G.D. McBain

Low-Reynolds-number fountain behaviour

Experimental evidence for previously unreported fountain behaviour is presented. It has been found that the first unstable mode of a three-dimensional round fountain is a laminar flapping motion that can grow to a circling or multimodal flapping motion. With increasing Froude and Reynolds numbers, fountain behaviour becomes more disorderly, exhibiting a laminar bobbing motion. The transition between steady behaviour, the initial flapping modes and the laminar bobbing flow can be approximately described by a function FrRe2/3 =C. The transition to turbulence occurs at Re > 120, independent of Froude number, and the flow appears to be fully turbulent at Re ≈2000. For Fr > 10 and Re 120, sinuous shear-driven instabilities have been observed in the rising fluid column. For Re 120 these instabilities cause the fountain to intermittently breakdown into turbulent jet-like flow. For Fr 10 buoyancy forces begin to dominate the flow and pulsing behaviour is observed. A regime map of the fountain behaviour for 0.7Fr 100 and 15Re 1900 is presented and the underlying mechanisms for the observed behaviour are proposed. Movies are available with the online version of the paper.

Natural convection with unsaturated humid air in vertical cavities

Abstract Simple formulae for the overall heat and moisture transport rates due to laminar natural convection in a rectangular cavity are obtained by scale analysis from the governing differential equations and a simplified picture of the flow. The two formulae contain a single unknown proportionality constant, which is determined by a least squares fit to the results of a series of numerical solutions. The relations apply for the case of isothermal vertical walls at constant, unsaturated relative humidity, and adiabatic, impermeable horizontal walls. The heat transfer formula agrees well with published data for the square cavity with zero humidity gradient.

Fully developed laminar buoyant flow in vertical cavities and ducts of bounded section

The fully developed flow in a vertical cavity or duct subject to horizontal heating is considered. Solutions of the Boussinesq equations are obtained for rectangular and elliptic sections, in terms of Fourier series and polynomials, respectively. Both generalize the familiar odd-symmetric cubic profile of the plane cavity. Uniqueness is demonstrated under the restriction that the flow is independent of height. For cavities with rectangular sections, it is predicted and verified that the flow in the plane of spanwise symmetry is practically independent of the span if this exceeds 1.7 times the breadth

Evaporation from an open cylinder

The rate of evaporation from the wetted floor of a tube open at the top to a relatively dry environment is investigated analytically, numerically and experimentally for the case of a light vapour. Though buoyancy forces ensure that the heavier external gas is always in motion, it is found that inside the tube both stagnation (diffusion-dominated evaporation) and convection are possible. In contrast to previous studies, axisymmetry of the postcritical flow is not assumed, leading to a reduction in the predicted critical Rayleigh number by a factor of 5 and much better agreement with experiment.

Heat and mass transfer across tall cavities filled with gas-vapour mixtures : the fully developed regime

Abstract Closed form expressions for the fully developed velocity, temperature and concentration profiles in a vertical channel are found by solving the equations for a cavity in the limit as the aspect ratio tends to infinity. We consider plane, steady, laminar, Boussinesq flow of an ideal gas-vapour mixture. The vertical walls are held at different constant temperatures and compositions, are impermeable to the gas and non-slip. The finite mass transfer effects of interfacial velocity and interdiffusion of enthalpy are included.

Instability of the buoyancy layer on an evenly heated vertical wall

The stability of the buoyancy layer on a uniformly heated vertical wall in a stratified fluid is investigated using both semi-analytical and direct numerical methods. As in the related problem in which the excess temperature of the wall is specified, the basic laminar flow is steady and one-dimensional. Here flows varying in time and with height are considered, the behaviour being determined by the fluids Prandtl number and a Reynolds number proportional to the ratio of two temperature gradients: the horizontal one imposed at the wall and the vertical one existing in the far field. For low Reynolds numbers, the flow is stable with variation only in the wall-normal direction. For Reynolds numbers greater than a critical value, depending on the Prandtl number, the flow is unstable and supports two-dimensional travelling waves. The critical Reynolds number and other properties have been obtained via linearized stability analysis and are shown to accurately predict the behaviour of the full nonlinear solution obtained numerically for Prandtl number 7. The stability analysis employs a novel Laguerre collocation scheme while the direct numerical simulations use a second-order finite volume method.

Convection in a horizontally heated sphere

Natural convection in horizontally heated spherical fluid-filled cavities is considered in the low Grashof number limit. The equations governing the asymptotic expansion are derived for all orders. At each order a Stokes problem must be solved for the momentum correction. The general solution of the Stokes problem in a sphere with arbitrary smooth body force is derived and used to obtain the zeroth-order (creeping) flow and the first-order corrections due to inertia and buoyancy. The solutions illustrate the two mechanisms adduced by Mallinson & de Vahl Davis (1973, 1977) for spanwise flow in horizontally heated cuboids. Further, the analytical derivations and expressions clarify these mechanisms and the conditions under which they do not operate. The inertia and buoyancy effects vanish with the Grashof and Rayleigh numbers, respectively, and both vanish if the flow is purely vertical, as in a very tall and narrow cuboid.

Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 2. The Stokes equations

We continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.

Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 1. Fields with divergence

It is shown how to decompose a three-dimensional field periodic in two Cartesian coordinates into five parts, three of which are identically divergence-free and the other two orthogonal to all divergence-free fields. The three divergence-free parts coincide with the mean, poloidal and toroidal fields of Schmitt and Wahl; the present work, therefore, extends their decomposition from divergence-free fields to fields of arbitrary divergence. For the representation of known and unknown fields, each of the five subspaces is characterised by both a projection and a scalar representation. Use of Fourier components and wave coordinates reduces poloidal fields to the sume of two-dimensional poloidal fields, and toroidal fields to the sum of unidirectional toroidal fields.

Spatial extrema of advected scalars

Abstract Scalar fields satisfying the stationary advection–diffusion equation with no source or sink terms cannot have strong local extrema. This can be deduced from the elliptical nature of the equation. Here, however, an alternative, original and more physically motivated proof is offered. It highlights the positive role of diffusion in preventing extrema and the inability of advection to create them. Application is made to the theory of energy transfer by species interdiffusion and some anomalous numerical solutions from the literature are identified.