Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation - Volume 15 Issue 1 - Henk A. Dijkstra, Fred W. Wubs, Andrew K. Cliffe, Eusebius Doedel, Ioana F. Dragomirescu, Bruno Eckhardt, Alexander Yu. Gelfgat, Andrew L. Hazel, Valerio Lucarini, Andy G. Salinger, Erik T. Phipps, Juan Sanchez-Umbria, Henk Schuttelaars, Laurette S. Tuckerman, Uwe Thiele

The axisymmetry-breaking three-dimensional instability of the axisymmetric flow between a rotating lid and a stationary cylinder is analysed. The flow is governed by two parameters – the Reynolds number Re and the aspect ratio γ (=height/radius). Published experimental results indicate that in different ranges of γ axisymmetric or non-axisymmetric instabilities can be observed. Previous analyses considered only axisymmetric instability. The present analysis is devoted to the linear stability of the basic axisymmetric flow with respect to the non-axisymmetric perturbations. After the linearization the stability problem separates into a family of quasi-axisymmetric subproblems for discrete values of the azimuthal wavenumber k. The computations are done using the global Galerkin method. The stability analysis is carried out at various densely distributed values of γ in the range 1 < γ < 3.5. It is shown that the axisymmetric perturbations are dominant in the range 1.63 < γ < 2.76. Outside this range, for γ < 1.63 and for γ > 2.76, the instability is three-dimensional and sets in with k = 2 and k = 3 or 4, respectively. The azimuthal periodicity, patterns, characteristic frequencies and phase velocities of the dominant perturbations are discussed.

A parametric study of multiple steady states, their stability, onset of oscillatory instability, and some supercritical unsteady regimes of convective flow of a Boussinesq fluid in laterally heated rectangular cavities is presented. Cavities with four no-slip boundaries, isothermal vertical and perfectly insulated horizontal boundaries are considered. Four distinct branches of steady-state flows are found for this configuration. A complete study of stability of each branch is performed for the aspect ratio A (length/height) of the cavity varying continuously from 1 to 11 and for two fixed values of the Prandtl number: Pr = 0 and Pr = 0.015. The results are represented as stability diagrams showing the critical parameters (critical Grashof number and the frequency at the onset of the oscillatory instability) corresponding to transitions from steady to oscillatory states, appearance of multi-roll states, merging of multiple states and backwards transitions from multi-roll to single-roll states. For better comparison with the existing experimental data, an additional stability study for varying Prandtl number (0.015 [les ] Pr [les ] 0.03) and fixed value of the aspect ratio A = 4 was carried out. It was shown that the dependence of the critical Grashof number on the aspect ratio and the Prandtl number is very complicated and a very detailed parametric study is required to reproduce it correctly. Comparison with the available experimental data for A = 4 shows that the results of a two-dimensional stability analysis are in good agreement with the experimental results if the width ratio (width/height) of the experimental container is sufficiently large. The study is carried out numerically with the use of two independent numerical approaches based on the global Galerkin and finite-volume methods.

A numerical investigation of steady states, their stability, onset of oscillatory instability, and slightly supercritical unsteady regimes of an axisymmetric swirling flow of a Newtonian incompressible fluid in a closed circular cylinder with a rotating lid is presented for aspect ratio (height/radius) 1 [les ] γ [les ] 3.5. Various criteria for the appearance of vortex breakdown are discussed. It is shown that vortex breakdown takes place in this system not as a result of instability but as a continuous evolution of the stationary meridional flow with increasing Reynolds number. The dependence of the critical Reynolds number Re _cr and frequency of oscillations ω _cr on the aspect ratio of the cylinder γ is obtained. It is found that the neutral curve Re _cr (γ) and the curve ω _cr (γ) consist of three successive continuous segments corresponding to different modes of the dominant perturbation. The calculated critical parameters are in good agreement with the available experimental and numerical data for γ < 3. It is shown that the onset of the oscillatory instability does not depend on the existence of a separation bubble in the subcritical steady state. By means of a weakly nonlinear analysis it is shown that the axisymmetric oscillatory instability sets in as a result of a supercritical Hopf bifurcation for each segment of the neutral curve. A weakly nonlinear asymptotic approximation of slightly supercritical flows is carried out. The results of the weakly nonlinear analysis are verified by direct numerical solution of the unsteady Navier-Stokes equation using the finite volume method. The analysis of the supercritical flow field for aspect ratio less than 1.75, for which no steady vortex breakdown is found, shows the existence of an oscillatory vortex breakdown which develops as a result of the oscillatory instability.

A series of time-dependent three-dimensional (3D) computations of a lid-driven flow in a cube with no-slip boundaries is performed to find the critical Reynolds number corresponding to the steady-oscillatory transition. The computations are done in a fully coupled pressure-velocity formulation on ${104}^3$ , ${152}^3$ , and ${200}^3$ stretched grids. Grid-independence of the results is established. It is found that the oscillatory instability of the flow sets in via a subcritical symmetry-breaking Hopf bifurcation at ${\text{Re}}_{\text{cr}}\approx 1914$ with the nondimensional frequency $\omega =0.575$ . Three-dimensional patterns in the steady and oscillatory flow regimes are compared with the previously studied two-dimensional configuration and a three-dimensional model with periodic boundary conditions imposed in the spanwise direction.