Turbulence transition in pipe flow [Elektronische Ressource] / vorgelegt von Holger Faisst

Sujets

• Physik

Informations

• Publié par: philipps-universitat_marburg
• Publié le: 01 janvier 2003
• Nombre de lectures: 15
• Langue: English
• Poids de l'ouvrage: 16 Mo

Extrait

Turbulence transition
in pipe flow
DISSERTATION
zur Erlangung des Doktorgrades
der Naturwissenschaften
(Dr. rer. nat.)
dem Fachbereich Physik
der Philipps-Universitat¨ Marburg
vorgelegt
von
Holger Faisst
aus Boblingen¨
Marburg/Lahn 2003Vom Fachbereich Physik der
Philipps-Universitat¨ Marburg als
Dissertation angenommen am: 08.07.2003
Erstgutachter: Prof. Dr. Bruno Eckhardt
Zweitgutachter: Prof. Dr. Peter Lenz
Tag der mundlichen¨ Prufung:¨ 22.07.2003Turbulence transition
in pipe flow
or, in Lord Reynolds’ words
A Numerical Investigation of the Circumstances
which determine whether the Motion of Water
shall be Direct or SinuousContents
¨1 Uberblick 1
1.1 Einleitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Direkte numerische Simulation . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Exakte koharente¨ Strukturen in der Rohrstromung¨ . . . . . . . . . . . . . . 6
1.4 Simulation des Turbulenzuber¨ gangs in der Rohrstromung¨ . . . . . . . . . . 7
2 Introduction 9
3 Laboratory transition experiments in pipe flow 11
4 A new spectral code for pipe flow 17
4.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Regularity constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Fourier-Legendre collocation . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Lagrange method of the first kind . . . . . . . . . . . . . . . . . . . . . . 23
4.6 Lagrange method of the second kind . . . . . . . . . . . . . . . . . . . . . 24
4.7 Search and continuation of travelling waves . . . . . . . . . . . . . . . . . 25
5 Physics of pipe flow and verification of the new pipe code 27
5.1 Linearized equations of motion . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Two-dimensional nonlinear equations of motion . . . . . . . . . . . . . . 30II CONTENTS
5.2.1 Constant-flux pipe flow . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Three-dimensional nonlinear equations of motion . . . . . . . . . . . . . . 34
5.3.1 Turbulent pipe flow at : a comparison with laboratory
and DNS literature data . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.2 Optimal resolution for transitional Reynolds numbers . . . . . . . 39
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 Exact coherent states in pipe flow 41
6.1 Earlier attempts to find coherent states in pipe flow . . . . . . . . . . . . . 42
6.2 Embedding method with transversal volume force . . . . . . . . . . . . . . 43
6.3 -vortex travelling waves . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.4 Search for -symmetric two-vortex travelling waves . . . . . . . . . . . . 57
6.5 Search for spiral -vortex travelling waves . . . . . . . . . . . . . . . . . 59
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7 Transition to turbulence in constant-flux pipe flow 63
7.1 Numerical lifetime experiments . . . . . . . . . . . . . . . . . . . . . . . 63
7.1.1 Sensitivity on initial conditions and on parameters . . . . . . . . . 67
7.1.2 Exponential distribution of lifetimes . . . . . . . . . . . . . . . . 72
7.2 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A Eigenvalue analysis 79
A.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.2 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
B Numerical convergence of travelling waves 85
C Scaling at a complex boundary: the Mandelbrot set 91

List of figures
3.1 The hall of fame: Gotthilf Hagen, Jean Poiseuille, Osborne Reynolds . . . . 11
3.2 Sketch of a section of a straight pipe of uniform circular cross-section with
the laminar Hagen-Poiseuille flow profile . . . . . . . . . . . . . . . . . . 12
3.3 The intermittent character of transition in pipe flow as sketched by Reynolds 13
3.4 Sketch of the experimental setup for transition experiments by Darbyshire &
Mullin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Laboratory transition experiments by Darbyshire & Mullin . . . . . . . . . 15
3.6 Time evolution of the streamwise centerline velocity at a fixed observation
point as a turbulent puff or slug passes by. . . . . . . . . . . . . . . . . . . 15
3.7 The parameter regions in which puffs and slugs occur; initial disturbance
level vs. Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.8 The propagation speed of puffs and slugs as a function of Reynolds number 16
5.1 ’Swear hand’ distribution of eigenvalues . . . . . . . . . . . . . . . . . . . 29
5.2 Non-normal linear and nonlinear evolution of a nearly optimal two-
dimensional disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3 Zikanov’s original data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.4 Constant-pressure-drop vs. constant-flux pipe flow . . . . . . . . . . . . . 34
5.5 Two-dimensional energy spectrum at . . . . . . . . . . . . . . 35
5.6 Experimental and numerical turbulent mean velocity profiles . . . . . . . . 37
5.7 Radial profiles of root-mean-square velocity fluctuations in wall units . . . 37