This note presents numerical results on the stability of large-amplitude gravity waves on deep water. The results are then used to predict new two-dimensional superharmonic instabilities. They are due to collisions of eigenvalues of opposite signatures, confirming the recent condition for instability of MacKay & Saffman (1986).

The flow in a two-dimensional curved channel driven by an azimuthal pressure gradient can become linearly unstable owing to axisymmetric perturbations and/or non-axisymmetric perturbations depending on the curvature of the channel and the Reynolds number. For a particular small value of curvature, the critical Reynolds number for both these perturbations becomes identical. In the neighbourhood of this curvature value and critical Reynolds number, non-linear interactions occur between these perturbations. The Stuart-Watson approach is used to derive two coupled Landau equations for the amplitudes of these perturbations. The stability of the various possible states of these perturbations is shown through bifurcation diagrams. Emphasis is given to those cases that have relevance to external flows.

When a shock moves in a non-uniform medium, its motion is influenced both by the non-uniformity ahead of the shock and also by a wave which overtakes the shock. In this paper the overtaking wave is examined in an analytic manner by considering its formation and subsequent propagation. This analysis is then combined with the known ‘freely propagating’ description of the shock motion, in which the overtaking wave is ignored, and results in a description of the shock motion in integral form. For the particular problem of a strong shock propagating in a medium with a power-law density variation, the freely propagating shock law has previously been compared with the available similarity solution. The estimate of the effect of the overtaking wave presented in this paper is shown to provide a significant improvement to the description of the shock motion in this instance.

In Part 1 of this series conservation principles for ring circulation and kinematic swirl angular momentum were developed for general axisymmetric incompressible viscous flow. These principles were then used to classify the four independent axial causes of swirl-free conically similar viscous flow. Part 2 provided a detailed analysis of the one-parameter swirl-free flows that are generated by each one of the axial singularities acting alone. The present paper extends, to swirling flow, the description of the axial singularities that drive axisymmetric viscous flow. In the special case of conically similar viscous flow, two independent half-line sources of swirl angular momentum suffice to complete the set of axial singularities that can generate such swirling flows. The individual strengths of the six independent axial causes provide a complete characterization of all conically similar viscous flows that can be generated in this way. This Part 3 completes the task of analysing in detail the independent one-parameter flows generated by axial causes by studying the flow caused by uniform production of kinematic swirl angular momentum on a half-axis. This flow demonstrates how swirl may induce an axial half-plane flow. For large swirl circulation strengths, swirl angular momentum diffuses and convects so as to fill slightly more than half the space with an almost constant density of swirl angular momentum. A well-developed internal boundary layer, in the form of an outward radial jet, then separates this region from one in which the flow is almost irrotational. The jet entrains two impinging convection fields. The angular location of the jet is determined by relating the axial component of moment of whirl produced at the origin to the strength of the swirling circulation singularity on the axis.

The infinite set of the Lundgren-Monin equations for the multi-point velocity distributions of fluid turbulence is closed by making use of the cross-independence closure hypothesis proposed by Tatsumi (Geometry and Statistics of Turbulence, 2001, p. 3), and the minimum deterministic set of equations is obtained as the equations for the one-point velocity distribution f, the two-point velocity distribution f(2) and the two-point local velocity distribution f(2)*. In practice, the two-point distributions f(2) and f(2)* are more conveniently expressed in terms of the velocity-sum and -difference distributions g+, g− and g+*, g−*, respectively.

As an outstanding result, the energy dissipation rate is expressed in terms of the distribution g− which is mainly contributed from small-scale turbulent fluctuations, making clear analogy with the ‘fluctuation-dissipation theorem’ in non-equilibrium statistical mechanics.

It is to be remarked that the integral moments of the equations for the distributions f and f(2) give the equations for the mean flow and the mean velocity procducts of various orders, which are identical with the corresponding equations derived directly from the Navier--Stokes equation. This results clearly shows the exact consistency of the cross-independence closure and gives an overall solution for the classical closure problem concerning the mean velocity products since they are derived from the known distributions.

Although the present work is confined to the two-point statistics of turbulence, the analysis can be extended to the higher-order statistics and even to turbulence in other fluids such as magneto and quantum fluids.

Results are presented of a preliminary numerical investigation into a three-dimensional laminar boundary layer. It is assumed that the flow is over a developable surface and the boundary conditions at the outer edge of the layer are chosen to be u = U0 + xU1, v = yV1. This choice enables the governing equations to be written in terms of two, and not three, independent variables, viz. x and z. However, the three-dimensionality of the problem gives rise to a coupling of the equations which, not unnaturally, is still present after the elimination of y.

For appropriate values of U1 and V1 it is found possible to integrate the equations approximately from the ‘birth’ of the boundary layer (x = 0) right up to a saddle point of attachment. Calculations have already been made for flow at such attachment points and the comparison of the present results with them is extremely good.

We report that the well-known Marangoni film thickening in surfactant-laden Landau–Levich–Bretherton coating flow can be completely suppressed by wall slip. The analysis is made by mainly looking at how the deposited film thickness varies with the capillary number $Ca$ ($\ll 1$) and the dimensionless slip length ${\it\Lambda}={\it\lambda}/R$ ($\ll 1$) in the presence of a trace amount of insoluble surfactant, where ${\it\lambda}$ is the slip length and $R$ is the radius of the meniscus. When slip effects are weak at sufficiently large $Ca$ (but still $\ll 1$) such that $Ca\gg {\it\Lambda}^{3/2}$, the film thickness can still vary as $Ca^{2/3}$ and be thickened by surfactant as if wall slip were absent. However, when slip effects become strong by lowering $Ca$ to $Ca\ll {\it\Lambda}^{3/2}$, the film, especially when surface diffusion of surfactant is negligible, does not get thinner according to the strong-slip quadratic law reported previously (Liao et al., Phys. Rev. Lett., vol. 111, 2013, 136001; Li et al., J. Fluid Mech., vol. 741, 2014, pp. 200–227). Instead, the film behaves as if both surfactant and wall slip were absent, precisely following the no-slip $2/3$ law without surfactant. Effects of surface diffusion are also examined, revealing three distinct regimes as $Ca$ is varied from small to large values: the strong-slip quadratic scaling without surfactant, the no-slip $2/3$ scaling without surfactant and the film thickening along the no-slip $2/3$ scaling with surfactant. An experiment is also suggested to test the above findings.

Chatwin (1970) has described the approach to normality of a cloud of solute which is injected into a pipe containing solvent in steady laminar flow. This paper is concerned with the modification of Chatwin's theory when there is a small density difference between the solvent and the dissolved solute. Asymptotic series are derived for the induced density currents and for the distribution of solute in the case when the molecular diffusivity is constant throughout the pipe's cross-section. These series lead to the asymptotic forms of the moments of the distribution, thereby describing some additional deviations from normality caused by buoyancy effects at large times. The theory predicts that the additional dispersion due to buoyancy effects is proportional to the square of the Rayleigh number and depends on the Péclet number of the flow. There is excellent agreement between the results and those previously obtained by the author (1976) from Erdogan & Chatwin's (1967) model of dispersing buoyant solutes. The results confirm that Erdogan & Chatwin's intuitive theory correctly models the significant features of the situation for large Schmidt numbers.

Particle image velocimetry (PIV) has been applied to the study of acoustic flow of liquid in a standing wave tube. Even though liquid compressibility is very small, the liquid must be treated as compressible in this case. With the finite compressibility of liquid in mind, a series of different standing wave modes can be formed by pressure waves emanated at specific driving frequencies from a bimorph piezo disk at the end of the tube. In this paper, the first three natural standing wave modes were visualized using 1 μm diameter fluorescent microspheres seeded in the liquid. The variation of the flow field in the acoustic boundary layer near the wall was measured using PIV. Water was first used as a working fluid. Experiments were then carried out with a glycerol–water mixture (50%–50% by volume) to examine the effect of viscosity change on the wave propagation and flow structure inside the tube. The experimental results are compared with theoretical model predictions.

We study the linear temporal hydrodynamic stability in the Rayleigh–Bénard problem for a compressible fluid (a perfect gas) under marginally super-adiabatic conditions, i.e. when the ambient temperature gradient only slightly exceeds the adiabatic gradient and then only within the fluid adjacent to the upper (cold) wall. The onset of convection in this limit demonstrates some unique features which differ qualitatively from those of the familiar Boussinesq approximation. Thus, the ensuing convection is effectively confined to a narrow domain of the fluid close to the upper wall and is characterized by large wavenumbers. Furthermore, these distinct attributes persist with diminishing temperature difference, implying that the prevailing generalized Boussinesq approximation (based on the use of the potential temperature gradient) is non-uniform in the present limit. This non-uniformity is resolved in terms of the small yet significant variations of fluid properties (which are commonly neglected). We comment on the analogy between the present problem and the Taylor–Couette problem for a viscous incompressible fluid within a narrow gap between counter-rotating cylinders. We briefly discuss the potential relevance of the present limit to some recent observations of the onset of convection within near-critical fluids.

We report new experiments with the ‘sliced-cylinder’ β-plane model of Pedlosky & Greenspan (1967) and Beardsley (1969), but with a much wider basin such that the western boundary current and its eddies occupy a small fraction of the basin width. These experiments provide new insights into nonlinear aspects of the flow: the critical conditions for boundary current separation and the transition from stable to unstable flow are redefined, and a further transition from periodic to chaotic eddy shedding under strong anticyclonic forcing is also found. In the nonlinear regimes the western boundary current separates from the western wall and shoots into the interior as a narrow jet that undergoes a rapid adjustment to join with the broad slow interior flow. In the unstable regimes this adjustment involves eddy shedding. Each transition occurs at a fixed critical value of a Reynolds number Reγ based on the velocity and width scales for a purely viscous boundary current: the flow is unstable for Reγ > 123±4 and aperiodic for Reγ > 231±5. The results provide evidence that the mechanism causing instability is shear in the separated jet rather than the breaking of a large-amplitude Rossby wave. A quasi-geostrophic numerical model applied to the laboratory conditions yields a stability boundary and detailed characteristics of the flow largely consistent with those determined from the experiments. It also reveals a strong dependence of the circulation pattern on basin aspect ratio, and shows that an adverse higher-order pressure gradient is responsible for western boundary current separation in this model. Eddy–eddy interactions and feedback of fluctuations from the eddy formation region to upstream parts of the boundary current contribute to aperiodic behaviour. As a result of eddy shedding, passive tracer from each streamline in the boundary current can be stirred across much of the width of the basin.

A study of the unique role of impurities in the initial stages of ionization relaxation in shock-heated argon, using a sampling mass spectrometer to determine the ionic products of the reaction, is described. The ions are extracted from the shock tube through a small orifice in the end wall after they have diffused through the dense thermal layer adjacent to the wall from the ionizing gas behind the reflected shock wave. The ion diffusion is analysed in detail to assess the possibility that the sampling process alters the reaction products. It is shown that this is unlikely because the impurities are in dilute concentration and the reaction is studied in its initial stages. This mode of sampling is compared with others.

The experiments were conducted in argon at temperature of 16,600 °K and pressure of 16 mmHg with an estimated impurity level of 300 parts per million. A surprisingly large number of different ions were detected during the initial stages of ionization. O+ and H+ were found in much greater amounts than any of the other products, each being about five times more abundant than A+. The results suggest that H2O is probably quite generally the most important impurity in thermal-ionization experiments, and that ionization ‘incubation’ is due to dissociation of molecular impurities (especially H2O) before ionization commences. Possible explanations of the well-known efficiency of small amounts of impurities in initiating ionization are discussed.

A statically stable stratification with buoyancy frequency N2(z) = z2 is found to cause large changes in the modal structure for viscous plane Couette flow (as compared with the case when N2(z) = 1) and it also has a strongly destabilizing effect on the flow. On minimizing with respect to both wavenumber and Richardson number, it is found that the flow is unstable if the Reynolds number is greater than about 183. A study of the Reynolds stress and vertical buoyancy flux shows that there is a large transfer of energy from the basic flow to the velocity disturbances and this is consistent with such a surprisingly low value of the minimum critical Reynolds number.

Marginal stability arguments are used to describe the rotation number dependence of torque in Taylor–Couette (TC) flow for radius ratios $\unicode[STIX]{x1D702}\geqslant 0.9$ and shear Reynolds number $\mathit{Re}_{S}=2\times 10^{4}$. With an approximate representation of the mean profile by piecewise linear functions, characterised by the boundary-layer thicknesses at the inner and outer cylinder and the angular momentum in the centre, profiles and torques are extracted from the requirement that the boundary layers represent marginally stable TC subsystems and that the torque at the inner and outer cylinder coincide. This model then explains the broad shoulder in the torque as a function of rotation number near $R_{\unicode[STIX]{x1D6FA}}\approx 0.2$. For rotation numbers $R_{\unicode[STIX]{x1D6FA}}<0.07$ the TC stability conditions predict boundary layers in which the shear Reynolds numbers are very large. Assuming that the TC instability is bypassed by some shear instability, a second narrower maximum in torque appears, in very good agreement with numerical simulations. The results show that marginal stability theory, despite its shortcomings in other cases, can explain quantitatively the non-monotonic torque variation with rotation number for both the broad maximum as well as the narrow maximum.

Numerical finite-difference results of the full axisymmetric incompressible Navier–Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a cylindrical container. The governing parameters are the Ekman number, E, the Rossby number, Ro, and the dimensionless height of the container, H (with respect to the diameter of the particle). The study concerns small values of E, Ro, and HE−1/2 and compares the numerical results with predictions of previous analytical (mostly approximate) studies. Special attention is focused on the drag force. First, developed (quasi-steady state) cases are considered. Excellent agreement with the exact linear (Ro = 0) solution of Ungarish & Vedensky (1995) is obtained when the computational Ro = 10−4. The effects of the nonlinear momentum advection terms are analysed and shown to be proportional to RoE−1/2. Next, the time-development for both (a) impulsive start and (b) start under a constant axial force are considered, and good qualitative agreement with previous analytical results (including the appearance of oscillations in case (b)) is indicated.

A comprehensive set of exact solutions to the first-order boundary-layer equations has been computed using the finite difference computer programme of Sells, with and without wall cooling. The effects of Prandtl number, wall cooling and Mach number on separation point location were studied. Values of displacement thickness, skin friction coefficient and Stanton number are displayed graphically for the supersonic flow over a circular concave arc, for a subsonic cooled cylinder and for the case of a linearly retarded velocity distribution. The influence of pressure gradient on recovery factor was studied. Velocity and temperature profiles are shown for four cold wall cases. The exact computer results show the errors in many of the more approximate methods available for the case where Ue = U∞(1 - X/L). The importance of second-order effects and the applicability of a first-order solution are discussed briefly.

In this study we have theoretically investigated the effect of parallel superposition of modulation on the stability of single-layer Newtonian and viscoelastic flows down an inclined plane. Specifically, a spectrally based numerical technique in conjunction with Floquet theory has been used to investigate the linear stability of this class of flows. Based on these analyses we have demonstrated that parallel superposition of modulation can be used to stabilize or destabilize flow of Newtonian and viscoelastic fluids down an inclined plane. In general at low Reynolds number Re (i.e. O(1)) and in the limit of long and O(1) waves the effect of dynamic modulation on the stability of viscoelastic flows is much more pronounced; however, relatively large modulation amplitudes are required to achieve significant stabilization/destabilization. In addition, the dependence of the most dominant modulation frequencies on Re and the Weissenberg number We have been identified. Specifically, it has been shown that for Newtonian flows low-frequency modulations are destabilizing and the most dominant modulation frequency scales with 1/Re. On the other hand, for viscoelastic flows in the absence of fluid inertia low-frequency modulations are stabilizing and the most dominant modulation frequency scales with 1/We. In finite-Re viscoelastic flows the most dominant destabilizing modulation frequency scales with 1/Re while the most stabilizing modulation frequency scales with 1/WeRe. Finally, it has been demonstrated that the mechanism of both purely elastic and inertial instabilities in flows down an inclined plane is unchanged in the presence of dynamic modulation.