The theory of two-layer hydraulics is extended to topography with co-varying width and height. When these variations of the non-dimensional width and total depth have a power law relationship, the solutions can still be presented in the Froude-number plane for both unidirectional and exchange flows. These differ from previous solutions, which were limited to treating width and height variations separately.

The subgrid-scale (SGS) physics in the convective atmospheric surface layer is studied using the SGS stress and SGS scalar flux. We derive the budget equations for the conditional mean SGS stress and SGS temperature flux and show that, for transport-equation-based SGS models, the budget terms must be correctly predicted by the SGS model in order for large-eddy simulation (LES) to reproduce the resolvable-scale velocity and temperature probability density functions. Field data from the Advection Horizontal Array Turbulence Study, which notably includes measurements of the fluctuating pressure and the advection of the velocity and temperature fields, are then used to analyse the budget equations. The results reveal the complex behaviour of the dynamics of the convective atmospheric surface layer. The budgets of the conditional mean SGS shear stress and SGS temperature flux are an approximate balance between the conditional mean production and pressure destruction, with the latter causing return to isotropy. The budgets of the normal SGS stress components are more complex. For strongly convective surface layers, energy is redistributed from the (smaller) vertical to the (larger) horizontal stress components during downdrafts, resulting in generation of anisotropy by the conditional mean SGS pressure–strain-rate correlation; wall pressure reflections can also enhance the anisotropy. The conditional mean SGS pressure transport, meanwhile, is a significant source of energy during updrafts as a result of the near-wall pressure minima. The vertical advection also plays a significant role in the transfer of SGS energy. For weakly convective surface layers, pressure transport is small while the SGS pressure–strain-rate correlation reverts to its usual role of causing return to isotropy. The results of the present study, particularly for the conditional mean SGS pressure–strain-rate correlation, provide new insights into the SGS physics first educed in a recent analysis of the mean SGS budgets by Nguyen et al. (J. Fluid Mech., vol. 729, 2013, pp. 388–422) and have important implications for near-wall models utilizing SGS transport equations in the convective atmospheric surface layer.

The propagation of acoustic energy from a sound source to the far field is a fundamental problem of acoustics. In this paper the use of computational fluid dynamics (CFD) to directly calculate the acoustic field is investigated. The two-dimensional, compressible, inviscid flow about an accelerating circular cylinder is used as a model problem. The time evolution of the energy transfer from the cylinder surface to the fluid, as the cylinder is moved from rest to some non-negligible velocity, is shown. Energy is the quantity of interest in the calculations since various components of energy have physical meaning. By examining the temporal and spatial characteristics of the numerical solution, a distinction can be made between the propagating acoustic energy, the convecting energy associated with the entropy change in the fluid, and the energy following the body. In the calculations, entropy generation is due to a combination of physical mechanisms and numerical error. In the case of propagating acoustic waves, entropy generation seems to be a measure of numerical damping associated with the discrete flow solver.

An attempt is made in this paper to tackle the problem of nonlinear wave resistance by formulating it in Fourier space and by deriving a nonlinear integral equation for the wave amplitude by an approach similar to the one leading to the Zakharov equation.

The procedure is illustrated for two simple examples of two- and three-dimensional travelling pressure distributions. A regular perturbation solution up to third-order terms in the slenderness parameter shows that the expansion is not uniform for small Froude numbers. A uniform, generalized, expansion is then constructed, with its leading term satisfying a new nonlinear integral equation. This rather simple integral equation, of a Volterra type, is solved numerically. The generalized wave drag is shown to be significantly larger than the one predicted by the regular perturbation expansion at small Froude number. The method adopted here has the advantage of singling out in a systematic manner the terms of the free-surface conditions which cause the small-Froude-number non-uniformity, and it is applicable to both two- and three-dimensional flows. The results are compared with existing approximate methods of computing wave drag at low Froude numbers. It is found that quasilinearized approximations may be quite accurate for the examples considered here.

In this paper steady flows of an incompressible, viscous, electrically conducting fluid are constructed from fundamental solutions of magnetohydrodynamics in which the applied magnetic field is parallel to the velocity at infinity. The flat plate and the sphere are considered as examples, and approximate solutions are presented for the limiting cases of large and small Reynolds and magnetic Reynolds numbers. The effects of currents in the body are also considered, and it is found that unless the magnetic Prandtl number is larger than unity, currents in the body have negligible effect on the flow.

To determine experimentally the mean value of a randomly fluctuating quantity, it may be necessary to measure the average value over a considerable interval of time. This problem arose in a recent study of the temperature fluctuations over a heated horizontal plate, and a system was used that depended on the counting of electrical pulses generated at a rate proportional to the quantity being measured. The advantage of this technique is that mean values may be measured over time intervals of almost unlimited length with little added difficulty for the experimenter. Circuits are described which measure: (a) the mean square of a fluctuating quantity and of its time-derivative, (b) the statistical distribution of the fluctuations, (c) the mean frequency of the fluctuation assuming a particular value, and (d) the mean product of two fluctuating quantities. Over the range of use, the stability and linearity of the calibrations is better than 1%, more than sufficient for work on natural convection. In its present form, the equipment responds uniformly to all frequencies below 100 c/s, but it would not be difficult to extend this range of response to higher frequencies.

We consider a semi-infinite (bounded above) plane-parallel layer of barotropic fluid in a constant gravitational field. We present a proof that flows of such a fluid cannot be time-independent in a reference frame wherein the flow's velocity field falls off asymptotically faster than the inverse of the radial distance, R. This includes all flows of finite kinetic energy as such flows must fall off faster than R−1.5. The unsteadiness is due in part to the dynamical expansion of a compressible fluid in motion; this expansion leads to a density deficit so that in the presence of gravity the flow rises buoyantly and cannot be steady in time. The non-existence of certain classes of steady uniform-fluid flows is also discussed.

We report here on the results of a series of experiments carried out on a turbulent spot in a distorted duct to study the effects of a divergence with straight streamlines preceded by a short stretch of transverse streamline curvature, both in the absence of any pressure gradient. It is found that the distortion produces substantial asymmetry in the spot: the angles at which the spot cuts across the local streamlines are altered dramatically (in contradiction of a hypothesis commonly made in transition zone modelling), and the Tollmien–Schlichting waves that accompany the wing tips of the spot are much stronger on the outside of the bend than on the inside. However there is no strong effect on the internal structure of the spot and the eddies therein, or on such propagation characteristics as overall spread rate and the celerities of the leading and trailing edges. Both lateral streamline curvature and non-homogeneity of the laminar boundary layer into which the spot propagates are shown to be strong factors responsible for the observed asymmetry. It is concluded that these factors produce chiefly a geometric distortion of the coherent structure in the spot, but do not otherwise affect its dynamics in any significant way.

The effect of surface curvature on the low-wavenumber-frequency spectrum of turbulence-induced surface pressure fluctuations is considered and an estimate for the associated flow noise is obtained. The form of the low-wavenumber cross-spectral density of pressure on the surface of an infinitely long, rigid circular cylinder of radius a due to a statistically stationary turbulent boundary-layer flow at a low Mach number is determined. Viscous effects are ignored. It is shown that, in contrast to the case of an infinite plane surface, the pressure spectrum is finite everywhere in the wavenumber plane (k, n/a) except for a logarithmic, integrable, singularity at the acoustic wavenumber corresponding to the axisymmetric mode (n/a = 0); k and n/a being, respectively, the downstream and circumferential wavenumber. For non-axisymmetric modes (|n/a| > 0), the spectrum has two finite peaks in the radiative domain |k| < ω/c; ω being the frequency and c being the sound speed. For ωa/c large, the peaks occur in the vicinity of the total acoustic surface wavenumbers κ = ± ω/c (κ = (k2 + n2 / a2)½) and the principal contribution which determines the peak characteristics can be identified as being due to creeping rays emanating from turbulent sources on the cylinder out of the line of sight of the associated receiver point. For large value of ωa/c, the point pressure spectrum and the associated radiated sound vary logarithmically with $(\omega a/c)^{\frac{2}{3}}$; corresponding estimates for cylinders of moderate and small radius are also obtained. For an almost plane cylinder, it is shown that the effect of curvature may be included by a suitable simple modification of the form of the pressure spectrum for an infinite plane surface.

In this paper the dynamics of geostrophic flows localized in a thin layer of continuously stratified fluid, which overrides a thick homogeneous layer are studied. The displacement of isopycnal surfaces is assumed large; the β-effect is strong, i.e. \[(R_0/R_e)\cot\theta\gtrsim\epsilon, \] where ε is the Rossby number, θ is the latitude; Re is the Earth's radius, and R0 is the deformation radius based on the total depth of the ocean. An asymptotic system of equations is derived and used to study the stability of zonal currents. Three sufficient conditions of stability are obtained, which restrict the slope of the interface between the stratified and non-stratified layers. The results obtained are applied to the subtropical and subarctic frontal currents in the Northern Pacific: the former was found to be stable, the latter was found to be unstable. However, the growth rate of the instability is very small (the effective time of growth is about 2 years).

This paper gives a modified Myers boundary condition in swirling inviscid flow, which differs from the standard Myers boundary condition by assuming a small but non-zero boundary layer thickness. The new boundary condition is derived and is shown to have the correct quadratic error behaviour with boundary layer thickness and also to agree with previous results when the swirl is set to zero. The boundary condition is initially derived for swirling flow with constant azimuthal velocity, but easily extends to radially varying swirling flow, with terms depending on the boundary layer model. The modified Myers boundary condition is then given in the time domain rather than in the frequency domain. The effect of the boundary layer profile is then considered, and shown to be small compared to the boundary layer thickness. The boundary condition is then used to analyse the eigenmodes and Green’s function in a realistic flow. Modelling the thickness of the boundary layer properly is shown to be essential in order to get accurate results.

An empirical model reduction method is performed on nonlinear transient convection patterns near the threshold, using a pseudo-inverse-based projective method called the pseudo-balanced proper orthogonal decomposition (PBPOD). These transient patterns are large-scale amplitude/phase modulations in convection rolls, obtained by prescribing selected spatial input-shape functions. For the nonlinear convection patterns modelled, PBPOD appears to be very effective. Using the nonlinear front example, PBPOD is compared with other existing methods, such as POD and linearized BPOD. The limitations of the methods are discussed. Using a complex prescribed input, complex disturbances are generated and the outputs of the open-loop responses are compared between the original and the reduced-order models. The agreement is good. A feedback-control study is performed using the nonlinear front example. The controller is built by the pseudo-balanced reduced-order model, with a low-order nonlinear estimator. Closed-loop simulations show that the nonlinear travelling fronts can be effectively damped out by the feedback-control actions.

We consider a viscous, incompressible fluid confined in a narrow annular channel rotating rapidly about its axis of symmetry with angular velocity Ω that itself precesses slowly about an axis fixed in an inertial frame. The precessional problem is characterized by three parameters: the Ekman number E, the Poincaré number ε and the aspect ratio of the channel Γ. Dependent upon the size of Γ, precessionally driven flows can be either resonant or non-resonant with the Poincaré forcing. By assuming that it is the viscous effect, rather than the nonlinear effect, that plays an essential role at exact resonance, two asymptotic expressions for ε ≪ 1 and E ≪ 1 describing the single and double inertial-mode resonance are derived under the non-slip boundary condition. An asymptotic expression describing non-resonant precessing flows is also derived. Further studies based on numerical integrations, including two-dimensional linear analysis and direct three-dimensional nonlinear simulation, show a satisfactory quantitative agreement between the three asymptotic expressions and the fuller numerics for small and moderate Reynolds numbers at an asymptotically small E. The transition from two-dimensional precessing flow to three-dimensional small-scale turbulence for large Reynolds numbers is also investigated.

A non-linear potential problem which describes the injection of a high speed jet into a uniform flow of lower total head is linearized when the jet injection angle is small. The resulting linear problem is solved using the Wiener–Hopf technique. Numerical results for the position of the streamline separating the jet from the free stream and the pressure coefficient along the upstream wall are obtained for various injection angles.

The magnetohydrodynamic and Boussinesq approximations are used on a fully ionized ideal fluid with a longitudinal magnetic field. The flow is in a direction normal to the gravity vector and all variations in velocity, density and magnetic field. The stability characteristics, mainly for normal-mode perturbations, are investigated. Two simple problems with discontinuous profiles are solved analytically. For a double shear layer, an appropriate range of magnetic field values destabilizes the flow. A long wave theory is presented and applied to several problems, some of which are destabilized by an appropriate magnetic field. Finally, the solution for continuous profiles is presented and shown to decay algebraically in time for any stable stratification.

Svendsen & Putrevu (1994) revealed that the much larger off-shore than vertical effective viscosity for longshore currents is the consequence of a shear dispersion mechanism. The multi-mode representation for the flow gives a mathematical framework within which a more general derivation can be made. It is shown that to a first approximation the horizontal shear dispersion tensor for velocity is the same as that for solutes.

In this theoretical report we explore the deformation and stability of a power-law non-Newtonian slender drop embedded in a Newtonian liquid undergoing a nonlinear extensional creeping flow. The dimensionless parameters describing this problem are: the capillary number $(Ca\gg 1)$, the viscosity ratio $(\unicode[STIX]{x1D706}\ll 1)$, the power-law index $(n)$ and the nonlinear intensity of the flow $(|E|\ll 1)$. Asymptotic analytical solutions were obtained near the centre and close to the end of the drop suggesting that only Newtonian and shear thinning drops $(n\leqslant 1)$ with pointed ends are possible. We described the shape of the drop as a series expansion about the centre of the drop, and performed a stability analysis in order to distinguish between stable and unstable stationary states and to establish the breakup point. Our findings suggest: (i) shear thinning drops are less elongated than Newtonian drops, (ii) as non-Newtonian effects increase or as $n$ decreases, breakup becomes more difficult, and (iii) as the flow becomes more nonlinear, breakup is facilitated.

Experimental data are presented for the vortex flow in a nominally two-dimensional circular cavity. The vortex is driven by a separated shear layer along an open section of the cavity circumference. It is shown that the core vortex flow is perturbed three-dimensionally. An inviscid analysis of an ideal core (solid body) vortex is given and it is shown that this flow contains a steady perturbation whose characteristics are almost exactly those identified in the experiments. Viscous effects reduce (by a few per cent) the spanwise wavelength of the perturbation and also lead, via spatial variations in Reynolds stress, to a modification of the core flow so that the radial profile of the circumferential velocity is ‘S’-shaped, rather than linear.

This study presents a detailed scaling analysis quantifying the transient behaviour of natural convection in a reservoir model induced by iso-flux surface heating. It is found that horizontal conduction, which has often been neglected in previous analyses, plays an important role in the development of the flow. Depending on the Rayleigh number, three possible pathways through which the flow develops towards the final steady state are identified. A thermal boundary layer initially grows downwards from the surface. When the thermal boundary layer reaches the sloping bottom and becomes indistinct, a horizontal temperature gradient establishes due to the increasing water depth in the offshore direction. A flow is then driven towards the offshore direction by a buoyancy-induced horizontal pressure gradient, which convects away the heat input from the water surface. On the other hand, the horizontal temperature gradient also conducts heat away. The flow behaviour is determined by the interaction between the horizontal conduction and convection. An interesting flow feature revealed by the present scaling analysis is that the region across which the thermal boundary layer encompasses the full water depth shrinks over time at a certain stage of the flow development. The shrinking process eventually stops when this region coincides with a conduction-dominated subregion. The present scaling results are verified by corresponding numerical simulations.