STUDIES IN ROTATING CONVECTION by Antonio M. Rubio A Dissertation Presented in Partial ...

141 Pages·2009·5.87 MB·English

STUDIES IN ROTATING CONVECTION
by
Antonio M. Rubio
A Dissertation Presented in Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
ARIZONA STATE UNIVERSITY
August 2009
STUDIES IN ROTATING CONVECTION
by
Antonio M. Rubio
has been approved
July 2009
Graduate Supervisory Committee:
J.M. Lopez, Chair
F. Marques
H.J.S. Fernando
W. Tang
M. Herrmann
ACCEPTED BY THE GRADUATE COLLEGE
ABSTRACT
This thesis presents an exploration and analysis of novel phenomena in Rayleigh-
B´enard convection in a rotating cylinder using direct numerical simulation and equiv-
ariant bifurcation analysis. The numerical method used is a second order predictor-
corrector method using Chebyshev collocation in the radial and axial directions and
a Fourier-Galerkin discretization in the azimuthal direction. This numerical method
is first applied to the problem of convection with a rotation rate sinuosoidally modu-
lated about a non-zero mean in a parameter regime for which the onset of convection
is to domain chaos. As laboratory experiments found the resulting flow to be axisym-
metric with radially inward traveling waves an analysis of the axisymmetric subspace
of the problem was conducted which showed that the emergence of the traveling
waves resulted from a symmetry-restoring saddle-node-on-an-invariant-circle (SNIC)
bifurcation. The exploration of the effect of modulated rotation on rotating convec-
tion was extended to the case of rapid rotation for which the onset of convection is
wall-localized. It was found that the onset of convection could be delayed to thermal
driving up to 20% beyond that required in the unmodulated case using modulation
amplitudes of only 1% of the background rotation rate. Additionally, the resulting
oscillatory boundary layers introduced a mean flow which can reverse the precession
of the thermal plumes over a large range of imposed frequencies and amplitudes.
Finally, an in-depth study of the origin of spatio-temporal chaos directly at onset is
presented. Inthisstudythecaseneglectingthecentrifugalforce(asiscommonlydone
in theoretical studies) is compared to the more realistic case incorporating centrifugal
force (which is necessarily evident in even the most carefully conducted laboratory
experiment). The two cases lead to two different routes to complexity; a detailed
generalized linear stability analysis is performed for each case. Novel patterns are ob-
served including ratcheting states with and without roll switching and comparisons
to existing laboratory experiments are drawn.
iii
ACKNOWLEDGEMENTS
I would like to acknowledge the efforts of the many professors who contributed
to my graduate education and especially Juan M. Lopez for helping me develop
scientifically and for giving me professional advice. I would also like to thank my
close collaborator and mentor Francisco Marques from the Departament de F´ısica
Aplicada at the Universitat Polit´ecnica de Catalunya for hosting me during my time
in Catalunya and his enthusiasm for fluids research has been a welcome inspiration
to me. I would like to thank our sister group in Barcelona for their warm hospitality
and enlightening discussions, especially Isabel Mercader and Oriol Batiste whose
numerical code made possible much of my research.
Furthermore, I would like to thank the More Graduate Education at Mountain
StatesAlliance(MGE@MSA)programandtheSloanFoundationformaterialsupport
including funds for computer equipment and travel grants that greatly enhanced my
studies. I would like to thank the Fulbright Program for the opportunity to study in
Catalunya and for the realization that I am a citizen of the world as well as of the
United States.
Finally, I would like to acknowledge the support in terms of computing time and
specialist help of the Ira A. Fulton High Performance Computing Initiative and the
NSF Teragrid which made possible the laborious computations detailed in this thesis.
iv
I dedicate this work to my mother and father, who have always supported me in
everything I’ve done.
v
TABLE OF CONTENTS
Page
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Early work on Rayleigh-B´enard convection . . . . . . . . . . . . . . . 2
The Taylor-Proudman theorem . . . . . . . . . . . . . . . . . . . . . 3
Recent developments in Rayleigh-B´enard convection with and without
rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Parameter regimes in rotating convection . . . . . . . . . . . . . . . . 6
Thesis plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Governing equations, symmetries and numerical method . . . . . . . . . . 10
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Nonlinear solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Generalized linear stability analysis . . . . . . . . . . . . . . . . . . . 15
3 Modulated rotating convection: Radially traveling concentric rolls . . . . . 17
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Governing equations, symmetries, and numerical technique . . . . . . 20
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Numerical technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Unmodulated RRBC: Ku¨pper–Lortz state . . . . . . . . . . . . . . . 26
Modulated basic state . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Onset of thermal convection: Axisymmetric pulsed target patterns . . 31
Traveling target pattern: The SNIC bifurcation . . . . . . . . . . . . 36
vi
Chapter Page
3.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Interacting oscillatory boundary layers and wall modes in modulated rotat-
ing convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Governing equations and numerical technique . . . . . . . . . . . . . 54
4.3 Background on unmodulated wall modes . . . . . . . . . . . . . . . . 59
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Wall mode quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Synchronous state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Modulated wall modes . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 Onset of Ku¨ppers-Lortz-like dynamics in finite rotating thermal convection 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Governing equations and numerical technique . . . . . . . . . . . . . 89
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Fr=0 linear stability analysis . . . . . . . . . . . . . . . . . . . . . . 93
Fr =0 secondary instabilities . . . . . . . . . . . . . . . . . . . . . . . 97
Fr =0.00882 linear stability analysis . . . . . . . . . . . . . . . . . . . 101
Fr =0.00882 secondary instabilities . . . . . . . . . . . . . . . . . . . 105
5.4 Experimental evidence of the centrifugally-driven large scale circulation108
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.1 Future outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
vii
LIST OF FIGURES
Figure Page
1.1 B´enard’s apparatus and a snapshot of hexagonal convective cells . . . . . 2
1.2 Regime diagram for rotating Rayleigh-B´enard convection . . . . . . . . . 6
1.3 Snapshots of flows at a variety of points in parameter space for rotating
Rayleigh-B´enard convection . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Example of time series of kinetic energies of different modal energies for a
linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Snap-shot of isotherms at mid-height of a Ku¨ppers–Lortz state . . . . . . 26
3.2 Snap-shots in a meridional plane of the stream function of basic states of
modulated rotating convection . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Contours of the angular momentum rv +Ω r2, radial velocity u, stream
0
function Ψ, and the temperature deviation Θ over one period for a basic
state for modulated rotating convection . . . . . . . . . . . . . . . . . . . 29
3.4 Contours of the radial velocity u, stream function Ψ and temperature de-
viation Θ in a meridional plane for developed modulated rotating convection 31
3.5 Snap-shots in a meridional plane of Θ for pulsed target states with A =
0.18 and Ω = Ω = 23.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 32
0 m
3.6 Snap-shots in a meridional plane of Θ for pulsed target states at A = 0.05
and Ω = Ω = 23.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
0 m
3.7 Snap-shots of isotherms at mid-height for axisymmetric pulsed target pat-
terns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.8 Snap-shots in a meridional plane of stream function for pulsed target states 34
3.9 Bifurcation diagrams at Ω = Ω = 23.6 for A = 0.05 and A = 0.18 and
0 m
corresponding heat transfer graphs . . . . . . . . . . . . . . . . . . . . . 35
3.10 Time series of Θ at Ω = Ω = 23.6, A = 0.05 and Ra = 2700 and 2705 36
m 0 m
viii
Figure Page
3.11 Reconstructed phase portraits of the basic state, pulsed target patterns
and one traveling target pattern along with time series of the Poincar´e
map and heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.12 Space-time plots of traveling target states . . . . . . . . . . . . . . . . . 39
3.13 Contours of temperature deviation in a meridional plane over one period
for a traveling wave state at Ra = 2844, Ω = Ω = 23.6 and A = 0.18 . 40
0 m
3.14 Contours of temperature deviation in a meridional plane over one period
for a traveling wave state at Ra = 2705, Ω = Ω = 23.6 and A = 0.05 . 41
0 m
3.15 Snapshots of u, Ψ and Θ over one period at Ra = 2705, Ω = Ω = 23.6
0 m
and A = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.16 Variation of traveling wave period with Ra and A. . . . . . . . . . . . . . 43
3.17 Schematic of the SNIC bifurcation on an invariant circle in a generic sys-
tem and a Z -symmetric system along with a schematic of a SNIC bifur-
2
cation on an invariant two-torus . . . . . . . . . . . . . . . . . . . . . . . 44
3.18 Regime diagram showing the pitchfork and SNIC bifurcation curves . . . 46
4.1 Convergence of spectral method for modulated wall modes . . . . . . . . 57
4.2 Isosurfaces of Θ for a wall mode at Ra = 5 104, Ω = 625 and A = 0 . 60
0
×
4.3 Space-time diagrams for modulated wall modes . . . . . . . . . . . . . . 61
4.4 Variation of Θ with A at Ω = 101.75 . . . . . . . . . . . . . . . . . . . 63
0 m
4.5 Variation with Ω of Θ and Nu 1 . . . . . . . . . . . . . . . . . . . . 63
m 0
−
4.6 Contours in a meridional plane of streamlines, vortex lines, azimuthal
vorticity and temperature perturbation at A = 0.05, Ω = 101.75, Ra =
m
4 104 and Ω = 625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
0
×
4.7 Profiles of azimuthal vorticity and relative axial vorticity for several syn-
chronous states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
ix
Figure Page
4.8 Estimatedtop,bottomandsidewallboundarylayerdepthsforsynchronous
states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.9 Instantaneous and time-averaged velocity and thermal perturbation mea-
surements for synchronous states . . . . . . . . . . . . . . . . . . . . . . 72
4.10 Contours of time-averaged stream function, temperature perturbation and
azimuthal velocity for a synchronous state at A = 0.03, Ra = 4 104,
×
Ω = 625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
0
4.11 Snapshots and 3D render of the temperature pertubation for a modulated
wall mode at A = 0.0075, Ω = 101.75, Ra = 5 104 and Ω = 625 . . . 77
m 0
×
4.12 Space-time diagrams of modulated wall modes at various points in param-
eter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.13 Angular position of the center of a plume of a modulated wall mode . . . 80
4.14 Variation of modulated wall mode precession rates with changing amplitude 81
5.1 Average Nusselt number variation with Ra (and ǫ) for Fr = 0 and Fr =
8.82 10−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
×
5.2 Marginal stability curves for the eigenvectors corresponding to Fourier
modes 0 to 10 of the linear stability analysis of rotating convection with
σ = 4.5, γ = 11.8 and Fr = 0 . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Critical thermal driving required for linear stability for eigenvectors cor-
responding to Fourier modes 0 to 34 at Ω = 19.7 for σ = 4.5, γ = 11.8
and Fr = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Snapshots of the first four eigenmodes to bifurcate from the conduction
state as Ra is increased from 2371.2 to 2371.7, for Ω = 19.7, σ = 4.5,
γ = 11.8 and Fr = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.5 Snap-shots of the temperature perturbation Θ for Ω = 19.7, σ = 4.5,γ =
11.8, Fr = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
x