Numerical Methods for Steady State/Hopf Mode Interactions

1997 ◽  
Vol 07 (03) ◽  
pp. 585-605 ◽  
Author(s):  
F. Amdjadi ◽  
P. J. Aston
Keyword(s):  
Steady State ◽  
Numerical Methods ◽  
Symmetry Breaking ◽  
Mode Interaction ◽  
Extended System ◽  
Test Function ◽  
Alternative Approach ◽  
Mode Interactions ◽  
Sivashinsky Equation ◽  
Two Parameter

Numerical methods for dealing with steady state/Hopf mode interactions using extended systems are considered. In particular, it is shown that such a mode interaction corresponds to a symmetry breaking bifurcation of a Hopf extended system as well as a Hopf bifurcation of a symmetry breaking extended system. Non-degeneracy conditions associated with these bifurcations are derived and interpreted in the context of the mode interaction. The alternative approach of using a single test function instead of a full extended system is considered in detail in one of the cases. Numerical results for a two-parameter version of the Kuramoto–Sivashinsky equation are presented to illustrate the theory.

GROUP THEORETICAL MODE INTERACTIONS WITH DIFFERENT SYMMETRIES

1994 ◽  
Vol 04 (01) ◽  
pp. 177-191 ◽  
Author(s):  
KARIN GATERMANN ◽  
BODO WERNER
Keyword(s):  
Group Theory ◽  
Symmetry Breaking ◽  
Irreducible Representations ◽  
Bifurcation Points ◽  
Mode Interactions ◽  
Different Types ◽  
Secondary Bifurcation ◽  
Two Parameter ◽  
Symmetry Breaking Bifurcation

In two-parameter systems two symmetry breaking bifurcation points of different types coalesce generically within one point. This causes secondary bifurcation points to exist. The aim of this paper is to understand this phenomenon with group theory and the inner-connectivity of irreducible representations of supergroup and subgroups. Colored pictures of examples are included.

THE 1:2 MODE INTERACTION IN RAYLEIGH–BÉNARD CONVECTION WITH WEAKLY BROKEN MIDPLANE SYMMETRY

2001 ◽  
Vol 11 (01) ◽  
pp. 27-41 ◽  
Author(s):  
ISABEL MERCADER ◽  
JOANA PRAT ◽  
EDGAR KNOBLOCH
Keyword(s):  
Steady State ◽  
Symmetry Breaking ◽  
Mode Interaction ◽  
Reflection Symmetry ◽  
Steady State Mode ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The effects of weak breaking of the midplane reflection symmetry on the 1:2 steady state mode interaction in Rayleigh–Bénard convection are discussed in a PDE setting. Effects of this type arise from the inclusion of non-Boussinesq terms or due to small differences in the boundary conditions at the top and bottom of the convecting layer. The latter provides the simplest realization, and captures all qualitative effects of such symmetry breaking. The analysis is performed for two Prandtl numbers, σ=10 and σ=0.1, representing behavior typical of large and low Prandtl numbers, respectively.

MODE INTERACTIONS WITH SPHERICAL SYMMETRY

1994 ◽  
Vol 04 (04) ◽  
pp. 885-904 ◽  
Author(s):  
S.B.S.D. CASTRO
Keyword(s):  
Steady State ◽  
Spherical Harmonics ◽  
Spherical Symmetry ◽  
Mode Interaction ◽  
Natural Representation ◽  
Mode Interactions ◽  
Study Mode ◽  
Bifurcation Problems ◽  
Steady State Bifurcation ◽  
Secondary Bifurcations

We study mode interaction steady-state bifurcation problems with spherical symmetry. Using the representation of O(3) in terms of spherical harmonics, we study the interactions of the modes of dimension 1, 3 and 5. When studying mode interactions involving the 3- and the 5-dimensional modes, we come across a very natural representation, which turns out to be that of SO(3) instead of O(3). Given that the study of problems with this latter symmetry has already been done, we then study problems with SO(3) symmetry. For all these problems, we stress the existence of secondary bifurcations giving rise to the existence of limit cycles and the occurrence of heteroclinic connections between equilibria.

scholarly journals A frequency quantum interpretation of the surface renewal model of mass transfer

2017 ◽  
Vol 4 (7) ◽  
pp. 170103 ◽  
Author(s):  
Chanchal Mondal ◽  
Siddharth G. Chatterjee
Keyword(s):  
Mass Transfer ◽  
Steady State ◽  
Frequency Distribution ◽  
Age Distribution ◽  
Gas Absorption ◽  
Bulk Liquid ◽  
Gas Transfer ◽  
Transfer Resistance ◽  
Surface Renewal ◽  
Two Parameter

The surface of a turbulent liquid is visualized as consisting of a large number of chaotic eddies or liquid elements. Assuming that surface elements of a particular age have renewal frequencies that are integral multiples of a fundamental frequency quantum, and further assuming that the renewal frequency distribution is of the Boltzmann type, performing a population balance for these elements leads to the Danckwerts surface age distribution. The basic quantum is what has been traditionally called the rate of surface renewal. The Higbie surface age distribution follows if the renewal frequency distribution of such elements is assumed to be continuous. Four age distributions, which reflect different start-up conditions of the absorption process, are then used to analyse transient physical gas absorption into a large volume of liquid, assuming negligible gas-side mass-transfer resistance. The first two are different versions of the Danckwerts model, the third one is based on the uniform and Higbie distributions, while the fourth one is a mixed distribution. For the four cases, theoretical expressions are derived for the rates of gas absorption and dissolved-gas transfer to the bulk liquid. Under transient conditions, these two rates are not equal and have an inverse relationship. However, with the progress of absorption towards steady state, they approach one another. Assuming steady-state conditions, the conventional one-parameter Danckwerts age distribution is generalized to a two-parameter age distribution. Like the two-parameter logarithmic normal distribution, this distribution can also capture the bell-shaped nature of the distribution of the ages of surface elements observed experimentally in air–sea gas and heat exchange. Estimates of the liquid-side mass-transfer coefficient made using these two distributions for the absorption of hydrogen and oxygen in water are very close to one another and are comparable to experimental values reported in the literature.

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