The problem of natural convection in a cavity of small aspect ratio with differentially heated end walls is considered. It is shown by use of matched asymptotic expansions that the flow consists of two distinct regimes: a parallel flow in the core region and a second, non-parallel flow near the ends of the cavity. A solution valid at all orders in the aspect ratio A is found for the core region, while the first several terms of the appropriate asymptotic expansion are obtained for the end regions. Parametric limits of validity for the parallel flow structure are discussed. Asymptotic expressions for the Nusselt number and the single free parameter of the parallel flow solution, valid in the limit as A → 0, are derived.

Numerical solutions of the full Navier-Stokes equations are obtained for the problem of natural convection in closed cavities of small aspect ratio with differentially heated end walls. These solutions cover the parameter range Pr = 6·983, 10 ≤ Gr 2 × 104 and 0·05 [les ] A [les ] 1. A comparison with the asymptotic theory of part 1 shows excellent agreement between the analytical and numerical solutions provided that A [lsim ] 0·1 and Gr2A3Pr2 [lsim ] 105. In addition, the numerical solutions demonstrate the transition between the shallow-cavity limit of part 1 and the boundary-layer limit; A fixed, Gr → ∞.

The steady motion of water in an enclosed rectangular cavity with differentially heated vertical end walls was studied experimentally, and the results are compared with the findings of parts 1 and 2. The depth-to-length ratios of the cavities were 102 and 1·9 × 102, and the Rayleigh number was allowed to vary sufficiently to enable a study to be made of the transition from a flow driven by the vertical wall boundary layers to one sustained by a longitudinal temperature gradient in the central sections of the cavity.

Assuming information about the mean velocity and vertical turbulent velocity, it is possible to calculate the flow direction wavenumber spectrum of pressure fluctuations ϕ(k1 δ)/τ02δ. The law of the wall plus Cole's wake function represented the mean velocity profiles. A scale-anisotropic model of R22 was used and the component intensity û2 was assumed to vary across the boundary layer in constant proportionality to the Reynolds stress. Calculated zero-pressure-gradient spectra rise as k11.5 at low wavenumbers. Curves for various Reynolds numbers are closely similar, and diverge only slightly around the peak in the spectrum. A high wavenumber spectrum ϕk1v/u*. u*/τ02v is independent of Reynolds number. The calculations reveal an overlap region in which ϕ ∼ k1−1. Imposing an equilibrium pressure gradient increases the spectrum at the low and mid wavenumbers, but has no effect in the overlap region. The spectrum peak for II = 6 is a factor 102 higher than for the zero-pressure-gradient layer. It is proposed that the convective velocity Uc(k1) has an overlap region. The overlap law is found to be \[ \frac{U_c}{u_{*}} = -\frac{1}{\kappa}\ln k_1\delta +\frac{1}{\kappa}\ln\frac{u_{*}\delta}{\nu}+A, \] where K and A are the same constants as in the mean velocity expression. Comparison with experiments shows very good agreement. A rough convective ‘wake’ function is formulated for the low-wavenumber range. Wavenumber spectra are converted to frequency spectra, and compared with experiments. Data from a zero pressure gradient and an adverse pressure gradient II = 3 show reasonable agreement with the calculations.

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.

The effects of viscosity and heat conduction on the propagation of an internal wave generated by a simple harmonic localized disturbance are considered for the case of an isothermal compressible atmosphere. A similarity solution of the linearized equations shows that the velocities decay and the wave width increases away from the disturbance. Superpositions of this solution show how a few waves of small wavelength attenuate rapidly whereas waves of larger wavelength can increase in amplitude as they propagate upwards before eventually attenuating.

A series of measurements of trailing vortices behind lifting hydrofoils is described. These measurements were made in the Caltech Free-Surface Water Tunnel, using a laser-Doppler velocimeter to measure two components of velocity in the vortex wake. Two different model planforms were tested, and measurements were made at several free-stream velocities and angles of attack for each. Velocity profiles were measured at distances downstream of the model of from five to sixty chord lengths. These measurements are the first results of a continuing experimental programme.

In § 3 of this paper, the theory of trailing vortices is discussed. The effects of ‘vortex wandering’ upon the measurements are computed, and the corrected results are seen to be in reasonable agreement with the theory.

The structure of a shock wave in a vibrationally relaxing gas undergoing reflexion from a plane wall is examined. The shock wave is assumed to be weak, and departures from thermodynamic equilibrium are assumed small; both an adiabatic and an isothermal wall are considered. The flow field is divided into three regions: a far-field region, an interaction region, and, for the isothermal-wall case, a thermal boundary layer. Different asymptotic expansions are determined for the various regions through the method of matched asymptotic expansions. In the region far from the wall, a non-equilibrium Burgers equation governs the motion and the incident and the reflected shock wave structures. During reflexion, a non-equilibrium wave equation applies; its first-order terms are equivalent to an acoustic approximation. Heat conduction to the wall is modelled by an isothermal wall boundary condition which requires the introduction of a thermal boundary layer adjacent to the wall. This thermal boundary layer is thin and the adiabatic-wall result provides the outer solution for treating this layer. This thermal layer affects the structure of the reflected wave.

The familiar Segré-Silberberg effect of inertia-induced lateral migration of a neutrally buoyant rigid sphere in a Newtonian fluid is studied theoretically for simple shear flow and for two-dimensional Poiseuille flow. It is shown that the spheres reach a stable lateral equilibrium position independent of the initial position of release. For simple shear flow, this position is midway between the walls, whereas for Poiseuille flow, it is 0·6 of the channel half-width from the centre-line. Particle trajectories are calculated in both cases and compared with available experimental data. Implications for the measurement of the rheological properties of a dilute suspension of spheres are discussed.

An extensive series of measurements of sphere drag coefficients has been made in an aeroballistic range for a broad range of Reynolds and Mach numbers. These measurements have been compared with those obtained in other test facilities. As a result of this comparison it has been possible to suggest reasons for many of the inconsistencies in the earlier measurements and to establish more accurate values of the sphere drag coefficient for M∞ [lsim ] 0·2 and 10−2 [lsim ] Re∞ [lsim ] 107.