The Taylor-Couette system is one of the most studied examples of fluid flow exhibiting the spontaneous formation of dynamical structures. In this book, the variety of time independent solutions with periodic spatial structure is numerically investigated by solution of the Navier-Stokes equations.

Inhalt

1 The Taylor Experiment.- 1.1 Modeling of the Experiment.- 1.1.1 Introduction.- 1.1.2 Mathematical Description of the Experiment.- 1.1.3 Narrow Gap Limit and Rayleigh-Bénard Problem.- 1.1.4 End Effects.- 1.2 Flows between Rotating Cylinders.- 1.3 Stability of Couette Flow.- 1.3.1 Equations of Motion for Axisymmetric Perturbations.- 1.3.2 Computation of Marginal Curves.- 1.3.3 Validity of the Principle of Exchange of Stability.- 2 Details of a Numerical Method.- 2.1 Introduction.- 2.1.1 Numerical Model.- 2.1.2 Numerical Methods.- 2.1.3 Validity of the Model.- 2.1.4 Stability.- 2.2 The Discretized System.- 2.2.1 Discretization in the Axial z-Direction.- 2.2.2 Discretization in the Radial r-Direction.- 2.2.3 Boundary Conditions.- 2.2.4 Final Version of the Equations.- 2.3 Computation of Solutions.- 2.3.1 Pseudo-Arclength Continuation and Newton iterations.- 2.3.2 Continuation in the Reynolds number Re.- 2.3.3 Continuation in the Wave Number k.- 2.3.4 Simple Continuation.- 2.3.5 Switching Branches.- 2.4 Computation of flow Parameters.- 2.4.1 Periodicity.- 2.4.2 Computation of um: = u(rm, zm; ?) at rm: = 1 + ?/2, zm = 0.- 2.4.3 Computation of the Torque.- 2.4.4 Computation of Kinetic Energies.- 2.4.5 Computation of the Stream function.- 2.5 Numerical Accuracy.- 2.5.1 Finite Differences.- 2.5.2 Truncation of the Fourier Series.- 2.5.3 Conclusions.- 3 Stationary Taylor Vortex Flows.- 3.1 Introduction.- 3.2 Computations with Fixed Period ? ? 2.- 3.2.1 A Narrow Gap Problem, ? = 0.95.- 3.2.2 A Wide Gap Problem, ? = 0.5.- 3.3 Variation of Flows with Period ?.- 3.3.1 Previous Results on Flows with Wavelengths ? ? 2.- 3.3.2 Continuous Change of Period.- 3.3.3 Flows near Re = 1.5 Recr.- 3.3.4 Flows for Re = 800 ? 3.65 Recr.- 3.4 Interactions of Secondary Branches.- 3.4.1 A Neighborhood of (Re24, ?24) and the Basic (2,4) Fold.- 3.4.2 Connections to the Rayleigh-Bénard Problem.- 3.4.3 The basic (n, 2n)-Fold for Higher Reynolds Numbers.- 3.4.4 The Basic 2-vortex Surface.- 3.5 Re = 2 Recr and the (n, pn) Double Points.- 3.6 Stability of the Stationary Vortices.- 3.6.1 Wavy Vortices.- 3.6.2 Eckhaus and Short-Wavelength Instabilities.- 4 Secondary Bifurcations on Convection Rolls.- 4.1 Introduction.- 4.2 The Rayleigh-Bénard Problem.- 4.2.1 Convection in Fluids.- 4.2.2 Boussinesq Approximation.- 4.2.3 The Rayleigh-Bénard Problem as Limiting Case of the Taylor Problem.- 4.3 Stationary Convection Rolls.- 4.3.1 The Basic Equations.- 4.3.2 Critical Curves of the Primary Solution.- 4.3.3 Pure-Mode Solutions.- 4.4 The (2,4) Interaction in a Model Problem.- 4.4.1 The Model Problem.- 4.4.2 Calculation of Secondary Bifurcation Points on the 2-Roll Solutions.- 4.4.3 Calculation of Secondary Bifurcation Points on the 4-Roll Solutions.- 4.4.4 The Perturbation Approach.- 4.5 The (2,6) Interaction in a Model Problem.- 4.5.1 Calculation of Secondary Bifurcation Points on the 2-Roll Solutions.- 4.5.2 Calculation of Secondary Bifurcation Points on the 6-Roll Solutions.- 4.5.3 Nonlinear Interactions between the Bifurcating Branches.- 4.6 Generalisations and Consequences.- 4.6.1 Other Interactions.- 4.6.2 Linear Superpositions of Pure-Mode Solutions.- 4.6.3 Secondary Bifurcations in the Taylor Problem Revisited.