Numerical computations of the steady, two-dimensional, incompressible, uniform velocity but stably stratified flow past a normal flat plate (of unit half-width) in a channel are presented. Attention is restricted to cases in which the stratification is weak enough to avoid occurrence of the gravity wave motions familiar in more strongly stratified flows over obstacles. The nature of the flow is explored for channel half-widths, H, in the range 5 [les ] H [les ] 100, for Reynolds numbers, Re, (based on body half-width and the upstream velocity, U) up to 600 and for stratification levels between zero (i.e. neutral flow) and the limit set by the first appearance of waves. The fourth parameter governing the flow is the Schmidt number, Sc, the ratio of the molecular diffusion of the agent providing the stratification to the molecular viscosity. For cases of very large (in the limit, infinite) Sc a novel technique is used, which avoids solving the density equation explicitly. Results are compared with the implications of the asymptotic theory of Chernyshenko & Castro (1996) and with earlier computations of neutral flows over both flat plates and circular cylinders. The qualitative behaviour in the various flow regimes identified by the theory is demonstrated, but it is also shown that in some cases a flow zone additional to those identified by the theory appears and that, in any case, precise agreement would, for most regimes, require very much higher Re and/or H. Some examples of multiple (i.e. non-unique) solutions are shown and we discuss the likelihood of these being genuine, rather than an artefact of the numerical scheme.

Even mild longitudinal wall curvature is known to produce significant effects on the time-averaged turbulent transport in a boundary layer. The present study was undertaken to study the manner in which the instantaneous structure of turbulence in the boundary layer responds to mild streamline curvature. Both convex and concave boundary layers with a boundary-layer thickness to wall radius ratio of about 0·01 were studied. Attention was directed mainly to two events characterizing the instantaneous turbulence structure. These were the so-called ‘bursting’ and ‘zero-crossing’. Quantitative data on the statistics of these events were obtained using a combination of analog instrumentation and visual counting (from continuous film records). These data were compared with data from flat-wall boundary layers obtained from similar signal-processing techniques. The results indicate that neither the individual nor the joint statistics of these events are significantly affected by curvature in the vicinity of the wall. On the other hand, curvature seems to affect appreciably at least some properties of these events at large distances from the wall. Careful examination of these results shows, however, that neither the process of turbulent production near the wall nor the turbulent dissipative process anywhere in the boundary layer is significantly affected by mild curvature. Apparent curvature effects on the instantaneous structure in the outer part of the boundary layer can be explained as being due to the strong effect of streamline curvature on the turbulent diffusion process. This explanation is consistent with previous observations of the time-averaged structure of the flow. The results of the present study indicate the need to re-examine some of the recent turbulence models for curved flows that involve modification of the production and dissipation terms rather than the diffusion term in the transport equations.

When a body of fluid with a vertical salinity and temperature gradient is bounded by a sloping boundary, convective instabilities are often observed. These can occur if the fluid is subjected to heating or the addition of solute at the boundary, or if the boundary is an insulator. These instabilities often take the form of long thin convection cells that are almost horizontal. We present a linear stability analysis of the background states associated with these different boundary conditions and derive criteria for their stability in terms of one non-dimensional parameter, Q. This parameter is related to the Rayleigh number and is a generalization of the similar parameter found by Kerr (1989) in his study of heating a salinity gradient from a vertical boundary. This analysis uses a quasi-static assumption that is valid when the vertical lengthscales of the instabilities are less than the horizontal lengthscales.

A systematic low-frequency theory is developed for the propagation of one-dimensional sound waves in a variable-area duct. The mean flow in the duct is assumed to be isentropic, compressible, and one-dimensional. Two applications are made of the theory. One concerns the reflexion coefficient from a pipe-nozzle combination, in which case comparisons are also made with some experimental data. In the second application, we consider the case of a sonic throat separating subsonic and supersonic flow. In this case, if the mean Mach number distribution in addition to being unity at the throat is also stationary at the throat, there is an axial ‘boundary-layer’ region in which the impedance of the sound wave changes from a fundamentally unsteady (reflexion-free) value at the sonic throat to the quasi-steady value away from the throat.

Two integral theorems are proved which are applicable to the motion of an incompressible fluid in three dimensions. From either of these theorems one can derive the monopole component of the pressure fluctuation at infinity when a bubble undergoes non-spherical oscillations. The results confirm and generalize some recent calculations of this effect (Longuet-Higgins 1989a). They also provide a basis for a physical discussion of the origin of the monopole terms.

The ignition of the supersonic boundary layer flow of a combustible mixture over a flat plate is studied through both direct numerical integration and activation energy asymptotics. Since ignition can be induced through either internally generated viscous heating or heat transfer from a hot wall, analyses are conducted for both an adiabatic wall and an isothermal wall whose temperature can be either higher or lower than the maximum frozen temperature in the flow. The analyses provide a description of the flow structure under various ignition situations, especially the extent of flow non-similarity and the interaction between the inner reaction region and the outer frozen regions. Explicit expressions for the ignition distance are obtained for all ignition situations, and the corresponding effects of the physical parameters on the ignition delay are also assessed. Specifically, it is demonstrated that, for low free-stream Mach number M∞, the ignition distance increases linearly with M∞ because of the decreased residence time, and for high M∞ it decreases exponentially with M∞ because of viscous heating. Results from the asymptotic analyses are found to compare well with those obtained from the direct numerical integration.

This paper describes the solution of Long's problem for steady rotationally symmetric swirling jets in a uniform viscous fluid. Long found these vortices in 1958 by assuming a similarity form of solution, and in 1961 solved the consequent problem in the boundary-layer limit, finding dual solutions. The overall pattern of the solutions to the problem for general values of the Reynolds number is described. The linear spatial stability of the flows to small steady disturbances is analysed and a few results presented. In particular, details of the solutions and their stability are given asymptotically for small and large values of the Reynolds number. The asymptotic results for the basic flow are linked by direct numerical integration of the flow at several finite positive values of the Reynolds number.

We consider instabilities in a fluid with a constant density gradient that is subject to arbitrarily oriented oscillatory accelerations. With the Boussinesq approximation and for the case of an unbounded fluid, transformation to Lagrangian coordinates allows the reduction of the problem to an ordinary differential equation for each three-dimensional wavenumber. The problem has three parameters: the non-dimensional amplitude R of the base-state oscillation, the non-dimensional level of background steady acceleration, which for some cases can be represented in terms of a local (in time) Richardson number Ri, and the Prandtl number Pr. Some general bounds on stability are derived. For Pr = 1 closed-form solutions are found for impulse (delta function) accelerations and a general asymptotic solution is constructed for large R and general imposed accelerations. The asymptotic solution takes advantage of the fact that at large R wave growth is concentrated at ‘zero points’. These are times when the effective vertical wavenumber passes through zero. Kelvin–Helmholtz instabilities are found to dominate at low R while Rayleigh–Taylor instabilities dominate at high. At high R, the uniform shear of the Kelvin–Helmholtz case tends to distort and weaken instability waves. With unsteady flows, Ri = ¼ is no longer an instability limit. Significant instabilities have been found for sinusoidal forcing for Ri up to 0.6.

An exact derivation is given of the dynamo equation in the presence of an arbitrary incompressible mean flow v0. A mixed representation is used specifying v0 in terms of a Lagrangian displacement, while Eulerian coordinates are employed for the turbulent velocity v superimposed on v0.

When the first-order-smoothing approximation is made (FOSA; valid when the turbulent velocity v has a short correlation time τc) the usual dynamo equation is recovered, except that the turbulent velocity v in the tensors αis and βisk is replaced by $\overline{\boldmath v}$. The bar represents the effect of advection and is expressed solely in terms of the Lagrangian coordinate specifying the mean flow v0. Thus the intuitive idea is confirmed that dynamo action depends only on velocity correlation functions measured at a point comoving with the mean flow. The result admits easy evaluation in actual model situations. This is illustrated with an example tailored to the solar dynamo. A shear in v0 causes a (kinematic) anisotropy in the tensors αis and βisk. This can be a large effect, which comes on top of the intrinsic (dynamical) anisotropy in the velocity correlation functions. Subsequently, the analysis is extended beyond FOSA up to arbitrary order, relevant for long correlation times τc on the basis of the work of Van Kampen (1974). It is shown that the same formalism is also applicable to the problem of turbulent transport of a scalar.

Conditions for applicability of the work are (1) very large magnetic Reynolds number, (2) incompressible flows v and v0, (3) stationary mean flow v0, and (4) correlation time τc [Lt ] period of the dynamo.

The dynamics of an asymmetric annulus of vorticity in an incompressible, inviscid twodimensional fluid are experimentally studied using a pure electron plasma. A strict fluid analogy requires that the plasma system behave like an ideal fluid in a frictionless cylindrical container. For certain parameters the asymmetric annulus undergoes a complex evolution which is quite different from that of a symmetric annulus. During the first ‘active’, phase the symmetries grow until the annulus collapses, leaving a large vortex at the device centre. In the second, ‘passive’, phase the remainder of the annulus winds around this central vortex into an ever tighter spiral. Finally, slow shear instabilities destroy the structure of the highly evolved spiral.

The evolution of long waves generated by a pressure disturbance acting on an initially unperturbed free surface in a channel of finite depth is analysed. Both off-critical and transcritical conditions are considered in the context of the fully nonlinear inviscid problem. The solution is achieved by using an accurate boundary integral approach and a time-stepping procedure for the free-surface dynamics.

The discussion emphasizes the comparison between the present results and those provided by both the Boussinesq and the related Korteweg–de Vries model. For small amplitudes of the forcing, the predictions of the asymptotic theories are essentially confirmed. However, for finite intensities of the disturbance, several new features significantly affect the physical results. In particular, the interaction among different wave components, neglected in the Korteweg–de Vries approximation, is crucial in determining the evolution of the wave system. A substantial difference is indeed observed between the solutions of the Korteweg–de Vries equation and those of both the fully nonlinear and the Boussinesq model. For increasing dispersion and fixed nonlinearity the agreement between the Boussinesq and fully nonlinear description is lost, indicating a regime where dispersion becomes dominant.

Consistently with the long-wave modelling, the transcritical regime is characterized by an unsteady flow and a periodic emission of forward-running waves. However, also in this case, quantitative differences are observed between the three models. For larger amplitudes, wave steepening is almost invariably observed as a precursor of the localized breaking commonly detected in the experiments. The process occurs at the crests of either the trailing or the upstream-emitted wave system for Froude numbers slightly sub- and super-critical respectively.

By specializing to the case of unit Lewis number and Prandtl number equal to ¾, a number of general results for the structure of a plane steady compressible flow field, within which chemical energy is being liberated by a simple Arrhenius type of combustion reaction, can be acquired by the use of elementary arguments. The field is of the semi-infinite variety, with a ‘flameholder’ presumed to exist at the origin of coordinates. In these circumstances it is necessary to specify the velocity gradient at inlet to the system or, equivalently, the pressure difference across the field. These quantities cannot be selected arbitrarily, and the nature and extent of the restrictions upon them is fully explored. Since the Mach number of the stream is hypothesized to be a quantity of order unity, local Damköhler numbers are always small. Therefore the field is shown to consist of relatively long regions within which the combustion activity takes place, with embedded thin domains of rapid, almost chemically inert, gasdynamical adjustment, whose dimension is typically that of the conventional shock wave. When the inlet Mach number is less than unity the gasdynamical adjustment domains are always adjacent to the origin, and this is also true under most supersonic inlet conditions.

However, there are some special circumstances for which the shock is detached from the flameholder and is established in the middle of the combustion activity. A specific example is provided by a shock within the induction domain.

These special circumstances are shown to be ultrasensitive to pressure difference across the whole domain. It is also shown that wholly supersonic combustion does exist, but only under similar conditions of extreme sensitivity to pressure difference.

The general arguments are supported and illuminated by asymptotic analysis based on the large activation energy of the Arrhenius reaction. Space precludes a full asymptotic treatment of the combustion activity but a companion paper that analyses these parts of the general field is being prepared in collaboration with D. R. Kassoy. Analysis of the shock within the induction domain, together with results from the case of subsonic inlet Mach numbers, shows that gasdynamical effects can prevent ignition by channelling combustion energy into kinetic energy of the flow at the expense of thermal energy.

The maximum angle of attack for unseparated flow over an airfoil of chord c with finite nose radius of curvature r is shown to be 0.818(r/c)½

We investigate electrophoresis of a tightly fitting sphere of radius $R-h_{0}$ on the axis of a circular tube of radius $R$, using lubrication theory and ideas due to Schnitzer & Yariv (Phys. Fluids, vol. 26, 2014, 122002). The electrical charge clouds on both the cylindrical wall and the surface of the sphere are assumed thin compared to the gap between the sphere and cylinder, so that charge clouds do not overlap and ion exclusion effects are minimal. Nevertheless, non-uniform pumping of counter-ions within the charge clouds leads to a change in the ionic concentration outside the charge clouds in the narrow gap between sphere and cylinder. The electro-osmotic slip velocities at the two surfaces are modified, leading to a decrease in the electrophoretic velocity of the sphere at low Péclet numbers and an increase in the velocity at high Péclet numbers. When the field strength $E_{0}$ is low, it is known that the electrophoretic velocity $U_{0}$ is proportional to $E_{0}(\unicode[STIX]{x1D701}_{s}-\unicode[STIX]{x1D701}_{c})$ which is zero when the zeta potential $\unicode[STIX]{x1D701}_{s}$ on the sphere surface is equal to the zeta potential $\unicode[STIX]{x1D701}_{c}$ on the cylinder. The perturbation to the above low field strength electrophoretic velocity (at high Péclet number) is predicted to be proportional to $E_{0}^{3}(\unicode[STIX]{x1D701}_{s}+\unicode[STIX]{x1D701}_{c})^{2}(\unicode[STIX]{x1D70E}_{s}+\unicode[STIX]{x1D70E}_{c})$, where $\unicode[STIX]{x1D70E}_{s}$ and $\unicode[STIX]{x1D70E}_{c}$ are the surface charge densities on the sphere and cylinder. The choice of materials with similar or identical zeta potentials (and surface charge densities) for the cylinder and sphere should therefore facilitate the observation of velocities nonlinear in the field strength $E_{0}$, since the reference linear electrophoretic velocity will be small.