An orthogonal coordinate system along a generic spatial curve has been introduced, and the Navier-Stokes equations for a steady incompressible viscous flow have been explicitly written in this frame of reference. As an application the flow in a helical pipe has been studied, and, formdii 5f curvature and torsion small compared with the radius of the pipe, the flow has been considered as a perturbed Poiseuille flow. The result is that for curvatures and torsions of the same order and for low Reynolds number the curvature induces on the flow a first-order effect on the parameter ε =κa, where κ is the curvature and a the radius of the pipe, while the effect of the torsion on the flow is of the second order in E. This last result disagrees with those of Wang (1981), who, adopting a non-orthogonal coordinate system, found a first-order effect of torsion on the flow.

This paper re-examines the theoretical arguments that indicate the structure of the pressure field induced on a flat surface by boundary-layer turbulence at low Mach number. The long-wave elements are shown to be dictated by the acoustics of the flow, and the limit of the acoustic range is the coincidence condition of grazing waves where the spectrum is singular and proportional to the logarithm of the flow scale. The surface spectrum is shown to be proportional to the square of frequency at low-enough frequency and to the square of wavenumber at those low wavenumbers with subsonic phase speed.

The similarity model successfully used by Corcos for the main convective elements of the field is used in this paper to model the turbulent sources of pressure, not the pressure itself, so that a Corcos-like description of the pressure spectrum is derived that is consistent with constraints imposed by the governing equations. This results in a fairly compact specification of the pressure spectrum with yet-undetermined constants, which must be derived from experiment. Despite an extensive search of published data on the pressure field, it is concluded that existing information is an inadequate basis for setting those constants and that new free field experiments are needed. Boundary layers formed on gliders or buoyant underwater bodies offer the most promising source of such data.

The paper concludes with a study of how large flush-mounted transducers discriminate against the local flow noise field and i t is shown that they do so at a rate of 9 decibels per doubling of transducer diameter. This different conclusion from Corcos’ correct 6 decibel rate for small transducers is entirely due to the low- wavenumber constraints on the spectrum, which are misrepresented in the simple similarity model. This result, which conforms with the constraints imposed by the weak compressibility of the fluid, is the same as that later suggested by Corcos for transducers that are large on the boundary-layer scale.

The generalized Lagrangian-mean (GLM) formulation of Andrews & McIntyre (1978a, b) offers alternative physical concepts and possible saving of effort in calculation, as compared with the more conventional Eulerian-mean approach. Though most existing applications of this theory concern waves on weakly sheared mean flows, it is also suitable for study of waves in strong shear flows. The hydrodynamic stability of parallel shear flows is examined from this point of view. An appreciation is gained of the roles of Stokes drift, pseudomomentum, energy and pseudoenergy in this context, such understanding being a necessary prerequisite for future developments. Several known results of linear stability theory, including the inflexion-point and semicircle theorems, are concisely rederived from the GLM conservation laws.

The instability of two-dimensional periodic flows to spanwise-periodic ‘longitudinal- vortex’ modes is examined. The undisturbed state comprises a parallel shear flow and a two-dimensional O(ε) wave field as encountered in, say, water-wave or hydrodynamic-stability theories.

When the mean shear is weak, of order ε2, the present theory reduces to that of Craik (1977) and Leibovich (1977b, 1980). For stronger but still weak shear, of order ε, it is established that the Craik-Leibovich instability mechanism is essentially unchanged, apart from scaling factors. For strong O(1) shear flows, the governing equations are derived by using, in part, a generalized Lagrangian-mean formulation. The resultant eigenvalue problem for the longitudinal-vortex instability is then more complex, but simplifies in the case of small spanwise spacing of the vortices, in the inviscid limit. An example is given of flows that exhibit instability in this limiting case. Such instability seems likely to occur for a wide class of periodic shear flows. Complementary physical interpretations of the instability mechanism are discussed.

This paper describes a new surface-streamline flow-visualization technique that is suitable for use in low-speed wind tunnels or other low-speed gas flows. The technique provides a permanent record of ink traces that show surface-streamline direction and shape. The visualization of the endwall surface flow of the horseshoe-vortex system formed around the base of a cylinder by a separating turbulent boundary layer is used as an example. The results of the new technique are compared with those obtained from a conventional flow-visualization technique. Good agreement was found between the two, with the new technique appearing to give more-accurate surface streamlines.

The stability of weakly compressible boundary-layer flow over a spring-mounted piston is examined theoretically by modelling the mean boundary layer at low Strouhal numbers by means of a step-function velocity profile. This constitutes a prototype problem for the treatment of the interaction of unsteady boundary-layer flow with a compliant surface, and the present discussion complements a recent analysis due to Ffowcs Williams and Purshouse by incorporating the influence of flow separation at the edges of the piston. This is effected analytically by application of the unsteady Kutta condition at both the leading and trailing edges of the piston. At high Reynolds numbers and in the case of light fluid loading i t is predicted that the separated flow can cause piston flutter. Stability criteria are derived for a rectangular piston of large aspect ratio.

A two-layer configuration of thermohaline convection is studied, with the principal aim of explaining the observed independence of the buoyancy-flux ratio on the stability parameter when the latter is large. Temperature is destabilizing and salinity is stabilizing, so diffusive interfaces separate the convecting layers. The convection is treated in the single-mode approximation, with a prescribed horizontal planform and wavenumber. Surveys of numerical solutions are presented for a selection of Rayleigh numbers R, stability parameters λ and horizontal wavenumbers a. The solutions yield a buoyancy flux ratio ζ that is insensitive to ζ, in accord with laboratory experiments. However ζ increases with increasing R, in contradiction to laboratory observations.

Truncated modal equations are used to study the time evolution of thermal convection. In the Boussinesq approximation these nonlinear equations are obtained by expanding the fluctuating velocity and temperature fields in a finite set of planforms of the horizontal coordinates. Here we report on numerical studies dealing with two or three modes with triad interactions. We have found rich time dependence in these cases: periodic and aperiodic solutions can be obtained, along with various steady solutions. Three-mode solutions reproduce the qualitative appearance of spoke-pattern convection as observed in experiments at high Prandtl numbers. Though the values of the periods of the time-dependent solutions do not agree with those of the experiments, their variation with Rayleigh number compares favourably. Except at the highest Rayleigh number we have considered (107), the theoretical Nusselt numbers agree well with experiment.

The onset of double-diffusive convection is discussed for a layer of fluid in which the vertical salinity gradient varies with depth and for which the thermal and saline Rayleigh numbers R and Rs are large. These conditions are similar to those that exist in a solar pond prior to the onset of any instability. It is shown that when convection occurs it takes the form of an overstable mode and is essentially confined to a narrow region of vertical extent $\sim R_{\rm s}^{-\frac{1}{14}} \times $ depth of the fluid layer, centred at the critical depth where the salt gradient is smallest. The leading terms in asymptotic expansions of the ratio R/Rs, the frequency of oscillation p and the horizontal wavenumber a are determined for R, [Gt ] 1. The results predicted by the theory are shown to be in good agreement with numerical results and with observations of solar ponds.

An investigation on novel lines is made into the problem of water waves according to the perfect-fluid model, with reference to wave motions in both two and three space dimensions and with allowance for surface tension. Attention to the Hamiltonian structure of the complete nonlinear problem and the use of methods based on infinitesimal-transformation theory provide a Systematic account of symmetries inherent to the problem and of corresponding conservation laws.

The introduction includes an outline of relevant elements from Hamiltonian theory (§ 1.1) and a brief discussion of implications that the present findings may carry for the approximate mathematical modelling of water waves (§1.2). Details of the hydrodynamic problem are recalled in §2. Then in §3 questions about the regularity of solutions are put in perspective, and a general interpretation is expounded regarding the phenomenon of wave-breaking as the termination of smooth Hamil- tonian evolution. In §4 complete symmetry groups are given for several versions of the water-wave problem : easily understood forms of the main results are listed first in §4.1, and the systematic derivations of them are explained in §4.2. Conservation laws implied by the one-parameter subgroups of the full symmetry groups are worked out in §5, where a recent extension of Noether's theorem is applied relying on the Hamiltonian structure of the problem. The physical meanings of the conservation laws revealed in §5, to an extent abstractly there, are examined fully in §6 and various new insights into the water-wave problem are presented.

In Appendix 1 the parameterized version of the problem is considered, covering cases where the elevation of the free surface is not a single-valued function of horizontal position. I n Appendix 2 a general method for finding the symmetry groups of free-boundary problems is explained, and the exposition includes the mathematical material underlying the particular applications in §§4 and 5.

The robustness of localized states that transport energy and mass is assessed by a numerical study of the Euler equation in two space dimensions. The localized states are the translating ‘V-states’ discovered by Deem & Zabusky. These piecewise- constant dipolar (i.e. oppositely-signed ± or ±) vorticity regions are steady translating solutions of the Euler equations. A new adaptive contour-dynamical algorithm with curvature-controlled node insertion and removal is used. The evolution of one V-state, subject to a symmetric-plus-asymmetric perturbation is examined and stable (i.e. non-divergent) fluctuations are observed. For scattering interactions, coaxial head-on (or ± on ±) and head-tail (or & on ±) arrangements are studied. The temporal variation of contour curvature and perimeter after V-states separate indicate that internal degrees of freedom have been excited. For weak interactions we observe phase shifts and the near recurrence to initial states. When two similar, equal-circulation but unequal-area V-states have a head-on interaction a new asymmetric state is created by contour ‘exchange’. There is strong evidence that this is near to a V-state. For strong interactions we observe phase shifts, ‘breaking’ (filament formation) and, for head-tail interactions, merger of like-signed vorticity regions.

I n this paper the streamwise growth of a passive contaminant cloud in the laminar flow within a uniform conduit is investigated. A probabilistic formulation, based on the Lagrangian motion of typical marked fluid molecules, is used to gain insight into the complex dispersion problem that exists at times that are significantly smaller (and often of more practical relevance) than those required for the asymptotic case discussed by Taylor (1953) to be valid.

Previous investigations of the small-time spread of a contaminant cloud in a tube by Lighthill (1966) and Chatwin (1976, 1977) were primarily concerned with a cloud near a tube axis as may be appropriate to an injection into the flow in arteries. When the contaminant is more uniformly spread over the conduit cross-section it is shown that, even at quite small times, the conduit boundary has a very pronounced influence on the streamwise contaminant distribution. Such a situation occurs, for example, when extracting sample fluid from a flow by means of a small-diameter sampling tube.

The streamwise spread of the contaminant cloud that results from an initial sheet of contaminant, spread uniformly over the conduit cross-section, is shown to depend critically on the Lagrangian mean-velocity history of a typical fluid molecule. This mean-velocity history function generally (and necessarily) is distinguished by a ‘hump’ whose location is determined by the proximity of the molecule's release position to the nearest conduit boundary. The ‘hump’ is a more-pronounced feature for release positions near a boundary than i t is for contaminant molecules released near the conduit centre, where the ‘hump’ becomes almost indiscernible.

The specific case of flow between parallel plates is investigated using a random-walk model of the process. A significant difference is found from the results of an analysis that excludes the influence of the conduit boundaries on the streamwise contaminant distribution at times $t = O[d^4/\kappa U^2]^{\frac{1}{3}}$, where U is the mean-flow velocity, κ is the molecular diffusivity and d is the plate separation distance.

First this paper describes in detail an apparatus for heat-transport measurements in shallow horizontal layers of fluid at low temperatures. Then high-precision results of convective heat transport as a function of the Rayleigh number R are presented for cylindrical cells of aspect ratio L = 2.08,4.72 and 57. The present paper concentrates on the long-time behaviour of Boussinesq systems. Non-Boussinesq effects, transient effects near the convective onset, and time-dependent states are described elsewhere (Walden & Ahlers 1981 Ahlers et al. 1981 Ahlers 1980b and references therein). The measurements show that the convective onset near the critical Rayleigh number Rc is sharp within the experimental resolution of about 0.1 % of the Nusselt number N even in laterally finite containers. Values of R and of the initial slopes of N(R), are obtained and compared with predictions for different flow patterns. Over a wider range of R and for L = 57 and 4.72, N was found within experimental resolution to be a unique, continuous function of R For L = 2.08, hysteretic transitions are revealed by N(R) near R ≈ 3 and R ≈ 10. For L = 4.72, the effect of impulsive heating was studied and revealed complicated, long-lived, but surprisingly repro- ducible transients.

Using an automated laser-Doppler scanning technique, we have performed an extensive study of pattern evolution and instabilities in a large Rayleigh-BBnard cell (20 by 30 times the layer depth) at moderate Prandtl number (2.5). This work differs from earlier experiments in that the Doppler mapping technique permits both the spatial structure and time evolution of the velocity field to be quantitatively studied, and runs lasting up to ten thousand vertical thermal diffusion times are presented. We observe and document the properties of three qualitatively different regimes above the critical Rayeigh number Rc. (i) Below 5Rc, there are a number of simple patterns in which the rolls align perpendicular to all lateral cell boundaries. The patterns are therefore curved and generally contain two boundary-related defects. A change in R induces patterns with many defects, which evolve toward the simple patterns over extraordinarily long times. These features seem to be consistent with theoretical models based on the competition between boundary, curvature and defect contributions to a Liapunov functional. However, stable states are not always reached. (ii) Above 5Rc the skewed-varicose instability of Busse & Clever causes progressively faster broadband time dependence with a spectral tail falling off approximately as the fourth power of the frequency. Doppler imaging shows that the fluctuations are caused by narrowing and pinching off of the rolls. (iii) Above 9Rc both the oscillatory and skewed-varicose instabilities cause local velocity fluctuations. However, substantial mean roll structure persists even over a 40 h period (one horizontal thermal diffusion time) at 1 5Rc. Velocity power spectra with two distinct maxima associated with the two instabilities are still resolved at 50Rc. Finally, we impose thermal inhomogeneities in order to pin the rolls, and show that the fluctuations are suppressed only if the local heat flux is a significant fraction of the convective heat transport per wavelength.

The fluid motion, temperature distribution and the mass-transfer problem of a binary gas mixture in a rapidly rotating centrifuge are investigated. The model centrifuge considered consists of a pair of concentric circular cylinders bounded on the top and bottom by horizontal end plates; the apparatus rotates rapidly about the axis of the cylinders. During steady operation a binary gas mixture containing species A and B is injected into and withdrawn from the centrifuge through axisymmetric slots located on the sidewalls. Solutions for the velocity, temperature and mass-fraction fields within the centrifuge are obtained for mechanically or thermally driven centrifuges. For the mass-transfer problem, a detailed analysis of the fluid-mechanical boundary layers is required, and, in particular, mass fluxes within the boundary layers are obtained for a wide range of source-sink geometries. Solutions to the mass-transfer problem are obtained for moderately and strongly forced flows in the container; the dependence of the separation (or enrichment) factor on centrifuge configuration, rotational speed and fraction of the volumetric flow rate extracted at the product port (the cut) are predicted.

Observations of inertial waves generated by uniform horizontal flow over ridges and truncated axisymmetric obstacles in a homogeneous fluid rotating about a vertical axis are discussed and compared with linear theory. The dependence of the flow on obstacle shape, Ro, H, E and E is investigated. Here Ro = U/2ΩL is the Rossby number, H = Ro(D/L), E = v/2ΩL2 is the Ekman number, and ε = h/L is the non-dimensional height of the obstacle, where U is the basic velocity, Ω is the angular frequency, L is a streamwise length, D is the depth of the fluid, h is the height of the obstacle, and v is the kinematic viscosity. Previous linear analysis of this problem has been for the limit H fixed, Ro→0, referred to here as the small-Ro limit. However, it is shown that certain linear terms neglected in the small-Ro limit can be important for finite Ro, and are included in the analysis given here. The observed flow is then well described by linear theory for H/ε [Gt ] 1, particularly in the case of two-dimensional flow over a ridge. However, for H/ε [Gt ] 1 the flow field is dominated by a vertical columnar motion, which is not adequately described by the analysis.

This paper presents a new nonlinear analytical method based on the Fourier-series expansion for cavity flows, by which we can systematically deal with curved bodies of arbitrary shape. Furthermore, in the present study, the momentum defect within the cavity wake is reasonably well estimated by the displacement effect through the momentum theorem. For two-dimensional symmetric flows around various bluff bodies, theoretical predictions are shown quantitatively and compared with experi- mental data, wherever possible.

We consider separated oscillatory flows in asymmetric channels when the Strouhal number is sufficiently great for the flow not to be quasi-steady, but where the flow is not dominated by viscous effects. In this range, we show that numerical calculations predict the expansion of vortices during a deceleration and we investigate these flows in several different geometries. These calculations are supported by experimental observations of the motion of small particles immersed in water.

This note is to show that in the presence of surface tension small progressive waves can always exist in water having an inertial surface composed of uniformly distributed floating particles, in contrast to the known result in the absence of surface tension that precludes propagation under a surface that is too heavy.

A formalism that accounts for inertial and diffusive effects in the dynamics of a dilute gas-particle suspension is introduced. The treatment is purely deterministic away from a very thin Brownian diffusion sublayer, while, within the sublayer, inertial effects are small, permitting a near-equilibrium expansion in powers of the Stokes number (particle relaxation time divided by flow characteristic residence time). This expansion provides phenomenological expressions for the particle velocity including two terms : the standard Brownian diffusion, and an additional inertial drift velocity which is closely related to the pressure diffusion term of the Chapman-Enskog expansion. As an example, the general formalism is applied in detail to the case of Stokes flow about a sphere, and sketched for the similar case of a cylinder. Two competing mechanisms are seen to affect the total rate of particle capture by the sphere : (if the stagnation-point region is considerably enriched in particles owing to the high compressibility of the particle phase, which leads to locally enhanced deposition; (ii) centrifugal forces tend to deplete the Brownian diffusion sublayer of particles, reducing diffusion rates away from the stagnation point to the surface. The first effect is seen to dominate over the second except in a very narrow zone of small Stokes numbers. Our method bridges the gap between Levich's solution for the ‘pure-diffusion’ limit and Michael's treatment in the ‘pure-inertia’ limit.