We report theoretical growth rates for the Rayleigh–Bénard instability when the fluid layer is contained by non-slip walls in a cylindrical geometry with diameter D and height L. Our results are for the growth rates of the first two axisymmetric modes as functions of the Prandtl number P and the aspect ratio γ≡D/2L. We have considered the two extreme cases of ideally insulating and ideally conducting side walls, and found that the growth rate is relatively insensitive to the choice of the thermal boundary conditions on the side walls. Our results are useful in understanding recent experimental measurements of the convective time-scale.

In a previous paper (Grimshaw 1979) the resonant over-reflection of internal gravity waves from a vortex sheet was considered in the weakly nonlinear regime. It was shown there that the time evolution of the amplitude of the vortex sheet displacement was balanced by a cubic nonlinearity. For one vortex sheet mode, symmetrical with respect to the interface, it was shown that a steady finite-amplitude wave was possible. For the other, asymmetric modes, a singularity develops in a finite time. In the present paper, that analysis is extended by replacing the vortex sheet with a thin shear layer of thickness α2, where α is the amplitude of the shear layer displacement. The effect of this extension is to introduce a linear growth rate term in the amplitude equation, which is otherwise unaltered. The linear growth rate can be computed from a formula due to Drazin & Howard (1966, p. 67). The effect on the modes is that the symmetric mode is linearly damped and requires sustained forcing to be observed, while the asymmetric modes are slightly destabilized by the linear term and, as in the vortex-sheet model, develop a singularity in finite time.

Experimental results are presented for steady, supercritical flow of a liquid in a thin-walled compliant tube which is in a state of partial collapse due to a negative transmural pressure. Particular attention is paid to the effects of longitudinal tension.

With a constant external pressure, friction acts to increase the area in the down-stream direction. With the tube tilted downward, friction may be so balanced by gravity forces as to result in an asymptotic approach to an equilibrium situation in which the area and velocity remain constant. When the downstream pressure is increased sufficiently, shock-like transitions to subcritical inflated states, with positive transmural pressure, occur. The longitudinal length scale of the shock is on the order of one to several tube diameters. The pressure rise across the shock lies between that for loss-free pressure recovery and that given by the Borda-Carnot sudden-expansion theory.

The presence of longitudinal tension causes a train of standing waves of area to appear upstream of a local area disturbance, such as a shock-like transition. The standing waves are superimposed upon the more gradual area changes associated with friction and gravity. The wave amplitude grows in the downstream direction. Theoretical interpretations of the observations are presented in the companion paper (part 2).

Theories are developed to explain the experimental observations of steady, supercritical flow in compliant, partially collapsed tubes, presented in the companion paper (part 1).

It is shown that the measured curves of area vs. distance are governed by a combination of (i) friction and gravity, which produce mean gradients of area, and (ii) longitudinal bending and tension forces, which produce standing waves of area superposed upon the mean gradients. The experiments confirm the one-dimensional theory for the mean gradients: (i) in the absence of gravity, friction causes a pressure rise and a positive mean gradient of area; (ii) a downward slope can cancel gravity and lead asymptotically to a uniform state having zero gradients of pressure and area.

The inviscid dispersion relationship for area waves due to longitudinal bending and tension is developed, based on a simple, approximate model for the mechanics of the tube. The phase velocity increases as the wavelength decreases, hence the group velocity exceeds the phase velocity. Consequently, in steady flows that are supercritical with respect to the infinite-wavelength phase velocity, energy can propagate upstream and standing waves of area may appear.

In the experiments of part 1, longitudinal tension predominated over longitudinal bending. The measured wavelengths of standing waves were found to be in general agreement with the dispersion relationship for tension-induced area waves. The observed streamwise growth of standing area waves is interpreted physically as the attenuation of waves radiating upstream from a source of disturbance such as a shock-like rapid increase of area. The rate of wave attenuation indicates that the skin-friction coefficient has a large out-of-phase oscillatory component. The observed steepness of shock transitions agrees with an approximate theory based on treating the forward portion of the shock as the rearward part of the standing wave train that the shock drives upstream.

Experimental results are reported for a U-shaped, free convection loop. The top of the loop is open to an isothermal reservoir. The horizontal leg and one vertical leg are heated at rates Q1 and Q2, respectively. The loop is filled either with water or a water-saturated porous medium. Symmetric heating and asymmetric heating favouring the ascending leg of the loop both yield stable flows. Asymmetric heating favouring the descending leg leads to stable flows when the ratio Q1/Q2 is above a critical value. Below this critical value, the flow is observed to oscillate with increasing amplitude until the direction of flow in the loop undergoes a reversal. A steady flow follows the reversal. Analytical results include a stability analysis and time-dependent, one-dimensional numerical calculations, both of which compare favourably with experiment.

The disturbance amplification mechanism is explained in terms of thermal anomalies which move through the loop with the material motion of the fluid. Since the heating and buoyancy generation processes are in phase in the heated, descending leg, a thermal anomaly can amplify as it flows through that leg. As the anomaly moves through the ascending leg, it initiates a subsequent anomaly of opposite sign in the descending leg. The result is an oscillating flow which, under appropriate conditions, can amplify.

The initial-value problem defined by two parallel vortex sheets of opposite sign is studied. Strictly two-dimensional, incompressible, nearly inviscid dynamics is assumed throughout. The roll-up of the sheets into a vortex street is simulated numerically using 4096 point vortices. Much longer runs than in previous work are performed, and it is found that only for a finite range of values of the ratio, h/λ, of sheet separation to perturbation wavelength, does a long-lived vortex street emerge. For h/λ [gsim ] 0·6 a pairing transition within each row intervenes. For h/λ [lsim ] 0·3 we find oscillatory modes.

Using up to 16384 point vortices, we also study the breakdown of the metastable street to a two-dimensional, turbulent shear flow. The vortex blobs that made up the street may merge with others of the same sign after the breakdown, but otherwise persist throughout the turbulent regime. Neither their disintegration nor amalgamation with vortices of opposite sign was observed. Using dimensional arguments we derive the relevant scaling theory, and show that it applies to a flow started from two random vortex sheets. The resulting turbulence is not self-similar. For the turbulent flow that follows from the breakdown of a regular vortex street two length scales with different power-law growth in time appear to be necessary. The important differences in the asymptotic structure of flows initialized from random and regular sheets leads us to question the idea of universality. The influence of the symmetry of the initial perturbation on the subsequent development is also considered.

The time-dependent deformation and burst of a viscous drop in an arbitrary shear flow at zero Reynolds number is studied. The viscosities of the drop and the suspending fluid are assumed to be equal. A numerical scheme to track the (non-axisymmetric) drop shape in time is presented, and used to investigate the deformation induced by two-dimensional shear and orthogonal rheometer flows. Steady deformations, critical flow rates and burst modes are determined, and compared with asymptotic (small and large) deformation theories, and with experiment.

The excitation of Tollmien-Schlichting waves in low subsonic flat-plate boundary layers by sound is investigated theoretically. The problem is formulated mathematically as an inhomogeneous boundary-value problem which is then solved by a Green's-function technique. It is found that the amplitude of the excited Tollmien-Schlichting wave satisfies an inhomogeneous first-order differential equation. The calculated wave amplitude according to this equation exhibits spatial oscillations in the region ahead of the lower branch neutral stable point of the boundary layer. This characteristic feature resembles that observed experimentally by Shapiro (1977). The theoretical value of the coupling constant between incident sound wave and excited Tollmien-Schlichting wave agrees favourably with measured data. Other predictions of the theory also seem to compare well with available experimental measurements.