P. G. Daniels

A singularity in thermal boundary-layer flow on a horizontal surface

A thermal boundary layer, in which the temperature and velocity fields are coupled by buoyancy, flow along a horizontal, insulated wall. For sufficiently low local Froude number the solution terminates in a singularity with rising skin friction and failling pressure. The structure of the singularity is obtained and the results are compared with numerical solutions of the horizontal boundary-layer equations

Convection in a vertical slot

A boundary-layer approximation is used to describe the convective regime in a laterally heated vertical slot at large Prandtl numbers. The determination of the core flow requires the solution of the vertical boundary-layer equations in a rectangle, subject to appropriate boundary conditions on each of the four walls. Solutions based on a spectral decomposition in the vertical direction allow a comparison with experimental and numerical results, and an appraisal of an approximate solution frequently used as a basis for stability studies. Both the numerical results and an approximate stability argument lead to a simple criterion for the appearance of multiple rolls in the slot which appears to be in good agreement with experiments.

On the long-wave instability of natural-convection boundary layers

This paper considers the stability of the one-dimensional boundary layer generated by sudden heating of an infinite vertical wall. A quasi-steady approximation is used to obtain the asymptotic form of the growth rate and phase speed of disturbances whose wavelength is comparable with the boundary-layer width. Results for the inviscid modes governed by Rayleighs equation are obtained for several values of the Prandtl number and are compared with solutions of the full stability equations. As the wavelength increases,the phase speed of disturbances approaches the maximum flow speed of the boundary layer and a five-tier structure extends across and outside the boundary layer. This intermediate regime,where viscous effects are important within a critical layer centred on the position of maximum flow speed,provides a link with an earlier long-wave analysis of the problem.

Nonlinear convection in a rigid channel uniformly heated from below

A weakly nonlinear theory is developed for convection in an infinite rigid horizontal rectangular channel uniformly heated from below. A combination of analytical and numerical techniques along and in the cross-section of the channel leads to the derivation of an amplitude equation governing the spatial and temporal evolution of the flow above the critical Rayleigh number. Results are obtained for general Prandtl numbers and a wide range of aspect ratios. Overall trends are confirmed by comparison with results for an idealized model with stress-free horizontal boundaries. For wide channels, where the aspect ratio is large, the limiting form of the amplitude equation is predicted by reference to the two-dimensional equation describing roll patterns in infinite layers. The connection with this well-developed theory is established for both rigid and stress-free horizontal boundary conditions.

On the evolution of thermally driven shallow cavity flows

The temporal evolution of thermally driven flow in a shallow laterally heated cavity is studied for the nonlinear regime where the Rayleigh number R based on cavity height is of the same order of magnitude as the aspect ratio L (length/height). The horizontal surfaces of the cavity are assumed to be thermally insulating. For a certain class of initial conditions the evolution is found to occur over two non-dimensional timescales, of order one and of order L(exp 2). Analytical solutions for the motion throughout most of the cavity are found for each of these timescales and numerical solutions are obtained for the nonlinear time-dependent motion in end regions near each lateral wall. This provides a complete picture of the evolution of the steady-state flow in the cavity for cases where instability in the form of multicellular convection does not occur. The final steady state evolves on a dimensional timescale proportional to iota(exp 2)/kappa, where iota is the length of the cavity, kappa is the thermal diffusivity of the fluid and the constant of proportionality depends on the ratio R/L. 20 refs.

On the boundary layer structure of differentially heated cavity flow in a stably stratified porous medium

This paper considers two-dimensional flow generated in a stably stratified porous medium by monotonic differential heating of the upper surface. For a rectangular cavity with thermally insulated sides and a constant-temperature base, the flow near the upper surface in the high-Darcy-Rayleigh-number limit is shown to consist of a double horizontal boundary layer structure with descending motion confined to the vicinity of the colder sidewall. Here there is a vertical boundary layer structure that terminates at a finite depth on the scale of the outer horizontal layer. Below the horizontal boundary layers the motion consists of a series of weak, uniformly stratified counter-rotating convection cells. Asymptotic results are compared with numerical solutions for the cavity flow at finite values of the Darcy-Rayleigh number.

Shallow cavity flow in a porous medium driven by differential heating

This paper describes steady flow through a porous medium in a shallow two-dimensional cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated. The main emphasis is on the situation where the Darcy-Rayleigh number R is large and the aspect ratio of the cavity L (length/depth) is of order R 1/2 . For a monotonic temperature distribution at the upper surface, the leading-order problem consists of an interaction involving the horizontal boundary-layer equations, which govern the flow throughout most of the cavity, and the vertical boundary-layer equations which govern the flow near the colder sidewall. This problem is solved using numerical and asymptotic methods. The limiting cases where L » R 1/2 and L « R 1/2 are also discussed.

On the boundary-layer structure of high-Prandtl-number horizontal convection

This paper describes the boundary-layer structure of the steady flow of an infinite Prandtl number fluid in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated and the upper surface is assumed to be either shear-free or rigid. In the limit of large Rayleigh number (R →∞ ), the solution involves a horizontal boundary layer at the upper surface of depth of order R −1/5 where the main variation in the temperature field occurs. For a monotonic temperature distribution at the upper surface, fluid is driven to the colder end of the cavity where it descends within a narrow convection-dominated vertical layer before returning to the horizontal layer. A numerical solution of the horizontal boundary-layer problem is found for the case of a linear temperature distribution at the upper surface. At greater depths, of order R −2/15 for a shear-free surface and order R −9/65 for a rigid upper surface, a descending plume near the cold sidewall, together with a vertically stratified interior flow, allow the temperature to attain an approximately constant value throughout the remainder of the cavity. For a shear-free upper surface, this constant temperature is predicted to be of order R −1/15 higher than the minimum temperature of the upper surface, whereas for a rigid upper surface it is predicted to be of order R −2/65 higher.