The spiralling flow within a curved tube of rectangular cross-section normally forms two distinct cells in the plane of the cross-section, but a transition to a different, four-cell flow is known to occur at a critical value of the axial pressure gradient. This paper shows that, for a square cross-section, the transition is a result of a complex structure of multiple, symmetric and asymmetric solutions. The structure is revealed by solving extended systems of equations for steady-state, fully developed, laminar flow, in a finite-element approximation, to locate exactly the positions of singular points. Continuation methods are used to trace the paths of these points as the aspect ratio and radius of curvature vary. For a square cross-section, multiple, symmetric flows of both two and four cells are found in two distinct ranges of axial pressure gradient q. In addition, a pair of asymmetric solutions arise from a symmetry-breaking bifurcation point on the primary flow branch. The paths of limit points and symmetry-breaking bifurcation points are obtained as the aspect ratio γ varies, and a transcritical bifurcation point is found at γ = 1.43. For larger aspect ratios the secondary four-cell branch is disconnected from that of the primary two-cell flow. The bifurcation set in the two parameters q and γ has a complex structure, with a number of higher-order singularities; in particular, the path of limit points is found to cross in the manner of an unfolded swallowtail catastrophe. A stability analysis shows that all multiple solutions except the two-cell type are unstable with respect to either symmetric or antisymmetric perturbations. For values of the aspect ratio less than 1.43 there is a range of axial pressure gradients for which there is no stable flow. The critical Dean numbers of the singular points are found to vary strongly at small values of radius of curvature β, but the bifurcation set remains qualitatively the same. Previous work is interpreted in the light of the present results.

SUMMARYA detailed case study is made of one particular solution of the 2D incompressible Navier-Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high-order finite-element method solving the unsteady equations by time-marching; (2) a high-order finite-element method solving both the steady equations and the associated linear-stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at Re=800 is steady-and it is stable, to both small and large perturbations.

Techniques of bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh-Bénard problem: Lapwood convection in a two-dimensional saturated porous cavity heated from below. Two particular aspects of the problem are focused upon: (i) the existence of multiple steady solutions and (ii) the influence of aspect ratio.Convection begins only when the applied temperature difference (say) exceeds a critical value defined by linear stability theory. The resulting convective flow pattern depends both on the magnitude of the temperature difference and on the aspect ratio of the cavity. A weakly nonlinear analysis reveals the roles played by so-called secondary bifurcations in determining the formation of further, anomalous patterns at fixed aspect ratio. In addition to giving rise to alternative stable flows for identical operating conditions, the secondary bifurcations are required for the modal exchanges which take place as the aspect ratio varies, a process which causes an abrupt change in preferred flow pattern at certain critical values of the aspect ratio.As a complement to and an extension of the weakly nonlinear analysis, numerical methods are used to determine the bifurcation processes and to elucidate the modal exchange mechanisms in both weakly and strongly convective flows. The effect of container size is studied by continuation methods to predict the variation of the critical Rayleigh number of the bifurcation points for aspect ratios in the range 0.5 to 2.0. In this way a stability map is obtained which shows the alternative patterns expected for particular operating conditions. The Nusselt number is computed and it is found that the alternative stable modes transfer significantly different amounts of heat through the medium.The study has provided new information on the existence and characteristics of, and interactions between, alternative steady modes of two-dimensional Lapwood convection. The results have important ramifications for the modelling and design of physical systems in which convective flow in a saturated porous medium is stimulated by an imposed unstable temperature gradient.

No abstract

Oscillatory convection has been observed in recent experiments in a square, air-filled cavity with differentially heated sidewalls and conducting horizontal surfaces. We show that the onset of the oscillatory convection occurs at a Hopf bifurcation in the steady-state equations for free convection in the Boussinesq approximation. The location of the bifurcation point is found by solving an extended system of steadystate equations. The predicted critical Rayleigh number and frequency at the onset of oscillations are in excellent agreement with the values measured recently and with those of a time-dependent simulation. Four other Hopf bifurcation points are found near the critical point and their presence supports a conjectured resonance between traveling waves in the boundary layers and interior gravity waves in the stratified