Pattern formation in spatially forced thermal convection [Elektronische Ressource] / vorgelegt von Stephan Weiß

Sujets

• Physik

Informations

• Publié par: georg-august-universitat_gottingen
• Publié le: 01 janvier 2009
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

Pattern Formation in Spatially
Forced Thermal Convection
Dissertation
zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades
,,Doctor rerum naturalium“
an der Georg-August-Universit¨at G¨ottingen
vorgelegt von
Stephan Weiß
aus Suhl
G¨ottingen 2009Referent: Prof. Dr. E. Bodenschatz
Koreferent: Prof. Dr. A. Tilgner
Tag der mu¨ndlichen Pru¨fung: 14. Oktober 2009Contents
1 Introduction 15
1.1 Turing’s Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Basic Pattern Formation . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Convection Driven by a Thermal Gradient . . . . . . . . . . . . . 21
1.4 Pattern Formation in Anisotropic Systems . . . . . . . . . . . . . 25
1.5 Temporal and Spatial Periodic Forcing of Pattern Forming Systems 27
2 Theoretical Overview 31
2.1 Boussinesq Equations . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Surface Corrugation. . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Inclined Layer Convection (ILC) . . . . . . . . . . . . . . . . . . . 36
2.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 The Experiment 45
3.1 Overview of the Gas Convection Apparatus . . . . . . . . . . . . 45
3.2 Surface Corrugation. . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Shadowgraphy . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.2 Image Processing . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.3 Some Remarks about Fourier Analysis . . . . . . . . . . . 65
3.3.4 Phase Demodulation . . . . . . . . . . . . . . . . . . . . . 69
4 Inclined and Noninclined Resonant Forcing 73
4.1 Forced Rolls and Amplitude Equation . . . . . . . . . . . . . . . . 73
4.2 Longitudinal Forcing . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Varicose Pattern . . . . . . . . . . . . . . . . . . . . . . . 80
34 CONTENTS
4.2.2 Harmonic Undulation and Crawling Rolls. . . . . . . . . . 84
4.2.3 Patterns Driven by Shear Instability . . . . . . . . . . . . 88
4.2.4 Patterns at Low Inclination Angles . . . . . . . . . . . . . 95
4.3 Transverse Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3.1 Rhombic Pattern . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.2 Hexarolls and Crawling Rolls . . . . . . . . . . . . . . . . 106
4.3.3 Bimodal Pattern . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.4 Scepter Pattern . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.5 Heart Pattern . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4 Forcing Oblique Rolls: ϕ =60° . . . . . . . . . . . . . . . . . . . 120
5 Forcing with Various Wave Numbers 129
5.1 Varicose Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2 Coherent Kinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Cross Rolls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4 Brickwall Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.5 The Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6 Optical Forcing - An Outlook 149
6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.2 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3.1 Spatial Periodic Forcing and the Amplitude Equation . . . 154
6.3.2 Forcing Close to Ra . . . . . . . . . . . . . . . . . . . . . 156c
6.4 More Power - The Use of an IR Laser. . . . . . . . . . . . . . . . 159
7 Summary 163
A Boussinesq Equations 167
A.1 Unforced ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.2 Forced ILC: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B Filling and emptying procedure for CS 1712
C Suplementary Material on CD-ROM 173CONTENTS 5
C.1 Movies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
C.2 RBC - Program to calculate fluid properties . . . . . . . . . . . . 1746 CONTENTSList of Figures
1.1 Turing pattern on the pelt of a jaguar . . . . . . . . . . . . . . . . 16
1.2 Neutral curve and neutral surface for the Swift-Hohenberg equation 20
1.3 Pitchfork bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Schematic of a convection cell with periodic fluid motion . . . . . 22
1.5 Neutral curve for Rayleigh B´enard convection . . . . . . . . . . . 24
1.6 Stability regions for ISR at different Prandtl numbers . . . . . . . 24
1.7 Inclined layer convection (ILC) . . . . . . . . . . . . . . . . . . . 26
1.8 Pitchfork bifurcation and imperfect bifurcation . . . . . . . . . . . 29
2.1 Side and top view of an inclined convection cell (schematic). . . . 37
2.2 Onset of straight rolls as a function of γ and ϕ . . . . . . . . . . . 40
2.3 Critical wave number as a function of γ and ϕ . . . . . . . . . . . 41
3.1 Sketch of the convection apparatus . . . . . . . . . . . . . . . . . 46
3.2 Shadowgraph image of the forced and the unforced cell . . . . . . 47
3.3 Overview of the experimental apparatus . . . . . . . . . . . . . . 50
3.4 Schematic of the bottom plate surface . . . . . . . . . . . . . . . . 51
3.5 Schematic drawing of the shadowgraph arrangement . . . . . . . . 57
3.6 Light rays and light intensity due to a sinusoidal refraction grating 59
3.7 Shadowgraph intensity in dependence on the axial distance . . . . 60
3.8 Shadowgraph intensity due to bottom plate texture . . . . . . . . 62
3.9 Standard deviation of the intensity in dependence on the shadow-
graph setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.10 Standard deviation of the shadowgraph signal for forced and un-
forced convection in dependence of the control parameter ε . . . . 65
3.11 Shadowgraphimagesoftheforcedandthereferencecellatdifferentε 66
78 LIST OF FIGURES
3.12 Consideration of the amplitude of specific modes . . . . . . . . . . 68
3.13 Demodulation of stripe pattern . . . . . . . . . . . . . . . . . . . 70
3.14 Demodulation of Varicose Pattern . . . . . . . . . . . . . . . . . . 71
4.1 Schematic of the convection cell for forced ILC . . . . . . . . . . . 74
4.2 Convection amplitudes for forced and unforced convection as a
function of the control parameter ε . . . . . . . . . . . . . . . . . 77
4.3 Phase diagram for longitudinal forcing . . . . . . . . . . . . . . . 79
4.4 Straight rolls in unforced and forced convection experiments . . . 80
4.5 Varicose pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6 Calculated varicose pattern . . . . . . . . . . . . . . . . . . . . . 82
4.7 Modulation of the phase field of varicose pattern . . . . . . . . . . 82
4.8 Spatio-temporalchaoticstateforhorizontalconvectionanditscor-
responding Fourier transform . . . . . . . . . . . . . . . . . . . . 83
4.9 Harmonic undulations . . . . . . . . . . . . . . . . . . . . . . . . 84
4.10 Undulation pattern and its phase field . . . . . . . . . . . . . . . 86
4.11 Undulation chaos and crawling rolls . . . . . . . . . . . . . . . . . 89
4.12 Time evolution of transverse bursts (TB) . . . . . . . . . . . . . . 90
4.13 Fourier transform transverse bursts . . . . . . . . . . . . . . . . . 91
4.14 Amplitude of transverse bursts as a function of time . . . . . . . . 92
4.15 Transverse rolls and their Fourier transform . . . . . . . . . . . . 93
4.16 Fourier-filtered transverse rolls after removing the forced mode ~q 94f
4.17 Time evolution of a longitudinal burst . . . . . . . . . . . . . . . 94
4.18 Single kink line, subharmonic resonances and kink cluster. . . . . 96
4.19 Varicose pattern at higher inclination angles . . . . . . . . . . . . 97
4.20 Crawling kinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.21 Localized crawling kinks . . . . . . . . . . . . . . . . . . . . . . . 99
4.22 Phase diagram (transverse forcing) . . . . . . . . . . . . . . . . . 101
4.23 Longitudinal rolls at highε . . . . . . . . . . . . . . . . . . . . . 102
4.24 Evolution of rhombic pattern with time at very small inclination
angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.25 Fourier transform of varicose instability during evolution of rhom-
bic pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.26 Rhombic pattern and its Fourier transform . . . . . . . . . . . . . 104LIST OF FIGURES 9
4.27 Evolution of rhombic pattern with increasing ε . . . . . . . . . . . 105
4.28 Rhombic pattern atγ = 5° and ε = 3.7 . . . . . . . . . . . . . . . 106
4.29 Hexarolls and its corresponding Fourier transform . . . . . . . . . 107
4.30 Calculated hexaroll pattern . . . . . . . . . . . . . . . . . . . . . 108
4.31 Hexarolls and the transition to crawling rolls . . . . . . . . . . . . 109
4.32 Bimodal pattern and its corresponding Fourier transform . . . . . 110
4.33 Mode evolution of bimodal pattern . . . . . . . . . . . . . . . . . 112
4.34 Bimodal pattern, calculated with Eq. 4.10 . . . . . . . . . . . . . 112
4.35 Scepter pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.36 Scepter pattern as a convolution . . . . . . . . . . . . . . . . . . . 114
4.37 Amplitude development of scepter pattern for several modes . . . 116
4.38 Fourier decomposition of scepter pattern . . .