. A computation domain

can be dicretized using different methods in different spatial directions. For a given

number of degrees of freedom (grid points) it can be said that generally spectral

methods yield more accurate results than finite difference methods. Spectral meth-

ods perform well with fairly smooth and regular geometries but cause problems (loss

of accuracy and efficiency) in more complicated features, as commonly encountered

in industrial flows. Also, due to the necessary transformation into Fourier space,

spectral methods are more costly than finite difference methods.

The code used in the present work, uses a spectral method in all three spatial

directions. Therefore this chapter will be limited to spectral methods, which will

be presented shortly.

49


Spectral methods are extremely accurate and non-dissipative tools for calculating

derivatives of descrete data sets, which is the main objective of a numerical method

when finding the solution to a differential equation. Using the complex representa-

tion given in equation 4.15, it can easily be seen that the derivative of an exponential

turns into a multiplication with its exponent. Furthermore, spectral methods enjoy

exponential convergence and thus makes it possible, if drafted correctly, to find a

highly accurate solution of a differential equation.

A spectral method approximates a function in physical space as a series sum of

orthogonal basis functions. The most common choice for an orthogonal basis

function are the Fourier series. They are used in homogeneous directions, since

the flow is assumed to be periodic in those directions, and as stated in [3] Fourier

series work best for periodic problems. The complex representation of the velocity

component u in Fourier space is given by

u(x, t) = u(κ0, t) + 2N

∑

κ=1

u(κ, t)eiκx (4.15)

with the coefficient

u(κ, t) =1

2π

∫ κmax

κ0

u(x, t)e−inxdx (4.16)

which represents component of the velocity u(x, t) in Fourier space. In the channel

experiment considered in the present work, the zero mode u(κ0, t) represents the

mean velocity profile. The fluctuation from the mean is given the second part of the

sum in equation 4.15. Furthermore, the wavenumbers are assumed to be symmetric

with respect to zero and thus only positive wavenumbers (κ = 1 to N) are considered

and multiplied by the factor 2. The largest wave number that is represented and

thus associated with the mean profile, is

κ0 =Nπ

L(4.17)

where L denotes the box size in the direction considered and N the number of

50


grid points (modes) with which the respective direction was discretized. The grid

spacing in physical space is given by

∆x =L

N(4.18)

For the non-periodic wall-normal direction (y) a Chebyshev polynomal expansion is

used in the code. A Chebyshev polynomal expansion is merely a Fourier cosine series

with a change of the variable. The Chebyshev polynomial series approximation of

the wall normal velocity component is given by

v(y, t) =N

∑

n=1

v(y, t)Tm(y) − 1

2v(0, t) (4.19)

The coefficient v is defined as

v(y, t) =2

N

N∑

i=1

v(yi)Tm(yi) (4.20)

where the Chebyshev polynomial Tm of degree m, defined over the interval [−1, 1]

is given as

Tm = cos (m · arccos(y)) (4.21)

and the zeros (collocation points) of the polynomial are located at

yi = cos

(

π(i − 12)

m

)

(4.22)

for i = 1...N .

In order to keep computing costs down, a fast fourier transform (FFT) algorithm is

used for the one-to-one mapping between the Fourier coefficients and the velocities

in physical space. Further cost can be avoided by using a pseudo-spectral method.

This avoids the costly computation of the non-linear terms of the Navier-Stokes

equations by transfering the velocity field back into physical space, computing the

non-linear terms and transfering the result back into Fourier space.

51


4.2.4 Spacial Resolution

The spatial resolution required, to resolve the smallest energy dissipating scales, is

determined by the flow physics (e.g. Reynolds number). The accuracy, with which

theses scales are represented is determined by the numerical method, while the grid

determines the scales that can actually be represented.

While the Kolmogorov length scale η = (ν3/ǫ)1/4 is commonly used as the smallest

scale that must be resolved, the relevant scales to obtain reliable statistics are

typically larger than that. As stated in [4] most of the dissipation occurs at scales

larger than 15η and therefore a rule of thumb states, the smallest scales must only

be in the order of the Kolmogrov scale and not necessarily equal to η.

The approximation for the Kolomogorov length scale

η = Lǫ/Re3/4 (4.23)

for the Reτ = 550 (Re = 11180) case, with Lǫ = h = 1, yields η = 0.0009, which

correspsonds to η+ = 0.495. As can be seen in table 5.1, the grid spacing in all three

spatial directions is therefore within the order of magnitude of the Kolmogorov

length scale. This makes it possible to accurately represent the physics at small

scales (vortices) which are crucial for the study turbulence.

4.2.5 Time Resolution

In turbulence not only the length scales are spread over a wide range of scales,

but also the time scales are required to be sufficiently small to account for the

smallest structures to be resolved in time. The requirement of time accuracy over

a wide range of scales does not permit large steps, as e.g. used in aerodynamic

flows, where implicit schemes allow for it. Large time steps in turbulent flow would

lead to large errors in the small scales. The range of frequencies that need to be

accurately represented are dictated by physics, while the numerical scheme used,

determines the frequencies resolved. Implicit time advancement is unconditionally

stable but also uses more resources than explicit time advancement. It therefore is

most attractive when the frequencies of the flow are far lower than those represented

52


by the discrete equations. On the other hand, to achieve numerical stability with

explicit time advancement, the time step needs to be smaller than it takes for a

fluid particle to cross one grid element, which leads to the restriction that the CFL

(Courant Friedrich Levi) number cannot exceed the value of 1. The CFL number

is defined as

CFL =dtu

dx≤ 1 (4.24)

In the present simulation a CFL number of 1 is used, accounting for stability but at

the same time obtaining an economic scheme. Since, due to physics of turbulence,

time steps are required to be small anyway, the most economical solution is to use

explicit time advancement. In the present DNS code a Crank-Nicholson-Scheme

(implicit) is used for the viscous term and a Adams-Bashforth-Scheme (explicit) for

the non-linear term. In order to acquire a stable time integration, the time step has

to be determined by choosing the smaller of the viscous or the convective time step

but since the viscous term is treated implicitely (unconditionally stable), the time

step is determined by the convective part and therefore the CFL number.

4.2.6 Error

Even though as stated earlier, the spectral method used in the code, have excellent

error properties, there is one sources of error, that ought to be pointed out briefly in

this section. For spectral methods the so called ”Aliasing“ appears when computing

the nonlinear terms pseudo spectrally. It causes different modes to become indistin-

guishable and the solution therefore contains modes that are actually not present

in the flow. The anti-aliasing technique used, expands the number of collocation

points by a factor of 3/2 before transforming them into physical space.

53

5 The Numerical Experiment

5.1 Computational Domain and Numerical Issues

Lx, Nx

flowLy, Ny

Lz, Nzx,u

z,wy,v

Figure 5.1: Computational Domain

The computational domain of the channel is depicted in figure 5.1. A π × π/2 box

and a 2π × π box at a friction Reynolds number Reτ = 550 as well as a 2π × π

box at a friction Reynolds number Reτ = 950 where used for the experiments. The

respective parameters and labels are summarized in table 5.1.

Table 5.1: Summary of box sizes

Name Reτ Nx Ny Nz Lx/h Ly/h Lz/h ∆x+ ∆y+wall ∆y+

center ∆z+

small box 550 128 257 128 π 2 π/2 13.5 0.1 6.7 6.7

big box 550 256 257 256 2π 2 π 13.5 0.1 6.7 6.7

big box 950 384 512 384 2π 2 π 15.5 0.03 7.7 7.8

The fully developed channel of the hight 2h channel, shown in figure 5.2, consists

of two boundary layers that have grown together. The flow is predominantly in

54

Diplomarbeit 5 The Numerical Experiment

streamwise (x) direction and the velocity varies mainly in wall-normal (y) direction.

The bottom wall is located at y = 0 and the top wall is located at y = 2h. The

width b and length L are considered to be large compared to h. Thus, the flow

is statistically independent of x and z (statistically stationary in x and z) and

therefore essentially one dimensional. Furthermore, the flow is symmetric about

the horizontal plane y = h which is used to improve the statistics.

L

flow

2h

bx,u

z,wy,v

Figure 5.2: Channel

The requirement of the numerical method used in DNS codes for turbulence sim-

ulation, is to accurately reproduce the evolution of the flow variables over a wide

range of length and time scales. The three-dimensional, unsteady nature of turbu-

lence therefore demands resolution in all three spacial directions and in time. The

numerical methods used, are described in section 4.

Due to the Chebychev polynomial expansion in wall normal direction (y), a non-

uniform grid spacing according to yi = 1−cos (θi) with θi = (i−1)πNy−1

for i = 1, 2, ..., Ny

was used. Thus the first mesh points near the wall are much closer than the mesh

points in the center of the channel. This is because it is near the wall, where the

smallest structures with low local Reynolds numbers are located and therefore high

dissipation occurs.

55


5.2 Experimental setup

In the numerical experiment the mean velocity profile of the streamwise velocity

component U was held fixed at all times. The mean velocity profile for a channel

flow is well known from previous simulations such as [5]. Since a four times larger

box was used in [5], its profile is assumed to be the correct profile for the channel.

The data can be found on the web page of the fluids laboratory of the Universidad

Politecnica de Madrid (http://torroja.dmt.upm.es). For the name of the server this

profile is referred to as the “Torroja”-profile in the present work. Another way to

obtain the correct mean profile of a particular channel is to run the channel without

fixing the provile until a converged state is reached. This is especially important

for smaller channels, since the converged velocity profiles of smaller channels differ

slightly from the converged velocity profile of bigger channels. The difference might

not seem significant when looking at the mean profile (figure 5.3), but when the

“Torroja” profile of the (8π x 4π)-channel was imposed on the “small” (π x π/2)-

channel, the plot of the total stresses, defined as the sum of the Reynold stress and

the viscous stress

τtotal = −〈uv〉 +∂U

∂y(5.1)

depicted in 5.3 clarifies that a channel accepts nothing else but its very own naturally

developed mean velocity profile.

As can also be seen in figure (5.3), the mean velocity profile of the “big” (2π × π)-

channel and the Torroja (8π × 4π)-channel yield very similar mean profiles. The

results of the total stress 5.3 look very close to the unfixed converged profile as well.

Those results suggests that the “small” (π×π/2)-channel is too small. Therefore all

the experiments for the analysis of the influence of a fixed mean profile were carried

out in the “big” (2π × π)-channel only.

To obtain plausible results, the channel was run for 7 ETT (eddy-turn-over-time)

before statistics where collected for another 13 ETT. The same inflow boundary

condition was used for all experiments. It was obtained from a converged condition

56


0.08 0.1 0.12

0.72

0.73

0.74

0.75

0.76

U

y/h

Big BoxSmall BoxTorroja profile

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

τ xy+

y/h

TorrojaBig Box fixed to TorrojaSmall Box fixed to TorrojaSmall Box fcnf

Figure 5.3: Converged mean velocity profiles of different channel sizes (left). Total Stress for

fixed simulations of different channel sizes (right), where “fcnf” stands for “fixed from

converged non-fixed” and therefore denotes the natural profile of the channel

(≈ 35ETT ) of the same channel without a fixed mean velocity profile.

5.2.1 Fixing the mean velocity profile

As mentioned in the previous chapter the flow field is calculated using a spectral

method. In a spectral representation the zero modes of the streamwise velocity

component represent the mean velocity profile. Instead of computing the zero

modes, they were imposed by reading a file which contained the mean velocity

profile in streamwise direction for every wall-normal position. This way the mean

velocity profile works like an imposed force on the right-hand-side (RHS) of the

Navier-Stokes equations. The smaller scales will try to deform it, but since it is

fixed, they eventually will have to adapt to it. The influence of different mean

profiles on various flow quantities as well as on turbulent structures was studied.

How the different mean profiles were obtained is described in the following chapter.

5.2.2 Natural and unnatural profiles

The so called “natural” profile (also the terms “true” and “correct” are used inter-

changeably) was obtained from the converged state solution of an (8π×4π)-channel

computed in [5].

57


In order to examine the influence of an unnatural mean velocity profile, the Cess

Formula [?] was used. The Cess Formula which gives an approximation of the total

(molecular plus turbulent) viscosity is defined as

νtot

ν=

1

2

[

1 +κ2h+2

9

[

2Y − Y 2]2 [

3 − 4Y + 2Y 2]2

[

1 − e−Y h+

A

]2]1/2

+1

2(5.2)

where Y is defined as Y = y/h. The mean velocity profile is then obtained by

integrating the momentum equation

∂U

∂y=

u2τ (1 − Y )

νtot

(5.3)

which can be expressed in wall units

U =

∫ y

0

u2τ (1 − Y )

νtot/ν

dy

ν

U+ =

∫ 1

0

uτ (1 − Y )

νtot/νdY

uτh

ν

U+ = Reτ

∫ 1

0

uτ (1 − Y )

νtot/νdY (5.4)

dU+

dY= Reτ

uτ (1 − Y )

νtot/ν(5.5)

With the definition of y+

y+ =yuτ

ν(5.6)

this yields

Y =y+ν

uτh

Y =y+

Reτ

dY = dy+

Reτ

(5.7)

By applying 5.7 on 5.4 it follows

58


dU+

dY=

dU+

d y+

Reτ

=dU+

dy+Reτ = Reτ

1 − Y

νtot/ν(5.8)

and therefore

dU+

dy+=

1 − Y

νtot/ν(5.9)

Integrating 5.9 yields the mean velocity profile in wall units

U+ = Reτ

∫ 1

0

1 − Y

νtot/νdY (5.10)

where νtot/ν is obtained from the Cess formula 5.2. Equation 5.10 was integrated

numerically. For h+ = 550 and h+ = 950, respectively, the two parameters κ and

A in 5.2 were fitted to the “Torroja” profile by a least square fit. The procedure

will be explained examplarily for the Reτ = 550 case. The “Torroja” profile was

approximated well by κ = 0.46 and A = 29.90 as depicted in figure 5.4. The

collocation points for the numerical integration are given by

y = 1 − cos

(

kπ

N − 1

)

(5.11)

with k = 1...N = 257 and

Y =y

h(5.12)

The mass flux Ub, the friction Reynolds number Reτ and the nominal energy

production P/u3τ were held constant. P is defined as

P =

∫ h

0

[

u2τ (1 − Y ) − ν

∂U

∂y

]

∂U

∂ydy (5.13)

where again Y = yh. With

∂U+

∂Y= Reτ

(

1 − Y

νt/ν

)

(5.14)

it follows

59


0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

y/h

U+

fitted data: A=29.9, kappa=0.46ref data (Torroja)

Figure 5.4: Fit of Cess formula to “Torroja” profile

P =

∫ h

0

[

u2τ (1 − Y ) − ν∂Y

Uuτ

huτ

]

∂YUuτ

huτ

hdY (5.15)

P

u3τ

=

∫ 1

0

[

(1 − Y ) − ∂YU+

Reτ

]

∂Y U+dY (5.16)

P

u3τ

=

∫ 1

0

[

(1 − Y ) −(

1 − Y

νt/ν

)]

Reτ

(

1 − Y

νt/ν

)

dY (5.17)

where ∂Y denotes the partial derivative with respect to Y . The linear approximation

A = 540κ − 218.5 was then fitted to the line Pu3

τ= const., that was obtained from

the fitted parameters κ = 0.46 and A = 29.90. The procedure is depicted in figure

5.5.

An unnatural mean velocity profile with the same nominal production, the same

friction Reynolds number and the same mass flux, but slightly different mean

velocity gradient was obtained by choosing a different combination of κ and A along

the line of constant production. For A = 51.50 and κ = 0.50 an unnatural velocity

profile, depicted in figure 5.6 and 5.7 (magenta line), respectively, was obtained.

The unnatural Ufalse profile was then mixed with the Torroja Utrue profile according

to

60


14.5

14.5

15.5

15.5

15.5

16.5

16.5

16.5

17.8

793

17.8

793

17.8

793

19

19

19

20

20

20

21

21

21

22

22

22

23

23

24

24

κ

A

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70

10

20

30

40

50

60

70

80Prod = const

A = 540 κ − 218.5

A = 29.9 ; κ = 0.46

A = 51.50 ; κ = 0.50

Figure 5.5: Variation of A and κ along constant production lines. The blue dot represents

the natural profile, while the blue cross shows the unnatural profile with the same

nominal production

U(β) = Utrue · (1 − β) + Ufalse · β (5.18)

by choosing the factor β such, that a first profile, with a slightly higher gradient of

the mean velocity (shear) near the wall (flat profile) compared to the natural case,

was obtained (β = 0.5) and a second profile, such that a slightly lower shear near

the wall (round profile) was obtained (β = −0.5). A third profile with β = −1.0

was used to increase the impact of a small near wall mean velocity gradient and

thus to clarify the results obtained from the previous cases.

The mean velocity profiles of the three β-cases β = 0 (true), β = +0.5 and β = −0.5,

together with the converged profile of the “big” box, are shown in figure 5.8. A

zoomed-in view is shown in figure 5.9 to clarify the differences.

It is clear from this plot that the converged profile of the “big” box and the Torroja

profile yield the same mean velocity profile and can thus be used interchangeably as

the real profile. In the present work the Torroja profile was chosen as the “natural”

profile since more data and a four times bigger box was used to obtain it.

61


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

y/h

U

wrong profile (β = 1)true profile (β = 0)β = 0.5β = −0.5

Figure 5.6: Variation of mean veloc-

ity profile

0.2 0.4 0.6 0.8 1

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

y/h

U

wrong profile (β = 1)true profile (β = 0)β = 0.5β = −0.5

Figure 5.7: Zoomed-in view of figure

on the left

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

U

y/h

TorrojaBig convergedBig Beta = 0.5Big Beta = −0.5

Figure 5.8: Mean velocity profiles

0.05 0.1 0.15 0.2 0.25

0.6

0.65

0.7

0.75

0.8

0.85

0.9

U

y/h

TorrojaBig convergedBig Beta = 0.5Big Beta = −0.5

Figure 5.9: Zoomed in view of left

graph

62


5.2.3 Influence on the Reynolds number

As stated in section 5.2.2 the friction Reynolds number Reτ was kept constant for

all cases. In the code the Reynolds number Re = 1ν

is imposed, which means, that

the friction Reynolds number Reτ would change if Re = 1ν

was not adapted.

With ρ = 1, the squared friction velocity uτ is defined as

u2τ = τw = ν

[

∂U

∂y

]

Wall

(5.19)

since the Reynolds stresses at the wall become zero. With Reτ known, the Reynolds

number of the channel was determined to

Re =1

ν=

Reτ

uτ

=Re2

τ[

∂U∂y

]

Wall

(5.20)

where[

∂U∂y

]

Wallwas determined from the respective mean velocity profile. Table

5.2 and 5.3, respectively, show the vorticity ωz at the wall, the friction velocity,

the Reynolds number Re, defined as the inverse of the viscosity, and the Reynolds

number based on the friction velocity for the three β-cases and the natural case at

Reτ = 550 and Reτ = 950, respectively.

Table 5.2: Flow quantities for different β-cases of Reτ = 550 channel

β = 0 β = 0.5 β = −0.5 β = −1.0[

∂U∂y

]

Wall26.7367 25.1994 28.1414 29.6086

uτ 0.0489 0.0460 0.0512 0.0538

Re 11180 12003 10750 10216

Reτ 546.7 549.9 550.0 549.9

5.3 Blending mean profiles

Another set of “unnatural” mean velocity profiles was obtained, by blending two

of the β profiles. The two profiles that were taken as the basis of the blending

63


Table 5.3: Flow quantities for different β-cases of Reτ = 950 channel

β = 0 β = 0.5 β = −0.5 β = −1.0[

∂U∂y

]

Wall42.3491 40.5477 44.1553 45.9553

uτ 0.0453 0.0427 0.0465 0.0484

Re 20598 22260 20440 19637

Reτ 934.0 950.0 950.0 949.9

are the β = −0.5 (or β = −1.0 respectively) and the β = 0 (Torroja) profile.

The motivation for the blending is to understand, what is the influence of the

“unnatural” profile in different parts of the flow. The Torroja profile, used in the

near wall region, was blended into the β = −0.5 (and β = −1.0 respectively) at

various y/h locations, thus obtaining six blends. Three shallow blends, using the

β = −0.5 profile and three strong ones, using the β = −1.0 profiles in the outer

part of the flow. The blending locations where kept constant between the β = −0.5

and the β = −1.0 cases. However, since the β = −1.0 profile deviates stronger

from the natural (Torroja) profile, the blendings for those cases perturb the flow

stronger than in the β = −0.5 cases. For the Reτ = 550 case both, the β = −0.5

blend and the β = −1.0 blend where analyzed, while due to limited resources, for

the Reτ = 950 case only data for the stronger β = −1.0 case was computed.

5.3.1 Blending technique

A Bezier curve, which is named after its inventor, Dr. Pierre Bezier who was an

engineer with the Renault car company and developed a curve formulation in the

early 1960s for shape design, was used to create a smooth blending function.

In order to blend two mean velocity profiles with a continuous first order derivative

(C1-continuity) a cubic Bezier curve was chosen. It is defined by four points starting

at P0, going towards P1 and arriving at P3 coming from the direction of P2. The

curve does not necessarily pass through P1 and P2. Those points only provide

directional information. The Bezier cuve interpolation is defined as

64


f(y) = (1 − y)3P0 + 3(1 − y)2yP1 + 3(1 − y)y2

P2 + y3P3 , y ∈ [0, 1]. (5.21)

Figure 5.10 shows how two mean velocity profiles were blended over a distance of

about 15% of the channel half hight, using a Bezier curve with its control points

plotted in red.

0.2 0.25 0.3 0.35

0.8

0.82

0.84

0.86

0.88

U

y/h

Torrojabeta = −0.5blendedBezier Points

Figure 5.10: Blending of two mean profiles

As shown in figures 5.11 and 5.12 the Bezier-blending function yields a continuous

first derivative. This is important, since for example the production term is calcu-

lated, using the first derivative dUdy

of the mean velocity profile and a discontinuity

would thus be disadvantageous.

5.3.2 Variation of blending loctation

Applying the blending technique to the same set of basis profiles, using the Torroja

profile in the near wall region and the more rounded profiles (β = −0.5 and β =

−1.0, respectively) towards the channel center, three blended profiles were created,

by varying the blending location according to tables 5.4 and 5.5, respectively.

Figure 5.11 shows the three blends as well as their first derivative of the β = −0.5

blends at Reτ = 550. Figure 5.12 depicts the three blends and their first derivative

of the β = −1.0 blends at Reτ = 950. This way the influence of the blending location

65


0.03 0.04 0.05 0.06 0.070.55

0.6

0.65

0.7

U

y/h

Buffer Blend

0.04 0.05 0.06 0.07 0.08

2

3

4

5

6

dU/d

y

y/h

Buffer Blend

0.15 0.2 0.25 0.3 0.35

0.8

0.85

0.9

U

y/h

Log Blend

0.15 0.2 0.25 0.3 0.350.3

0.4

0.5

0.6

0.7

0.8

dU/d

y

y/h

Log Blend

0.6 0.7 0.8 0.9 10.95

1

1.05

U

y/h

Outer Blend

0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

dU/d

y

y/h

Outer Blend

Torrojabeta = −0.5blended

Figure 5.11: β = −0.5 blendings and their first derivative for Reτ = 550

and thus the influence of an increased shear in different parts of the channel was

investigated. As can also be seen in figures 5.11 and 5.12, respectively, the blending

locations were held constant for both Reynolds numbers, Reτ = 550 and Reτ = 950.

However, since the near the wall quantities scale in wall units the Buffer Blend for

the Reτ = 550 case will be closer to the wall than in the Reτ = 950 case.

As discussed above, the Reynolds number Re = 1ν, used in the code, solely depends

on the gradient of the mean velocity profile at the wall. It is therefore held constant

at Re = 11180 and Re = 20580, respectively, for all cases. Also, as for the other

66


0.03 0.04 0.05 0.06

0.6

0.65

0.7

U

y/h

Buffer Blend

0.04 0.05 0.06 0.07

1

2

3

4

dU/d

y

y/h

Buffer Blend

0.2 0.25 0.3 0.35

0.8

0.85

0.9

U

y/h

Log Blend

0.2 0.25 0.3 0.35

0.3

0.4

0.5

0.6

0.7

dU/d

y

y/h

Log Blend

0.6 0.7 0.8 0.90.96

0.98

1

1.02

1.04

U

y/h

Outer Blend

0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

dU/d

y

y/h

Outer Blend

Torrojabeta = −1.0blended

Figure 5.12: β = −1.0 blendings and their first derivative for Reτ = 950

cases, the mass flow was kept constant.

Table 5.4: Summary of blending cases for 550 channel

Name Near wall profile Center profile y/h y+

Buffer Blend Torroja β = −0.5 / β = −1.0 0.05 ≈ 30

Log Blend Torroja β = −0.5 / β = −1.0 0.25 ≈ 140

Outer Blend Torroja β = −0.5 / β = −1.0 0.77 ≈ 420

67


Table 5.5: Summary of blending cases for 950 channel

Name Near wall profile Center profile y/h y+

Buffer Blend Torroja β = −1.0 0.05 ≈ 50

Log Blend Torroja β = −1.0 0.25 ≈ 240

Outer Blend Torroja β = −1.0 0.77 ≈ 730

68

6 Results

The results of the numerical experiments, introduced in the previous chapter, ana-

lyzed using various post-processing techniques, are presented in the present chapter.

The chapter is structured as follows. First the results of fixing the mean profile to

the natural case and the unnatural cases, referred to as the β-cases, are presented

for both Reynolds numbers, followed by the results for the blending of two mean

profiles. It follow the results of several post processing techniques, such as the

energy balance and the wall normal energy distribution of the fluctuating kinetic

energy, the results for a linear stability analysis for transient growth, as well as the

results for the analysis of turbulent structures. Finally the results of the release of

the fixed β = −1.0-cased will be presented.

The statistical and spectral results of the fixed Torroja profile and the data from

the “Torroja” database (unfixed converged 8π × 4π-channel) coincide and thus, for

reasons of clarity, only the results of the fixed Torroja profile are shown in most plots.

It was however checked in each individual case that there is agreement between those

two data sets.

6.1 Statistics for β-cases

Figure 6.1 shows the effect of a fixed streamwise mean profile on the Reynold

stresses, the streamwise and wall normal velocity fluctuations as well as on the

total stress. The cases depicted are β = 0 (natural profile), β = +0.5 (flat profile

with increased shear near the wall) and β = −0.5 (round profile with decreased

shear near the wall).

69

Diplomarbeit 6 Results

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

−<uv

>+

y/h

Reynolds stress

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

u+

y/h

Streamw. vel. fluc.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

v+

y/h

Wall−norm. vel. fluc.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

τ xy+

y/h

Total Stress

Torroja 550Flat 550Round 550

Figure 6.1: Statistics for fixed mean velocity profile at Reτ = 550

The fixed natural profile yields identical results as the unfixed case within the

statistical uncertainty. However, the only slight increase in the gradient of the

mean velocity profile (β = +0.5 profile) yields a strong increase of the Reynold

stresses −〈uv〉 by nearly a factor of two near the wall. The wall normal location of

the maximum is fairly constant around y+ = 20. For the rounder profile (β = −0.5)

the magnitude of the maximum of the Reynolds stress stays fairly constant, though

the peak moves outwards to a location at about y+ = 120. This way, the flat

profile moves most of the turbulent kinetic fluctuating energy towards the wall,

while the round profile moves the energy outwards towards the channel center.

This phenomenon is further analyzed and explained in section 6.6.

70


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

−<uv

>+

y/h

Reynolds stress

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

u+

y/h

Streamw. vel. fluc.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

v+

y/h


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

τ xy+

y/h

Total Stress

TorrojaRound m05Round m1

Figure 6.2: Statistics for β = −0.5 and β = −1.0 profiles

Figure 6.2 compares the statistics for the β = −0.5 and β = −1.0 profiles. It is clear

from those results that the β = −1.0 profile essentially yields the same qualitative

results than the β = −0.5 profile, only more pronounced, meaning the extra shear

that is moved away from the wall, with respect to the β = −0.5 profile, increases

the deviation of each quantity from the natural profile.

Figure 6.3 compares the results for the Reynolds number Reτ = 550 with the results

of the Reτ = 950 cases, normalized in wall units (left graph) and outer units (right

graph). The location of the maximum of the intensities, and thus the Reynolds

stresses, scale in wall units. However, the effect of a unnatural profile near the wall

results in more pronounced reaction for the higher Reynolds number cases, while in

the channel center the Reτ = 950 cases come somewhat closer to the natural profile

and recovers the typical linear trend of the Reynolds stress towards the channel

center.

71


0 100 200 300 400 5000

0.5

1

1.5

−<uv

>+

y+

Reynolds stress

Torroja 550Flat 550Round m05 550Round m1 550Torroja 950Flat 950Round m05 950Round m1 950

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

−<uv

>+

y/h

Reynolds stress

Torroja 550Flat 550Round m05 550Round m1 550Torroja 950Flat 950Round m05 950Round m1 950

Figure 6.3: Comparison of Reynolds stresses for various fixed cases at Reτ = 550 and Reτ = 950

normalized in wall units (left) and in outer units (right)

As those experiments suggests, the fluctuations are highly sensitive to minor de-

viations of the mean profile from its natural shape. The rounder β = −0.5-

profile results in lower stream- and spanwise intensities near the wall, though higher

intensities away from the wall. The opposite is the case for the flatter β = +0.5-

profile as confirms figure 6.1. Fixing the mean velocity profile is equivalent to adding

a volume force on the RHS (right hand side) of the momentum equation. The

intensities, created by the incorrect velocity gradient, create a strong accelerating

force trying to adjust the mean profile to its natural value, while the forcing term

pulls it back to the given fixed shape. The nature of the extra forcing term will

further analyzed in section 6.6.

In figure 6.4 the structures created by the incorrect profiles are examined for the

Reτ = 550 cases. The isotropy coefficients of the fluctuations in streamwise direction

(Iu) and spanwise direction (Iw) are respectively defined as

Iu =u2

K

Iw =u2

K(6.1)

where K denotes the total turbulent kinetic energy given as

72


10−2

10−1

100

0.8

1

1.2

1.4

1.6

1.8

up2 /K

y/h

Isotropy Coefficient Iu

10−2

10−1

100

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

wp

2 /K

y/h

Isotropy Coefficient Iw

0 0.5 10

0.1

0.2

0.3

0.4

0.5

−<uv

>+ /(

up+ vp

+ )

y/h

Structure coefficient

0 0.05 0.1 0.15 0.20.3

0.35

0.4

0.45

0.5

−<uv

>+ /(

up+ vp

+ )

y/h

Structure coefficient (zoomed in)

Fixed to TorrojaBeta+05Beta−05

Figure 6.4: Isotropy coefficients and structure coefficients of the transverse Reynolds stresses

for the three β cases

K =1

2

(

u2 + v2 + w2)

(6.2)

Iu and Iw vary between 0 and 2 and can be interpreted according to

Iu =

0 structures entirely contained within the y − z plane

2/3 isotropic flow

2 purely streamwise motion

(6.3)

and respectively

73


Iw =

0 structures entirely contained within the x − y plane

2/3 isotropic flow

2 purely spanwise motion

(6.4)

Both isotropy coefficients suggest that the near wall structures, created by the

the flatter β = +0.5-profile increase the isotropic state by creating structures in

spanwise direction. The higher shear in combination with the inhibiting wall results

in small structures that are naturally more isotropic than larger structures.

The rounder β = −0.5-profile stays interestingly close to the natural case, suggesting

that the structures are essentially similar to the ones created by the natural profile.

In the channel center, both incorrect profiles result in a less isotropic flow than the

natural case. This makes sense for the round profile, whose increased shear in the

channel center creates large anisotropic structures.

The structure coefficient of the transverse Reynolds stress depicted in figure 6.4 is

defined as

−uv

uv(6.5)

It suggests that the fluctuations created by the incorrect profiles are essentially

similar to the natural ones in the middle of the channel but differ near the wall. The

structure coefficient changes for wall distances smaller than y/h = 0.45. Coming

from the middle of the channel towards the wall the flatter β = +0.5-case first

stays fairly constant but below the natural case. It then increases close to the wall

and exceeds the structure coefficient of the natural case at the y+ = 15 location.

In agreement with the results of the fluctuation in figure 6.2, this shows that the

increased intensities near the wall result from the formation of structures with higher

Reynolds stresses.

The structure coefficient of the rounder β = −0.5-profile drops to a minimum at

about y+ = 40 and then increases to the maximum at y+ = 15 but stays below the

natural case. The fluctuations created by the rounded profile away from the wall

do most likely not extend all the way to the wall and therefore do not create any

shear stresses close to the wall.

74


102

103

101

102

103

y+ = 15

λ+x

λ+ z

Torroja fixedFlatRound

102

103

101

102

103

y+ = 166

λ+x

λ+ z


102

103

101

102

103

y+ = 416

λ+x

λ+ z


Figure 6.5: 2D streamwise velocity spectra at various wall distances. Contour level CL = 0.4 of

the maximum of the natural case and CL = 0.9 of the maximum of each individual

case

101

102

103

101

102

103

y+ = 15

λ+x

λ+ z

101

102

103

101

102

103

y+ = 166

λ+x

λ+ z


101

102

103

101

102

103

y+ = 416

λ+x

λ+ z

Figure 6.6: 2D streamwise vorticity spectra at various wall distances. Contour level CL = 0.4 of

the maximum of the natural case and CL = 0.9 of the maximum of each individual

cas

6.2 Spectral results of β-cases

The two dimensional energy spectra for three different wall distances of the stream-

wise velocity fluctutions and the streamwise vorticity fluctuations are depicted in

figure 6.5 and 6.6 respectively. To show the spectral results at wall normal locations

y+ = 15 and y+ = 166 was decided for the maximum of the Reynolds stresses, that

occur at those locations for the respective cases, as can be seen in figure 6.1. The

location y+ = 416 was chosen for the information it yields on the effect of the fixed

mean profiles on the channel center.

75


As already mentioned above, changing the mean profile changes the size of the

structures that are created. For the flat profile, where the shear is moved towards

the wall, the near wall structures (y+ = 15) become narrower, shorter and about 50%

more intense. This is true for all three velocity spectra (only streamwise direction

shown here). For the round profile, in agreement with the statistics results, the

structures near the wall become less intense and the energy is moved away from the

wall towards the channel center at y+ = 416, where the energy increases in all three

velocity components with respect to the natural case. As show the contour levels

(CL = 0.9) near the peak of the respective spectrum, the entire spectrum is not

distorted but completely shifted.

Interestingly the influence of changing the mean profile does not only effect the

large scale structures but changes the small scale structures (such as vorticity) in

the same manner. Figure 6.6 shows the two dimensional energy spectrum of the

streamwise vorticity. Just like the large scale fluctuations of the velocities, the

vorticity fluctuations get more intense near the wall for the flat profile while energy

is added by the round profile in the channel center. This indicates that by changing

the mean shear not only structures of similar sizes, but structures at both ends of

the cascade are affected in the same manner, suggesting that the energy, that is fed

into the large scales by the mean shear is passed down the self similar cascade locally.

Figure 6.7 shows the pre-multiplied kinetic energy spectra splitted up into its three

components for the same three wall distances y+ = 15, y+ = 166 and y+ = 416 for

the three β-cases: Natural, flat and round.

Near the wall (y+ = 15), the wave number κ of the maximum of the flat profile

increases for all three components suggesting that most of the additional energy, that

is created near the wall, is added to the smaller scales (higher wave numbers) than in

the natural case. The largest increase in terms of the wave number can be observed

in the spanwise direction. Also, the relative increase of the extra energy added is

about 50% larger in spanwise and wall normal direction where the energy doubles

with respect to the natural case. In streamwise direction the energy increases only

76


1 10 1000

0.5

1

1.5

2y+ = 15

κ

κ E

(κ)

TorrojaFlatRound

1 10 1000

0.1

0.2

0.3

0.4y+ = 166

κ

κ E

(κ)

TorrojaFlatRound

1 10 1000

0.05

0.1

0.15

0.2y+ = 416

κ

κ E

(κ)

TorrojaFlatRound

Figure 6.7: Pre-multiplied 1D spectra of TKE spitted up into its components. Solid line:

streamwise. Dotted line: spanwise. Dashed line: wall normal

by 50%.

For the round profile near the wall (y+ = 15), the maximum of the energy drops

below the natural profile and is moved towards lower wave numbers in all three

components. The extra energy in the large wave numbers comes most likely from

bigger structures, that are created in the outer channel region and extent all the

way to the wall. This is confirmed by looking at the energy distribution away from

the wall (y+ = 166) and (y+ = 416), which make clear that more larger and intense

structures are created in the outer channel region, where the mean velocity gradient

is stronger than in the natural case. The largest structures do not even fit into the

box of the present simulation and it has to be left to simulations with larger box

sizes to determine the nature of the largest structures.

For the wall distance y+ = 166, the maximum of the energy fed into the fluctuations

by the flat profile drops below the natural case, suggesting that all the extra energy

from the increased mean shear in the near wall region stays near the wall. This

supports the assumption in [15] that the near wall region is essentially autonomous.

The increased shear in the channel center of the round profile creates large structures

and thus moves the energy towards lower wave numbers.

The fact that the additional energy for the round profile is added predominantly to

the larger scales was further investigated, by considering the zero modes of the two

homogeneous directions kx = 0 and kz = 0 separately. The zero-zero mode of the

energy spectrum is defined as the mean velocity profile.

Because the modes in stream- and spanwise direction kx = 0 and kz = 0, re-

77


Streamwise

0 100 200 300 4000

100

200

300

400

500<U>2 Mode (1,1)

y+ Position

Ene

rgy

0 100 200 300 4000

0.5

1

1.5kx = 0 Modes (1,2:end)

y+ Position

Ene

rgy

0 100 200 300 4000

0.02

0.04

0.06

0.08

0.1

kz = 0 Modes (2:end,1)

y+ Position

Ene

rgy

0 100 200 300 4000

2

4

6

8

10non−zero modes (2:end,2:end)

y+ Position

Ene

rgy

TorrojaFlatRound

Figure 6.8: Energy of streamwise direction seperated by modes

Wallnormal

0 100 200 300 4000

100

200

300

400

500<U>2 Mode (1,1)

y+ Position

Ene

rgy

0 100 200 300 4000

0.02

0.04

0.06

0.08kx = 0 Modes (1,2:end)

y+ Position

Ene

rgy

0 100 200 300 4000

0.02

0.04

0.06

0.08kz = 0 Modes (2:end,1)

y+ Position

Ene

rgy

0 100 200 300 4000

0.5

1

1.5non−zero modes (2:end,2:end)

y+ Position

Ene

rgy

TorrojaFlatRound

Figure 6.9: Energy of wall normal direction seperated by modes

spectively, are commonly omitted in the plot for the 2D energy spectra, they are

regarded separately. As the results of the 2D spectra suggest, those large structures

would contain some of the extra energy added to the flow by the unnatural profiles.

By looking at the results shown in figures 6.8 through 6.10 it is confirmed that

the energy is moved towards larger scales, and especially into the zero modes

78


Spanwise

0 100 200 300 4000

100

200

300

400

500<U>2 Mode (1,1)

y+ Position

Ene

rgy

0 100 200 300 4000

0.05

0.1

kx = 0 Modes (1,2:end)

y+ Position

Ene

rgy

0 100 200 300 4000

0.05

0.1


y+ Position

Ene

rgy

0 100 200 300 4000

0.5

1

1.5

2

2.5non−zero modes (2:end,2:end)

y+ Position

Ene

rgy

TorrojaFlatRound

Figure 6.10: Energy of spanwise direction seperated by modes

(largest structures contained in the channel), by the increased shear in the channel

center. The combination of an increased shear and no walll, which would inhibit

the formation of larger eddies, results in large structures that extend all the way

to the wall. Their signature can be seen in the streamwise (figure 6.8) and the

spanwise (figure 6.10) direction where the zero modes in the near wall region contain

substantially more energy for the round case than in the natural case.

The flat profile where the shear is move towards the wall contains less energy in

the zero modes. The shear is increased near the wall, which due to the closeness of

the wall creates smaller structures. This can be seen in the graph for the non-zero

modes (smaller structures) of all three directions (figures 6.8 through (figure 6.10)).

The energy in the zero modes drops, while the energy in the smaller scales close to

the wall increases. Away from the wall the shear is less than in the natural case

and therefore less energy is fed into the flow.

79


6.3 Results of blended cases

Figures 6.11 and 6.12 show the statistical results of blending two different mean

profiles at three wall normal locations as explained in chapter 5.3. The quantitative

results of the β = −0.5 blends and the β = −1.0 blends are essentially the same,

however the β = −1.0 blends disturb the flow stronger, since a “more incorrect”

profile in the outer part of the flow was used. Therefore, as was to be expected, the

results of the β = −1.0 blends are more pronounced.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

−<uv

>+

y/h

Reynolds stress

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

u+

y/h

Streamw. vel. fluc.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

v+

y/h


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

τ xy+

y/h

Total Stress

TorrojaRoundBuffer BlendLog BlendOuter Blend

Figure 6.11: Statistics of β = −0.5 blendings at Reynolds number Reτ = 550

The effect of the blending itself can be seen in the structure coefficient of the

β = −1.0 blend depicted in figure 6.13. As shown in figure 5.12, the gradient

of the mean velocity profile for the Buffer Blend and the Log Blend decreases in

order to blend the two profiles, while it increases at the blending location for the

Outer Blend. The structure coefficient in figure 6.13 reacts accordingly. A local

drop in the structure coefficient can be seen for the Buffer Blend and the Log Blend

80


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

−<uv

>+

y/h

Reynolds stress

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

u+

y/h

Streamw. vel. fluc.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

v+

y/h


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

τ xy+

y/h

Total Stress


Figure 6.12: Statistics of β = −1.0 blendings at Reynolds number Reτ = 550

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

−<uv

>+ /(

u+ v+ )

y/h

Structure coefficient of uv


Figure 6.13: Structure coefficient of transverse Reynolds stress for β = −1.0 blendings at

Reynolds number Reτ = 550

while it increases at the location of the Outer Blend. This locality suggests that

the additional structures created by the blending stay where they were created and

81


0 100 200 300 400 5000

0.5

1

1.5

−<uv>+

y+

Reynolds stress

Buffer Blend 550Log Blend 550Outer Blend 550Buffer Blend 950Log Blend 950Outer Blend 950

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

−<uv>

y/h

Reynolds stress


Figure 6.14: Comparison of Reynolds stresses for the three blended cases at Reτ = 550 and

Reτ = 950 normalized in wall units (left) and in outer units (right)

enter the cascade process at this wall normal distance.

Going back to figure 6.12, up to y+ ≈ 50, the fluctuations stay close to the near

wall Torroja profile. The buffer blend is blended at y+ ≈ 30, but interestingly

its fluctuations deviate from the Torroja profile not significantly below y+ ≈ 60.

However above the y+ ≈ 60 threshold, also the other two blends (Log and Outer

Blend), that are blended at location y/h ≈ 0.25 and y/h ≈ 0.78, respectively deviate

from the Torroja case, indicating that the larger structures, created by the round

profile, into which all three are eventually blended, extend all the way to the wall.

This explains the fact that the strongest difference can be observed for the Buffer

Blend, which uses the round profile for 95% of the channel hight, while the Outer

Blend, which uses the round profile only for roughly 15% in the channel center, is

less affected. In both figures 6.12 and 6.11 the Reynolds stress of the Buffer Blend is

perturbed quite locally. Before, just as after the perturbation by the blending, the

slopes (which corresponds to the force imposed on the flow) of the blended case and

its outer “mother profile” coincide. The Log Blend as well switches quite locally,

while the Outer Blend takes a larger wall normal intercept to switch from the inner

“mother profile” to the outer one.

The results of analyzing the energy in the zero modes of the streamwise direction

82


Streamwise

0 100 200 300 4000

100

200

300

400

500<U>2 Mode (1,1)

y+ Position

Ene

rgy

0 100 200 300 4000

0.5

1

1.5

kx = 0 Modes (1,2:end)

y+ Position

Ene

rgy

0 100 200 300 4000

0.05

0.1

0.15


y+ Position

Ene

rgy

0 100 200 300 4000

2

4

6

non−zero modes (2:end,2:end)

y+ Position

Ene

rgy

TorrojaBeta m1Buffer Blend m1Log Blend m1Outer Blend m1

Figure 6.15: Energy in zero modes of streamwise direction for blending cases

are shown in figure 6.15. As was readily mentioned above, the zero-zero mode is

defined as the mean velocity profile and the modes in stream- and spanwise direction

kx = 0and kz = 0, respectively, are regarded separately. The results confirm the

findings from section 6.1, that the energy is moved towards larger scales, away from

the smaller scales, by increasing the shear in the out channel region. The effect is

stronger for the profiles that blend earlier, strongly suggesting that it is the larger

shear of the round profile that is responsible for moving energy towards larger scales.

Those large eddies than extend all the way to the wall, where they influence the

near wall flow as was readily observed in the statistics and the spectral results.

Figures 6.16 and 6.17 show the 2D spectral results of the two basis profiles, the

Torroja and the round profile, as well as the results for all three β = −1.0 blends

for the streamwise velocity component and the vorticity component ωz at various

wall distances for the Reynolds number Reτ = 550.

The small scale structures created by the Log Blend are predominantly local and

adjust fast to local profile. As can be seen in the 2D spectra of the streamwise

velocity component depicted in figure 6.16. The blending takes place around y+ =

83

Dip

lom

arb

eit6

Resu

lts

102

103

102

103

y+ = 9

λ+ z

102

103

102

103

y+ = 15

102

103

102

103

y+ = 30

102

103

102

103

y+ = 60

102

103

102

103

y+ = 105

λ+x

λ+ z

102

103

102

103

y+ = 166

λ+x

102

103

102

103

y+ = 222

λ+x

102

103

102

103

y+ = 416

λ+x

TorrojaBeta m1Buffer BlendLog BlendOuter Blend

Figu

re6.16:

2Dsp

ectralresu

ltsof

the

streamw

isevelo

citycom

pon

entfor

various

wall

distan

ces

given

inw

allunits

ata

contou

rlev

elC

L=

0.4of

the

max

imum

ofth

eTorro

jacase

atR

eτ

=550

84

Dip

lom

arb

eit6

Resu

lts

101

102

10310

1

102

103

y+ = 9

λ+ z

101

102

10310

1

102

103

y+ = 15

101

102

10310

1

102

103

y+ = 30

101

102

10310

1

102

103

y+ = 60

101

102

10310

1

102

103

y+ = 105

λ+x

λ+ z

101

102

10310

1

102

103

y+ = 166

λ+x

101

102

10310

1

102

103

y+ = 222

λ+x

101

102

10310

1

102

103

y+ = 416

λ+x

TorrojaBeta m1Buffer BlendLog BlendOuter Blend

Figu

re6.17:

2Dsp

ectralresu

ltsof

the

span

wise

vorticity

compon

ent

ωz

forvariou

sw

all

distan

cesgiv

enin

wall

units

ata

contou

rlev

elC

L=

0.4of

the

max

imum

of

the

Torro

jacase

atR

eτ

=550

85


140. The spectrum at y+ = 105 still coincides well with the inner Torroja case,

while the spectrum at y+ = 166 already matches the round β = −1.0 profile into

which it is blended. This suggests that either the small scales have smaller time

scales than the larger structures within which they are contained and thus adapt

faster to the local equilibrium, or the eddies are long and thin without a very big

wall normal extension.

The buffer blend separates much later (y+ ≈ 140) from the near wall Torroja profile

than its blending location (y+ ≈ 30) would suggest. It must be the influence of the

large structures from the channel center that extent all the way to the wall. The

Outer Blend again switches quite locally.

By looking at the large scales (large λx and large λz, respectively) in figure 6.16, it

once more becomes clear that energy is moved towards larger structures by moving

the shear outwards and that all three blends tend towards the round profile even

before the blending has occurred. There is more energy in the large scales for the

blended profiles, explaining where the extra energy moves that was observed in the

Reynolds stress statistics in figure 6.12. The closer to the wall the blending occurs,

the greater is the increase of extra energy in the large scales, confirming that the

round profile increases the amount of energy in the large scales.

That suggests that the mean profile preferably interacts with scales of similar sizes

but since all scales at a given wall distance show the same reaction to a changed mean

profile (see spanwise vorticity component ωz in figure 6.17), it can be assumed that

the cascade process acts locally and faster than the time scale of the large eddies.

6.4 Intersection Point

An interesting observation is the fact, that the profile of the Reynolds stress of the

natural case intersects with the profiles of the Reynolds stresses of the unnatural

cases in one point at y+ ≈ 75, scaled in wall units, and depicted in figure 6.18. This

is true for both Reynolds numbers. Most likely the mean shear and therefore the

86


0 50 100 150 2000

0.5

1

1.5

−<uv

>+

y+

Reynolds stress

Torroja 550Flat 550Round 550Torroja 950Flat 950Round 950

0 50 100 150 2000

0.5

1

1.5

−<uv

>+

y+

Reynolds stress


Figure 6.18: Intersection of Reynolds stress for the three β cases (left) and the blended cases

(right) of both Reynolds numbers scaled in wall units

0.05 0.1 0.15 0.2 0.25 0.30.5

0.6

0.7

0.8

y/h

U

Inter caseTorrojaβ = 0.5β = −0.5β = −1.0

0.05 0.1 0.15

3

4

5

6

7

8

9x 10

−3

dU/d

y

y/h

Figure 6.19: Right: Mean profile of the β-cases and the “intersection” case. Left: The mean

shear of the same cases.

production of turbulent kinetic energy at this wall normal location is constant for

all cases. Also the blended profiles (right graph in figure 6.18) show this common

intersection point at the very same wall normal location even though the blending

disturbes the flow quite close to the intersection point (Buffer Blend).

In order to investigate this phenomenon further, a profile (Reτ = 950) that would

not intersect in the same wall normal distance was designed, using the Cess formula

from equation 5.2 (A = 19.10 and κ = 0.35) and keeping as for all other simulations

the friction Reynolds number Reτ , the mass flux Ub and the nominal production

P/uτ constant. The profile of the mean velocity and the mean shear together

with the β-cases are depicted in figure 6.19. The total stress for the four cases is

depicted in figure 6.20. As can be seen in the zoomed in view of the total stress

(right graph), the intersection point was moved further outwards by changing the

87


0.2 0.4 0.6 0.8 10

0.5

1

1.5

(dU

+ /dy)

+ <

uv>

+

y/h

Torrojaβ = +0.5β = −0.5β = −1.0Inter case

0 0.05 0.1 0.15 0.2 0.250.6

0.7

0.8

0.9

1

1.1

1.2

1.3

(dU

+ /dy)

+ <

uv>

+

y/h

Torrojaβ = +0.5β = −0.5β = −1.0Inter case

Figure 6.20: Right: The total stress of the β-cases and the “intersection” case. Left: Zoomed

in view.

mean shear. This suggests that the intersection of the total stress is an artifact,

caused by the intersection of the mean shear and therefore could be avoided by

changing the design of the mean profile.

6.5 Normalization

A new local normalization quantity is proposed in the following chapter to collapse

the unnatural profiles and the natural profile.

In the unfixed case, with the Reynolds stress τ ∼ u2 and thus the dissipation

D ∼ u3

L= τ3/2

L, the energy finds a state of equilibrium according to

∂E

∂t= P + D = S · τ − τ 3/2

L(6.6)

where the fluctuations u and v, created by the shear of the mean velocity profile

S = ∂U∂y

, create the Reynolds stress −〈uv〉, which adjusts the mean shear (feedback

mechanism).

In the fixed case, however, the mean profile cannot change and therefore the energy

can only be modified by changing the Reynolds stress τ and the equation 6.6

becomes

∂τ

∂t= P + D = τ − τ 3/2

L(6.7)

88


The stress has therefore to change when the profile is fixed to an unnatural profile,

which became obvious in the results of the previous chapters.

Following this reasoning a new velocity scale is introduced. The common velocity

scale in a turbulent channel flow is the friction velocity, which is in the order of the

velocity fluctuations. The friction velocity is given as uτ =√

τw, where τw is the

wall stress. This scaling is useful when the total stress τtotal, given as the sum of

the viscous stress and the Reynolds stresses

τtotal = −ν∂U

∂y− 〈uv〉 (6.8)

results in a straight line according to

τtotal = u2τ

(

1 − y

h

)

(6.9)

Thus uτ gives a constant global scaling factor for the entire channel. Since, as shown

in the statistical results of sections 6.2 and 6.5, the total stress for unnatural profiles

does not satisfy a straight line, a local scaling factor is proposed in the current work.

Figure 6.21 depicts a local scaling factor u∗

τ = f(y) for various unnatural cases and

the natural case. It is calculated from

u∗

τ =

√

τtotal

1 − yh

(6.10)

This way, the local condition of the Reynolds stresses for every wall normal position

is taken into account. As figure 6.21 shows, the scaling factor for the natural profile

results in a constant value, whereas for the unnatural profile it varies significantly.

Figures 6.22 through 6.27 show the streamwise, wall normal and spanwise velocity

fluctuations, respectively, for the two sets of unnatural profiles normalized the

conventional way with uτ (left graphs) and normalized with the new local quantity

u∗

τ (right graphs). The fluctuations for all unnatural and natural profiles collapse

when normalized with u∗

τ . Close to the wall, where the influence of the viscous stress

dominates, the scaling works only for the wall normal component of the intensities.

The near wall deviation was to be expected, since very close to the wall, due to

89


Figure 6.21: Total stress√

τtotal for various unnatural profiles

the influence of viscosity, the nature of the boundary layer is essentially different.

This strongly suggests that the mechanism with which energy is fed into velocity

fluctuation is predominantly local and not an interaction between eddies at different

wall distances.

Those results once more support the assumption of the fact that the cascade process

happens locally. As suggest the spectral results, large structures created by the

mean shear in the channel center penetrate the near wall region due to their size

but intensities are determined locally and neither through interaction of structures

at different wall distances, nor through a feedback mechanism where an acceleration

and deceleration of the mean profile takes place according to the strength of the

local intensities.

Figure 6.28 shows the spanwise velocity fluctuations of three of the β-cases, nor-

malized with uτ = const. (left) and uτ = f(y) (right) at both Reynolds numbers,

Reτ = 550 and Reτ = 950, respectively. Plotted in wall units it can be seen, that

the wall normal location, from where on outwards the intensities collapse, scales in

wall units. The black line at y+ ≈ 100 indicates the wall normal location of the

scaling threshold.

90


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

u+

y/h0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

3.5

u+

y/h

Torroja

Flat

Round m05

Figure 6.22: Streamwise velocity fluctuations of β-cases, normalized with uτ = const. (left)

and uτ = f(y) (right) at Reτ = 550

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

v+

y/h0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5v+

y/h

TorrojaFlatRound m05

Figure 6.23: Wall normal velocity fluctuations of β-cases, normalized with uτ = const. (left)


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

w+

y/h0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

w+

y/h

Torroja

Flat

Round m05

Figure 6.24: Spanwise velocity fluctuations of β-cases, normalized with uτ = const. (left) and

uτ = f(y) (right) at Reτ = 550

91


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

u+

y/h0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

u+

y/h

TorrojaRound m1Buffer BlendLog BlendOuter Blend

Figure 6.25: Streamwise velocity fluctuations of blended cases, normalized with uτ = const.

(left) and uτ = f(y) (right) at Reτ = 550

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

v+

y/h0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5v+

y/h

Torroja

Round m1Buffer BlendLog BlendOuter Blend

Figure 6.26: Wall normal velocity fluctuations of blended cases, normalized with uτ = const.

(left) and uτ = f(y) (right) at Reτ = 550

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

w+

y/h0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

w+

y/h

Torroja

Round m1Buffer BlendLog BlendOuter Blend

Figure 6.27: Spanwise velocity fluctuations of blended cases, normalized with uτ = const. (left)


92


0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

w+

y+0 100 200 300 400 500

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

w+

y+

Torroja 550Flat 550Round 550Torroja 950Flat 950Round 950

Figure 6.28: Spanwise velocity fluctuations of β-cases, normalized with uτ = const. (left) and

uτ = f(y) (right) at Reτ = 550 and Reτ = 950 plotted in wall units. The black

line at y+ ≈ 100 indicates the wall normal location from where on the intensities

collapse.

6.6 Energy Balance

As mentioned above, fixing the mean velocity profile adds a forcing term to the

RHS of the Navier-Stokes equations. Its nature and the influence of the fixed mean

profiles on the energy budgets of the Reynolds stresses will be discussed in the

current section.

6.6.1 Forcing Term

The forcing term pulls the mean velocity profile into the given shape while the

fluctuating velocity field tries to deform it to its natural value. The energy that is

added to the flow in this process is analyzed in this section to answer the question

where and in what way it enters the energy balance.

The instantaneous flow, given in equation 2.11 is

[

∂ui

∂t+ uj

∂ui

∂xj

]

= −1

ρ

∂p

∂xi

+ ν∂2ui

∂x2j

+ fi (6.11)

where the extra forcing term fi is added to account for the force that in every time

93


step pulls the mean velocity profile back into its given place when it wants to adjust

to its natural value. The Reynolds decomposition for the velocity field and the

forcing term respectively is given by

ui = Ui + ui

fi = Fi + fi (6.12)

To the equation for the averaged velocity, which is computed by applying 6.12 on

6.11 and averaging it, the mean forcing term Fi is added. This yields

DUi

Dt= − ∂p

∂xj

+ ν∇2Ui −∂uiuj

∂xj

+ Fi (6.13)

The equation for the instantaneous fluctuating velocity is obtained by subtracting

6.11 from 6.13

[

∂ui

∂t+ Uj

∂ui

∂xj

]

= − ∂p

∂xi

+ ν∂2ui

∂x2j

−[

uj∂Ui

∂xj

]

−{

uj∂ui

∂xj

−⟨

uj∂ui

∂xj

⟩}

+ fi (6.14)

To quantify the forcing term, equations 6.13 and 6.14 are considered in Fourier

space. As mentioned before, the zero mode of the Fourier notation represent the

mean velocity profile, while the fluctuations are represented in the higher modes.

The forcing term in streamwise direction pulls the zero mode of the instnataneous

streamwise velocity component to the given value. The forcing term in spanwise

direction pulls the instantaneous value of the zero mode of the spanwise velocity

component to zero. The wall normal mean velocity profile is zero for mass conser-

vation. Therefore the forcing term f is a function of the wall normal distance y and

has entries only in the zero modes for u = U and w = W = 0 but no fluctuating

terms f = 0. If now the stress tensor⟨

f u⟩

is calculated by applying the Reynolds

decomposition from equation 6.12. This yields

⟨

fiuk

⟩

= 〈(F + f) (U + u)〉

= 〈FU + fU + Fu + fu〉 (6.15)

= 〈FU〉 (6.16)

94


Herefrom it becomes clear that only the term 〈FU〉 prevails, since the zero modes of

the fluctuating forcing term f are zero as well as the zero modes of the fluctuating

velocity u are zero. The resulting term F goes into the equation for the averaged

velocity given in 6.13.

The resulting tensor

〈FU〉 =

⟨

FxU FxV FxW

FyU FyV FyW

FzU FzV FzW

⟩

(6.17)

reduces to a vector

〈FU〉 =

⟨

FxU

0

0

⟩

(6.18)

since Fy = 0, V = 0 and W = 0. To see where the extra energy is fed into the

fluctuations equation 6.14 is considered.

Since the forcing term fi drops out, the only way the extra energy from the force can

enter into the fluctuations is through the mean velocity gradient in the production

term[

uj∂Ui

∂xj

]

.

To quantify the energy that is produced by the various profiles the Reynolds stress

equation is considered. In order to obtain the equation for the kinetic energy

equation, the equation for the fluctuating velocity 6.14 is multiplied by uk and

averaged. It follows

[⟨

uk∂ui

∂t

⟩

+ Uj

⟨

uk∂ui

∂xj

⟩]

= −⟨

uk∂p

∂xi

⟩

+ν

⟨

uk∂2ui

∂x2j

⟩

−[

〈ukuj〉∂Ui

∂xj

]

−⟨

ukuj∂ui

∂xj

⟩

(6.19)

Since both i and k are free indices, they can be interchanged and after some

rearragement and the decomposition of the velocity deformation rate tensor (strain

rate tensor) into its symmetric part and antisymmetric part according to

95


∂ui

∂xj

= sij + ωij =1

2

[

∂ui

∂xj

+∂uj

∂xi

]

+1

2

[

∂ui

∂xj

− ∂uj

∂xi

]

(6.20)

as well as the fact that the double contraction of a symmetric tensor with an anti-

symmetric tensor is identically zero, yields the equation for the Reynolds stress

〈uiuj〉 to

D 〈uiuj〉Dt

= Pij + ǫij + Tij + ΠSij + Πd

ij + Vij (6.21)

following [17], where the various terms on the right hand side are referred to as

production Pij, dissipation ǫij, turbulent diffusion Tij, pressure strain ΠSij, pressure

diffusion Πdij and viscous diffusion Vij. They are given as

Pij = −〈uiuj〉∂Uj

∂xk

− 〈uiuj〉∂Ui

∂xk

(6.22)

ǫij = −2ν

⟨

∂ui

∂xk

∂uj

∂xk

⟩

(6.23)

Tij =∂ 〈uiujuk〉

∂xk

(6.24)

ΠSij =

⟨

p

(

∂ui

∂xj

+∂uj

∂xi

)⟩

(6.25)

Πdij = − ∂

∂xk

[〈pui〉 δjk + 〈puj〉 δik] (6.26)

Vij = ν∂2 〈uiuj〉

∂x2k

(6.27)

It becomes clear again, that the extra energy seen in the statistics of total stresses,

is added to the flow through the gradient of the mean profile (production term)

from where it is then moved into the other components. Thus, the extra energy can

be computed by subtracting the total stresses of each unnatural case from the total

stress of the natural case.

The Reynolds stresses of the natural case, the round (β = −0.5) and the flat

(β = +0.5) case are shown in figure 6.29 on the left. In the middle the subtraction

of the two unnatural cases from the natural cases depict the force which tries to

pull the profile to its natural value. The multiplication with the respective mean

profile of each case yields the energy added to or taken from the flow at a given wall

96


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

τ tot

y/h

Total Stress

TorrojaFlatRound

0 0.2 0.4 0.6 0.8 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

F

y/h

Force = ∆ Total Stress

Torroja−TorrojaFlat−TorrojaRound−Torroja

0 0.2 0.4 0.6 0.8 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

FU

y/h

F*U

(Torroja−Torroja)*U(Flat−Torroja)*U(Round−Torroja)*U

Figure 6.29: Total stresses for Torroja profile and the unnatural β = −0.5 and β = +0.5 profiles

(left. The difference between total stresses (middle). Extra energy added by the

unnatural profiles (right)

distance, depending on the sign.

Corresponding to the mean shear, the round profile feeds energy into the flow only

in the near wall region, while in the outer part of the flow a lack of energy over

compensates the added energy, resulting in a negative overall energy balance.

For the flat profile the shear is greater away from the wall, compared to the natural

profile, thus adding a substantial amount of extra energy in the outer part of the

flow, while not loosing the same amount near the wall. Therefore it results in a

positive overall balance. As was seen in the previous sections, this extra energy is

added into the largest scales (stream- and spanwise zero modes) and is then passed

down the energy cascade locally where it equally effects smaller scales and therefore

increases likewise the dissipation locally.

6.6.2 Reynolds stress budget

Following [17], the Reynolds stress budget of the component 〈uiuj〉, given in equa-

tion 6.21 was computed for four of the fixed mean velocity profiles at Reτ = 550,

by averaging over 20 fields each. Note, that due to homogeneity in streamwise and

spanwise direction only four terms of the budget are non-zero. The budget of the

unfixed channel budget, labeled“Torroja Ref data”was added to the plot to compare

and validate the data. The results for the streamwise and wall-normal components

of the Reynolds stress budget normalized with yu3

τare shown in figures 6.30 and 6.31,

respectively. As for the energy spectra, the pre-multiplication of the budget terms

97


with y adds the advantage of making areas proportional to the integrated energy,

when displayed with logarithmic abscissa.

10−3

10−2

10−1

100

−12

−10

−8

−6

−4

−2

0

x 10−3 Dissipation

y *

D *

nu

2 / ut

au4

y/h10

−310

−210

−110

0

0

0.005

0.01

0.015

0.02

0.025

y *

P *

nu

/ uta

u4

y/h

Production

10−3

10−2

10−1

100

−10

−8

−6

−4

−2

0

x 10−3

y *

PS

* n

u / u

tau

4

y/h

Pressure Strain

10 20 30 40 50 60 70−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

PD

* n

u / u

tau

4

y+

Pressure Diffusion

10 20 30 40 50 60 70−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

TD

* n

u / u

tau

4

y+

Turbulent Diffusion

10−3

10−2

10−1

100

−4

−3

−2

−1

0

1

2x 10

−3

y *

VD

* n

u2 /

utau

4

y/h

Viscous Diffusion

Torroja fixedFlat p05Round m05Round m1Torroja Ref data

Figure 6.30: Premultiplied budget of the streamwise component 〈u1u1〉 of four fixed mean

profiles and the unfixed case at Reτ = 550

10−3

10−2

10−1

100

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

x 10−3 Dissipation

y *

D *

nu

2 / ut

au4

y/h10

−310

−210

−1

−0.01

−0.005

0

0.005

0.01

y *

P *

nu

/ uta

u4

y/h

Production

10−3

10−2

10−1

100

−1

0

1

2

3

4

x 10−3

y *

PS

* n

u / u

tau

4

y/h

Pressure Strain

0 20 40 60

−0.05

0

0.05

0.1

PD

* n

u / u

tau

4

y+

Pressure Diffusion

0 20 40 60

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

TD

* n

u / u

tau

4

y+

Turbulent Diffusion

10−3

10−2

10−1

100

−1.5

−1

−0.5

0

0.5

1

x 10−4

y *

VD

* n

u2 /

utau

4

y/h

Viscous Diffusion

Torroja fixedFlat p05Round m05Round m1Torroja Ref data

Figure 6.31: Premultiplied budget of the wall normal component 〈u2u2〉 of four fixed mean

profiles and the unfixed case at Reτ = 550

Note that the pressure diffusion term and the turbulent diffusion term are not

premultiplied and displayed in wall units for the buffer and logarithmic layer only,

while the other terms are plotted in outer units.

98


The results show a local behavior of the budget terms. As for the statistics and the

spectral results, the fixed Torroja profile matches very well the unfixed case. All

near wall terms increase for the flat β = +0.5 profile, which moves the shear towards

the wall. For the two rounder profiles, β = −0.5 and β = −1.0, for which the shear

is moved towards the channel center, all quantities increase in the channel center

with respect to the natural case. However, the flat profile yields a stronger impact

on the production term and therefore on all other terms. A possible explanation is

that the flat profile has an increased shear at a more sensitive near-wall wall normal

location, where also in the natural case turbulence is created, while the rounder

profiles move the production of turbulent kinetic energy outwards, where usually

less and in the very center zero turbulent kinetic energy is produced.

The local response of all terms to the imposed mean shear lets suggest once more

that the mechanism by which energy is fed into the flow and passed down the

cascade is predominantly local.

Again, an interesting observation is the readily mentioned intersection of the profiles

at a wall normal location of around y/h = 0.14 or y+ = 75, respectively. That the

intersection in the production terms translates into a likewise very well defined

intersection of the dissipation terms, leads as readily mentioned above, to the

assumption that the energy is passed down in a quite local cascade process, rather

than being diffused by any transport mechanisms.

99


6.7 Linear stability analysis

Even though the mean velocity profile of a channel flow is linear stable, as can be

seen in figure 6.32, since the imaginary parts of all eigenvalues are negative, linear

perturbations can grow substantially before they decay by extracting energy from

the mean flow. This transient energy amplifications was linked by [8] to coherent

structures in wall-bounded turbulence.

For an instable profile only one eigenvalue has to become positive. Transient growth

however is the result of the combination of multiple stable eigenvalues, which for the

current streamwise and spanwise perturbation wave numbers α ≈ 1.75, β ≈ 0.30 is

depicted in the light green box in figure 6.32.

Two peaks of the transient growth were found in natural channel flows by [8].

One in the viscous layer and one in the outer layer. The peak in the viscous

layer identifies the sublayer streaks while in the outer layer it identifies the large

scale global structures, that span the full channel. The minimum in between the

two most amplified modes corresponds to the structures in the log-layer. In the

present work linear analysis of the most amplified transient modes was used to

obtain further insight into the dynamics of the nonphysical turbulent flow with a

fixed mean velocity profile.

In general it is assumed that the small structures in the viscous layer equilibrate

with their large-scale environment. By fixing the mean profile, and especially by

fixing it to wrong values it was tried to gain further insights into those mechanisms.

Also, the cycle mechanisms, that are involved in the deformation of the mean profile,

are of interest to the present work, since it would be disrupted or at least falsified,

by fixing the mean profile to the correct or to a wrong profile, respectively. Thus,

by using the linear stability analysis it is expected to gather further insight into

what kind of structures are created by the various profiles and how the interaction

mechanisms work.

The procedure will be shortly explained in the following section before the results

100


0 0.2 0.4 0.6 0.8 1 1.2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

cr

ci

Figure 6.32: Spectrum of Eigenvalues for a given pertubation: α ≈ 1.75, β ≈ 0.30 or λx = 11h,

λz = 2h, respectively

for the mean profiles of the present work are presented. For further details, however,

it is referred to [8]. The procedure was basically adapted from [8], applying only

minor changes in the computation of the eddy viscosity. In order to quantify the

differences of the original approach of [8] and the approach taken in this work, a

comparison in the means of a sensibility study was carried out and is documented

in the last part of this section.

6.7.1 Linear Model

The linearized dynamics of small-amplitude perturbations to the mean profile U(y)

of the channel are governed by

∂u

∂t+ U(y)

∂u

∂x+ v

∂U

∂y= −∂p

∂x+ νtot

∂2u

∂x2+ νtot

∂2u

∂z2+

∂

∂y

(

νT∂u

∂y

)

(6.28)

where νtot denotes the total eddy viscosity which was obtained from the momentum

equation

νtot =u2

τ (1 − Y )∂U∂y

(6.29)

101


10−2

10−1

100

101

1020

5

10

λz/h

Max

. am

plif.

λx = 0.5h

TorrojaBeta=0.5Beta=−0.5

10−2

10−1

100

101

1020

5

10

λz/h

Max

. am

plif.

λx = 4h


10−2

10−1

100

101

1020

5

10

λz/h

Max

. am

plif.

λx = 11h


Figure 6.33: Maximum linear transient amplification of perturbations of streamwise

wavelengths λx = 0.5h (left), λx = 4h (middle), λx = 11h (right)

where Y is defined as Y = y/h, since the mean velocity profile and its gradient ∂U∂y

is known or can be computed from the imposed profile, respectively.

6.7.2 Results

The results of the transient linear stability analysis for a Reynolds number Reτ =

550 are depicted in figure 6.33. According to [8] the peaks in the three graphs

in figure 6.33 correspond to turbulent structures. The figures from left to right

show the maximum linear transient amplification of perturbations over the spanwise

wave lengths λz, for streamwise wave lengths λx = 0.5h, corresponding to short

streamwise structures, for streamwise wave lengths λx = 4h, corresponding to

somewhat larger streamwise structures and for streamwise wave lengths λx = 11h,

corresponding to large streamwise structures.

All three plots show that for the flat profile, the shorter and narrower structures

at around λ+z = 100 are more amplified than in the natural case. For the round

profile, shorter and narrower structures are less amplified than in the natural case,

whereas long structures at λz = 2h, corresponding to the structures in the channel

center, are more amplified.

Those findings confirm the results obtained from the statistics of the respective

cases. The higher shear of the flat profile creates shorter and narrower structures

near the wall, while the round profile creates larger and wider structures away from

the wall.

102


6.7.3 Sensibility study of the linear model

As described in [8] the linear stability analysis was originally carried out using the

eddy viscosity νT calculated from the Cess formula [19]

νtot

ν=

1

2

[

1 +κ2h+2

9

[

2Y − Y 2]2 [

3 − 4Y + 2Y 2]2

[

1 − e−Y h+

A

]2]1/2

+1

2(6.30)

which was also used to obtain the unnatural profile (β = +1.0) that was then mixed

with the natural Torroja profile as described in chapter 5.

The different method of computation of the eddy viscosity, described above and

used in the current work was chosen, since the fitting of the various correct and

incorrect mean velocity profiles to the Cess formula turned out to be of unsatisfying

accuracy. The bad fittings were due to the fact, that in order to obtain the incorrect

profiles a β = +1.0 profile was mixed with the Torroja profile applying the mixing

parameter β. However the unnatural profiles, analyzed in the current work, were

chosen such that they were leaning towards the other side: β = −1.0 and β = −0.5,

which were then hard to fit with the Cess approximation.

Nonetheless, to see the impact of an incorrect eddy viscosity (corresponding to an

incorrectly suggested dissipation) versus an incorrect mean velocity (corresponding

to an incorrectly imposed production), four analysis for the three cases β = 0

(Torroja), β = +0.5 and β = −0.5 where carried out, mixing the calculation

technique of the eddy viscosity and the mean velocity gradient according to table

6.1 where the label “Direct” refers to the calculation technique described in 6.7.1

and the label “Cess” refers to the technique described below.

Using the Cess formula the parameters A and κ were fitted by a least square fit to

the respective mean velocity profile, which was obtained by integrating the mean

momentum equation

∂U

∂y=

u2τ (1 − Y )

νtot

(6.31)

The values for the fitted parameter can be found in table 6.2.

103


Table 6.1: Summary of cases for transient linear stability analysis

Name Reτ νT Profile

C1 550 Cess Cess

C2 550 Direct Cess

C3 550 Cess Direct

C3 550 Direct Direct

Table 6.2: Summary of fitted parameter for transient linear stability analysis

Name A κ

Torroja 29.6206 0.4557

β = +0.5 38.5003 0.4682

β = −0.5 23.9392 0.4450

The least square fits of the β = −0.5 and the Torroja profile do not result in very

good approximations while the fit for the β = +0.5 is acceptable.

The results of the sensitivity analysis are shown in figure 6.34. For short and narrow

structures the wrong production of case C3 results in a slightly lower maximum

amplification, though similar qualitative results. For long and wide structures which

are associated with the channel center, the differences become more obvious and

10−2

10−1

100

101

1020

5

10

λz/h

Max

. am

plif.

λx = 11h

nut = direct, u = directnut = Cess, u = directnut = direct, u = Cessnut = Cess, u = Cess

10−2

10−1

100

101

1020

5

10

λz/h

Max

. am

plif.

λx = 11h


10−2

10−1

100

101

1020

5

10

λz/h

Max

. am

plif.

λx = 11h


Figure 6.34: Comparison calculation technique of maximum linear transient amplification of

perturbations of streamwise wavelengths λx = 11h. Left: Torroja profile, middle:

β = +0.5, right: β = −0.5

104


especially so for the round (β = −0.5) profile which makes sense since the worst fit

was obtained for this profile.

By taking a closer look at the results of the (β = −0.5) profile, it can be seen that an

incorrect dissipation (dashed red line) predicts a stronger maximum amplification for

large structures in the channel center only. However, when an incorrect production

(dotted blue line) is used, the maximum amplification drops for the small near wall

structures as well as for the large structures associated with the channel center.

Interestingly the two effect cancel out each other to a certain extend and when

both, the incorrect production and the incorrect dissipation are used, the maximum

amplification (dashed green line) is somewhat closer to the correct case (solid black

line).

It can be summarized, that maximum amplification is sensitive to the calculation

technique and a bad fit of the Cess parameters can lead to erroneous results.

However, the qualitative features could be reproduced by using even badly fitted

mean profiles.

105


6.8 Coherent Structures

In this section the influence of a fixed mean velocity profile on the formation

of coherent structures, namely clusters, is studied. The identification method of

vortices, which form a particular kind of coherent structures, is briefly outlined,

following the results of the analysis.

To identify coherent structures in turbulent flows, the model proposed in [13], based

on [14], was used in the current work. According to [14], a coherent structure

is defined as a region in space where the velocity gradient tensor A = ∇u is

dominated by its rotational part. Using this definition and expressing it in terms

of the discriminant D the velocity gradient tensor, expressed in terms of the tensor

invariants of A, R and Q

D =27

4R2 + Q3, (6.32)

the condition translates into D > 0. To visualize the structure, a point x is

considered being a part of a vortex cluster if

D(x) > α

√

D′2(y) (6.33)

where α is the threshold parameter which can be chosen to visualize vortices of

different intensities and

√

D′2(y) is the standard deviation of D in planes parallel

to the wall. The value used in the present work to identify the vortices individually

is α = 0.02, as proposed and reasoned by [13]. The advantage of using

√

D′2(y)

instead of D(x) > α is, that the inhomogeneity from the wall can be reduced. If

D(x) > α was used, and α was set to depict coherent structures near the wall, the

channel center would appear empty. For each case the respective standard deviation

was computed individually, since D varies with the sixth order of the mean shear(

dUdy

)6

. Figure 6.35 shows the standard deviation over the wall normal distance y/h

for the all four cases analyzed. The data was obtained by averaging over 20 flow

fields. As expected, the standard deviation changes accordingly to the mean shear.

Figure 6.36 shows the coherent structures of instantaneous flow fields in the con-

106


10−3

10−2

10−1

100

10−5

100

105

1010

std(

D)

y/h

Torrojap05m05m1

Figure 6.35: Standard deviation of D over the wall normal coordinate y/h for the four Reτ =

550-cases analyzed

verged state for the four flow configurations at Reτ = 550: The unfixed channel,

the channel fixed to the correct mean profile as well as the channel fixed to the

β = −0.5 and the β = +0.5 mean profile, respectively. The turbulent structures

in the various channels look very similar and it therefore can be concluded, that

turbulence develops in a similar manner when the mean velocity profile is fixed. The

clusters in the different flow fields look different since instantaneous fields are used,

but severe differences could not observed. The differences must lie in the details,

which were analyzed, using various post processing techniques, such as for example

described in [13]. Together with the results they are presented in the following

sections. All plots were obtained by averaging over 20 flow fields of the respective

cases.

First the clusters in each flow field were counted. The respective histograms are

depicted in figure 6.37. Two types of criteria were evaluated. First, structures were

distinguished whether their upper boundary ymax is located within the buffer layer

or in the outer layer. This criteria translates into ymax < 100+ for clusters located

in the buffer layer and ymax > 100+ which are located in the outer layer or have

at least their upper boundary outside of the buffer layer. The upper row in figure

6.37 depicts the results of the analysis. The flatter p05 case (shear moved towards

the wall) shows a strong increase in clusters close to the wall, while in the channel

center a similar number as in the natural fixed case is obtained. The opposite is true

107


Figure 6.36: Visualisation of clusters in instantaneous flow fields at Reτ = 550 using threshold

α = 0.02 top left, unfixed. Top right fixed Torroja. Bottom left, fixed β = +0.5

and bottom right fixed β = −0.5

108


Torroja p05 m05 m10

0.5

1

1.5

2

2.5

3x 10

4

Num

ber

of C

lust

ers

Below y + = 100

Torroja p05 m05 m10

2

4

6

8

10x 10

4

Num

ber

of C

lust

ers

Above y + = 100

Torroja p05 m05 m10

1

2

3

4

5x 10

4

Num

ber

of C

lust

ers

Small Clusters

Torroja p05 m05 m10

1

2

3

4

5

6

7

8x 10

4

Num

ber

of C

lust

ers

Large Clusters

Figure 6.37: Histograms of clusters at Reτ = 550 for cases Torroja, β = +0.5, β = −0.5 and

β = −1.0

for the two rounder m05 and m1 cases. According to the shear, less clusters were

counted near the wall while in the channel center the number of clusters increases.

This suggests that the mean shear directly influences the number of clusters being

created.

The second criterion, evaluated distinguishes clusters for their size. Small structures

were defined according to their volume Vclus to Vclus < 10E−05, and large structures

had therefore to satisfy Vclus > 10E − 05. As depicted in the lower row of figure

6.37, the flatter p05 case creates larger numbers of smaller structures, while the

rounder m05 and m1 cases create higher numbers of large structures.

The streamwise and spanwise joint probability density functions (p.d.f.) of the

sizes of clusters for all four cases at Reτ = 550 are depicted in figure 6.38. They

109


101

102

103

10410

1

102

103

∆ x+

∆ y+

Torrojap05m05m1 ∆ y+ = 1/45 ∆ x+1.3

∆ y+ = 1/2 ∆ x+1

∆ y+ = 1 ∆ x+0.8

101

102

103

10410

1

102

103

∆ z+

∆ y+

Torrojap05m05m1 ∆ y+ = 0.9 ∆ x+

∆ y+ = 2 ∆ x+0.8

101

102

103

101

102

103

∆ x+

∆y+

Torroja

p05

m05

m1

∆ y+= 1.8 ∆ x

+0.8

101

102

103

101

102

103

∆ z+

∆y+

Torroja

p05

m05

m1

∆ y+= 1.5 ∆ x

+0.9

Figure 6.38: Streamwise and Spanwise joint probability density functions of attached (upper

row) and detached clusters (lower row) for all four cases Torroja, β = +0.5, β =

−0.5 and β = −1.0 at Reτ = 550. The magenta line indicates the channel center.

110


are divided into two groups: Attached and detached clusters. Attached clusters

(upper row) have their lower wall-normal boundary ymin located below y+ = 20.

All clusters that do not fulfill this criterion, are considered detached clusters (lower

row).

Detached clusters, which make up about 80% of the total number of clusters, seem

not to be affected by the change of the mean shear, while attached clusters do show

a response.

An increased shear in the channel center (cases m05 and m1) increases the cluster

hight ∆y and length ∆x in both wall parallel directions, while the flat case, for

which the shear is moved towards the wall (p05), decreases both, hight ∆y and

length ∆x in both wall parallel directions.

For the round cases in streamwise direction, the increase of the sizes of attached

clusters in ∆y is stronger than the increase in ∆x, which results in a 30% increase of

the slope from 1 to 1.3 with respect to the Torroja case (dashed blue line in figure

6.38). The slope of the flat case drops by 20% with respect to the Torroja case

(black dotted line in figure 6.38). In spanwise direction, only the flat case seems to

have an effect on the slope and just like in streamwise direction it drops by 20%

with respect to the Torroja case.

The increase and decrease, respectively, of the slope was related to a change in the

shape of the clusters in [13]. It suggests that the shear has a direct influence on the

shape of the clusters, however further analysis is however needed to confirm this

finding. In spanwise direction no change in shape was observed.

Those results confirm the results presented above, where it was found, that the

rounder profiles move energy towards larger structures (into zero modes). The

increased shear in the channel center of the round profiles create structures that

even cross the channel center (magenta line in figure 6.38), while the flat profile and

the Torroja profile do not create structures of that size.

Figure 6.39 depicts the joint probability density functions (p.d.f.) for attached

clusters at Reτ = 950. The plots only partly confirm the results obtained from

the Reτ = 550 cases. While in spanwise direction no change can be observed, in

111


101

102

103

10410

1

102

103

∆ x+

∆ y+

TorrojaFlat p05Round m05Round m1∆ y+ = 1/15 ∆ x+1.3

∆ y+ = 1/2 ∆ x+0.9

101

102

103

10410

1

102

103

∆ z+

∆ y+

TorrojaFlat p05Round m05Round m1 ∆ y+ = 0.7 ∆ x+

Figure 6.39: Streamwise and Spanwise joint probability density functions of attached clusters

for all four cases Torroja, β = +0.5, β = −0.5 and β = −1.0 at Reτ = 950. The

magenta line indicates the channel center.

streamwise direction only the round cases seem to change their shape. The change

of the sizes for the respective cases, however, could be confirmed.

It can be concluded that by changing the mean profile and thus the mean shear,

tall turbulent structures in streamwise and spanwise direction change in size. The

effect is independent of the Reynolds number. Whether or not a change in shape

actually occurs has to be determined in future analysis.

112


6.9 Release of fixed mean profile

The release of one of the fixed mean velocity profile was analyzed and the results are

presented in the current section. 10 fields, of the β = −1.0 case were released after

they had reached a converged state solution (20 ETTs) with a fixed mean profile,

to investigate the adaption process after the release of the extra energy that was

stored in the fixed mean profile.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

−<uv

>+

y/h

Reynolds stress

Beta = −1.0 case. T=0after approx 0.2 ETTafter approx 0.3 ETTafter approx 0.4 ETTafter approx 0.5 ETTafter approx 1 ETTafter approx 1.8 ETTafter approx 2.75 ETT

Figure 6.40: Reaction of the Reynolds stress to the release of the streamwise mean velocity

profile after it was fixed to the unnatural β = −1.0 case for 20 ETT

Figure 6.40 shows the reaction of the Reynolds stress to the release of the mean pro-

file. Time is measured in ETTs. A fairly stable condition of an unfixed channel flow

is reached after about 1.8 ETTs. The interesting part, however, is the immediate

response of turbulence. Fluctuations near the wall as well as in the channel center

increase initially in all three components as depicted in figure 6.41. Even though

not shown here, the streamwise and spanwise zero modes of all three components

show the same behavior, thus excluding the possibility of a mere shift of energy

from the largest scales, which have been learned to increase their energy content for

the rounder profiles, into higher modes (smaller structures).

113


0 1 21.4

1.6

1.8

2

2.2

2.4u

+

ETT

y/h = 0.4y+ = 30

0 1 20.7

0.8

0.9

1

1.1

1.2

v+

ETT

y/h = 0.4y+ = 30

0 1 20.9

1

1.1

1.2

1.3

1.4

1.5

w+

ETT

y/h = 0.4y+ = 30

Figure 6.41: Reaction of the streamwise, wall normal and spanwise velocity fluctuations to the

release of the streamwise mean velocity profile

Figure 6.42 shows the energy Einp that is fed into the flow by the mean profile

Einp(t) = 2Reτu2τ (t)Ub(t) (6.34)

where t denotes the time after the release measured in ETTs, the total energy of

the flow E

E =

∫ h

−h

1

2

(

u2 + v2 + w2)

dy, (6.35)

as well as its derivative with respect to time dE/dt and the fraction (dE/dt)/Einp.

As shown in figure 6.42, the fraction of the time derivative of the total energy in the

flow and the Energy input is much smaller than 1, suggesting that the flow merely

takes the extra energy from the fixed mean profile and distributes it in the flow.

The initial fluctuations can therefore be considered part of this transient adaption

process.

114


0 0.5 1 1.5 2 2.5−2

0

2

4

6

8

10

12

ETT

Einp

= 2 * 550 * uτ2 * Ub

E=∫−hh 0.5*(u2+v2+w2)

dE/dt(dE/dt) / E

inp

Figure 6.42: Energy Einp that is fed into the flow by mean shear. Total energy of flow E. Time

derivative dE/dt and fraction (dE/dt)/Einp

115

7 Discussion and Conclusions

To investigate systematically the effect of a fixed mean profile, several direct numer-

ical simulations with mean profiles fixed to various unnatural and blended cases, as

well as to the natural case, were carried out and the results of their post processing

were presented.

One of the central problems of wall-bounded turbulence is how structures of different

sizes, associated with different wall distances, adjust their relative intensities to

balance the mean momentum transfer in wall-normal direction. As mentioned in

the introduction, it is understood that Reynolds stresses and the mean velocity

gradient (mean shear) interact with each other to produce turbulence, but the way

how they do it is still not completely understood [7].

If a feedback mechanism between the Reynolds stresses and the mean shear exists,

which would suggest that locally weak structures, with weak Reynolds stresses,

result in a local acceleration of the mean velocity profile, which leads to local

enhancement of the velocity gradient and thus to the strengthening of the local

fluctuations, it would be disturbed by fixing the mean velocity profile and thus the

mean velocity gradient.

Interestingly, this feedback mechanism seems not to exist, since when the correct

profile is imposed, all statistical and spectral results are essentially identical, within

the statistical uncertainty, to the case when the profile is left unfixed and can evolve

according to its own equation of motion.

If however, only a slightly incorrect profile is imposed on the flow as a fixed mean ve-

116

Diplomarbeit 7 Discussion and Conclusions

locity profile, the fluctuations react highly sensitive. The statistical results showed,

that the increase or decrease of the intensities, respectively, coincides with the

gradient of the mean velocity profile. An increased mean velocity gradient results in

higher intensities and therefore higher production of turbulent kinetic energy, that

is, as was reasoned, the only way in which the extra energy from the changed mean

shear can enter the fluctuations. Following the production, it was shown, that the

dissipation peaks and drops quite locally, where the mean shear is changed.

Those results suggests, that the mean velocity gradient seems to determine the

energy that locally flows into fluctuations of a given size, which grow until they

activate an energy transfer, that is strong enough to balance the production. The

local production likewise balances the local dissipation, which is proportional to the

cube of the fluctuation intensities. This model is supported by the Reynolds budget

as well as by the results of the two dimensional energy spectra, which show, that for

a given wall distance not only the large scale fluctuations, but all scales, all the way

down to quantities like the vorticity, are effected in the same way, by the change in

the mean shear.

Rather than a feedback mechanism, the results suggest, that the intensities are

determined in a uni-directional causal chain, in which the mean shear determines

them locally and not by an interaction among structures at different wall distances.

An increased mean velocity gradient results in increased production of fluctuations,

which likewise results in an increase in vorticity and therefore dissipation, suggesting

that the whole cascade process takes place quite locally.

Furthermore, by using linear stability analysis of transient growth, it was shown,

that the changed mean shear could directly be related to the amplification of

structures at that given wall distance. This finding also supported the suggestion

of a local rather than interactive determination of the energy that is fed into the

fluctuations.

117


Further insights into the interaction of mean shear and fluctuations were gained

by blending two profiles, using the natural profile in the near wall region and an

unnatural (increase mean shear) in the channel center. The increased shear in the

channel center as well as the blending itself created large structures, that extend all

the way to the wall and induce Reynolds stresses in other parts of the channel. The

small structures of the two near wall blends adapted fast to the respective “mother

profile”supporting the locality of the mechanism by which energy is fed into the flow.

Those findings were confirmed by dividing the energy spectrum in the zero-zero

mode (mean profile), the zero modes (large scales) and the non zero modes (smaller

scales). All fixed cases with an increased shear in the channel center (round and

blended) show the tendency to increase the energy in the zero modes kx and kz of

the wall parallel planes, for the entire channel hight, while the flat case with less

shear in the channel center decreases its energy content in the large scales. This

suggest, that the large structures, that are created by the increased shear in the

channel center extent all the way to the wall and also implies that the mean profile

preferably interacts with scales of similar sizes. However, since all scales at a given

wall distance show the same reaction to a changed mean profile (fluctuations as well

as vorticity), it can be assumed that the cascade process acts locally and faster than

the time scale of the large eddies.

A new local velocity scale was introduced, using the square root of the total stress

for a given wall-normal distance, thus taking into account the local situation of the

Reynolds stresses. The fluctuations for all unnatural and natural profiles collapse

to the nominal value of the reference profile. Close to the wall, where the viscous

stress dominates, the scaling does not work for all components of the intensities.

By comparing the results of both Reynolds number cases, the threshold was found

to be y+ ≈ 100. This once more strongly suggests that the mechanism with which

energy is fed into velocity fluctuation is local, rather than an interaction between

eddies at different wall distances and once more support the assumption that the

intensities are determined locally and neither through interaction of structures at

118


different wall distances, nor through any kind of feedback mechanism.

The analysis of coherent structures (clusters) showed that turbulence develops at

first sight seemingly natural. The instantaneous field plots of the discriminant of

the velocity gradient tensor look very similar. A closer look however shows, that

by changing the mean shear, the size as well as possibly the shape of tall attached

clusters are affected. Detached clusters seem not to be affected by the changed

mean shear. This suggests that the mean shear only has an affect on large scales,

but leave smaller scales unaffected and therefore implies that it does predominantly

interact with structures of similar sizes.

An increased shear in the channel center favors the formation of large attached

“inactive” energy containing eddies. An increased shear close to the wall, however,

favors the formation of smaller, more isotropic and thus “active” structures that can

directly take part in the isotropic cascade process.

Realeasing the mean profile after the channel reached a converged state in the

unnatural configuration, showed that the adaption process happens within about 2

to 3 ETTs. The extra energy, released after releasing the mean profile settles to the

nominal value after initial fluctuations.

119

Bibliography

[1] POPE, Stephen B. 2000 Turbulent Flows, Cambridge University Press

[2] KIM, J. & MOIN, P.& MOSER, R.D. 1986 Turbulence statistics in fully

developed channel flow at low Reynolds number, J. Fluid Mech. (1987), vol.

177, pp. 133-166

[3] BOYD, John P. 2000 Chebyshev and Fourier Spectral Methods, Second

Edition, Dover Publications

[4] JIMENEZ, Javier. 2004 Turbulence and Vortex Dynamics

[5] DEL ALAMO, J.C. & JIMEENEZ, J. & ZANDONADE, P. & MOSER, R.D.

2004 Scaling of the energy spectra of turbulent channels, J. Fluid Mech. vol

500, pp 135-144

[6] JIMENEZ, Javier. 2007 Recent developments on wall-bounded turbulence,

RACSAM Vol. 101(2), pp. 187-203

[7] GEORGE, William, Lectures in Turbulence for the 21st Century, Department

of Applied Mechanics Chalmers University of Technology Gothenburg, Sweden

[8] DEL ALAMO, Juan-Carlos & JIMEENEZ, Javier. 2006 Linear energy

amplification in turbulent channels, J. Fluid Mech. 559, 205-213.

[9] SCHMID, P. J. & HENNINGSON, D.S. 2001 Stability and Transition in Shear

Flows. Springer Volume 142

[10] JIMEENEZ, Javier. 2011 Cascades in wall-bounded turbulence, Annu. Rev.

Fluid Mech. 2012 44

120

Diplomarbeit Bibliography

[11] HOYAS, Sergio & JIMEENEZ, Javier. 2006 Scaling of the velocity fluctuations

in turbulent cahnnels up to Reτ = 2003. Phys. Fluids 18:011702

[12] JIMEENEZ, Javier & Genta Kawahara. 2010 Dynamics of wall-bounded

turbulence

[13] DEL ALAMO, J.C. & JIMEENEZ, Javier & Zandonade,& Moser R.D. 2006

Self-similar vortex clusters in the logarithmic region. J. Fluid Mech., 561:329 -

358

[14] CHONG, M. S., PERRY & A. E. & CANTWELL, B. J. 1990 A general

classification of three-dimensional flow fields. Phys. Fluids A 2, 765777.

[15] FLORES, Oscar & JIMEENEZ, Javier. 2006 Effect of wall-boundary

disturbances on turbulent channel flows. J. Fluid Mech., 566, 357-376

[16] LOZANO DURAN, Adrian. 2010 Evoluticion temporal de racimos de

torbellinos en un canal turublente. Projecto Fin de Carrera. UPM.

[17] HOYAS, Sergio & JIMEENEZ, Javier. 2008 Reynolds number effects on the

Reynolds-stress budgets in turbulent channels. Phys. of Fluids. 20:101511-1

[18] KOLMOGOROV, A. N. 1941 Dissipation of energy in isotropic turbulence.

Dokl. Akad. Nauk. SSR, 32:19 -21

[19] CESS, R.D. 1958 A survey of the literature on heat transfer in turbulent tube

flow. Report 8-0529-R24. Westinghouse Research.

121

Diplomarbeit - UPM · Diplomarbeit (Proyecto Fin de Carrera) An inverse RANS simulation of a...

Documents

Transcript of Diplomarbeit - UPM · Diplomarbeit (Proyecto Fin de Carrera) An inverse RANS simulation of a...