When advection dominates diffusion, there are special directions (rays), distinct from but closely related to the contaminant flux vector, along which information is carried. The geometry of these ray paths is found to depend in a simple way upon the advection-diffusion vector ½u/D, where u is the flow velocity and D the cross-stream diffusivity. Simple models are used to show that for a steady point discharge the contaminant concentration is greatest in shallow water and towards the outside of bends.

This is the final part in the work on steady gravity–capillary waves by the author. On extending the work of Pierson & Fife (1961), the phenomenon of Wilton's ripples is resolved. These singularities of the traditional solution procedure are merely consequences of the non-uniformity in the ordering of the Fourier coefficients of the wave profile. Results presented here include wave properties and profiles. Near-resonant waves are also considered. Good agreement is found between this work and previous papers in the series as well as with other authors. An appendix contains all the results in numerical form. Minor algebraic errors in Wilton's original work are corrected.

Closed-form expressions for the turbulent mean reaction rate and its covariance with the temperature are derived for premixed and non-premixed combustion. The limit of large activation energies is exploited for a chemical reaction rate that, by virtue of coupling functions, depends on the mixture fraction and a non-equilibrium progress variable only. The probability density function (p.d.f.) formulation with an assumed shape of the p.d.f. is used; a beta-function distribution is assumed for the progress variable. The mean reaction rate is expressed in terms of the mean and the variance of the temperature and, for non-premixed combustion, of the mixture fraction. The reaction kinetics are represented by the non-dimensional activation energy and the laminar flame velocity. For non-premixed systems the possibility of local extinction by flame stretch is considered.

This paper describes an experimental study of free convection in an enclosed rectangular cavity, the end walls of which are maintained at uniform but different temperatures. The experiments are carried out for a variety of Rayleigh numbers, R = αgΔTh4/κνl, and aspect ratios, L = l/h, for fluids with Prandtl number σ ≥ 10. For R ∼ O(103) it is shown that the basic structure of the flow field is a single two-dimensional cell for 0·25 ≤ L ≤ 9. When R > O(104) the boundary layers on the vertical walls control the flow field, but the basic overall structure remains unicellular. At greater values of R secondary vortices appear for all L ≥ 0·5. As R increases the intensity and then the number of these vortices increases. Measurements of the end-wall boundary-layer profiles at different values of R and L confirm Gill's boundary-layer analysis. The effects of variations of viscosity with temperature are discussed in the context of the observed boundary-layer profiles.

Core shear profiles and mass flux measurements are also reported. For L = 1 the observed shear profiles are in good agreement with numerical solutions of the Boussinesq equations. However, when L > 1 the observations suggest that the horizontal boundary layers have a significant effect on the core flow field. The stream function is demonstrated to be L-dependent in the boundary-layer regime, where variations due to R are second order. Similarities between the results of the present work and earlier observations by Elder and by Seki, Fukusaka & Inaba for tall slender cavities (L [Lt ] 1) are discussed.

The interaction between a regular wavetrain and an adverse current containing an arbitrary distribution of vorticity, in two dimensions, is studied using a linear theory. The model is used to predict the wavelength and the particle velocities under the waves and these are found to agree well with experimentally obtained data for a number of current profiles. Surprisingly accurate predictions, for the profiles considered, were also obtained from an irrotational wave–current model in which the constant current has a value equal to the depth-averaged mean of the measured current profile. The changes in the wave amplitude as the current magnitude increases are predicted using an irrotational slowly varying model with good agreement being found between theory and experiment.

The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k−3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.