The inviscid instability of Stokes layers is investigated. The Stokes layer is shown to support inviscid Floquet modes at sufficiently high values of the disturbance wavenumbers. Other non-Floquet modes are investigated and are shown to be the likely cause for instability in Stokes layers. These modes intersect with a viscous continuous spectrum of disturbances and it is this interaction which enables free-stream disturbances to penetrate into the boundary layer and amplify exponentially. The relevance of the inviscid theory to previous viscous instability calculations for Stokes layers is discussed.

Linear stability of flow in a diverging–converging channel is considered. The flow may develop under either the fixed mass or the fixed pressure gradient constraint. Both cases are considered. It is shown that under certain conditions the divergence–convergence of the channel leads to the formation of a secondary flow in the form of streamwise vortices. It is argued that the instability is driven by centrifugal effect. The instability has two modes and conditions leading to their onset have been identified. These conditions depend on the amplitude and the length of the channel diverging–converging section and can be expressed in terms of a critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number required to create the instability for the specified amplitude of the variations of the channel opening are also given. It is shown that the flow developed under the fixed mass constraint is slightly more unstable than the flow developed under the fixed pressure constraint. This difference increases with an increase of the amplitude of the channel divergence–convergence.

An experimental and theoretical study on the effect of boundary layer suction on the laminar–turbulent transition process has been carried out. In the study an asymptotic suction boundary layer was established in a wind tunnel with a free-stream velocity of 5.0 m s$^{-1}$. Wall-normal suction (suction velocity 1.44 cm s$^{-1}$) was applied over a large area and the boundary layer was nearly constant over a length of 1800 mm. Measurements were made both with and without suction so comparisons between the two cases could easily be made. Measurements of the development of the mean velocity distribution showed good agreement with theory. The Reynolds number based on the displacement thickness for the suction boundary layer was 347. Experiments on both the development of forced Tollmien–Schlichting (TS) waves and boundary layer disturbances introduced by free-stream turbulence were carried out. Spatial linear stability calculations for TS-waves, where the wall-normal velocity component is accounted for, were carried out for comparison with the experiments. This comparison shows satisfactory agreement even though the stability of the asymptotic suction profile is somewhat overpredicted by the theory. Free-stream turbulence (FST) was generated by three different grids, giving turbulence intensities at the leading edge of the plate between 1.4% and 4.0%. The FST induces disturbances in the boundary layer and it was shown that for the present suction rate the disturbance level inside the boundary layer is constant and becomes proportional to the FST intensity. In all cases transition was prevented when suction was applied whereas without suction the two highest levels of grid turbulence gave rise to transition. Despite a twofold reduction in the boundary layer thickness in the suction case compared to the no suction case the spanwise scale of the streaky structures was almost constant.

We study advection–diffusion of a passive scalar, $T$, by an incompressible fluid in a closed vessel bounded by walls impermeable to the fluid. Variations in $T$ are produced by prescribing a steady non-uniform distribution of $T$ at the boundary. Because there is no flow through the walls, molecular diffusion, $\kappa$, is essential in ‘lifting’ $T$ off the boundary and into the interior where the velocity field acts to intensify $\bnabla T$. We prove that as $\kappa \to 0$ (with the fluid velocity fixed) this diffusive lifting is a feeble source of scalar variance. Consequently the scalar dissipation rate $\chi$ – the volume integral of $\kappa |\bnabla T|^2$ – vanishes in the limit $\kappa \to 0$. Thus, in this particular closed-flow configuration, it is not possible to maintain a constant supply of scalar variance as $\kappa\to 0$ and the fundamental premise of scaling theories for passive scalar cascades is violated.

We also obtain a weaker bound on $\chi$ when the transported field is a dynamically active scalar, such as temperature. This bound applies to the Rayleigh–Bénard configuration in which $T=\pm 1$ on two parallel plates at $z=\pm h/2$. In this case we show that $\chi \leq 3.252\times (\kappa \varepsilon/\nu h^2)^{1/3}$ where $\nu$ is the viscosity and $\varepsilon$ is the mechanical energy dissipation per unit mass. Thus, provided that $\varepsilon$ and $\nu/\kappa$ are non-zero in the limit $\kappa\rightarrow 0$, $\chi$ might remain non-zero.

In the atmospheric surface layer, the wavelength of the peak in the vertical velocity spectrum $\Lambda_w$ decreases with increasing stable stratification and proximity to the surface and this dependence constrains our ability to perform high-Reynolds-number large-eddy simulation (LES). Near the ground, the LES filter cutoff $\Delta_f$ is comparable to or larger than $\Lambda_w$ and as a result the subfilter-scale (SFS) fluxes in LES are always significant and their contribution to the total flux grows with increasing stability.

We use the three-dimensional turbulence data collected during the Horizontal Array Turbulence Study (HATS) field program to construct SFS fluxes and variances that are modelled in LES codes. Detailed analysis of the measured SFS motions shows that the ratio $\Lambda_w/\Delta_f$ contains the essential information about stratification, vertical distance above the surface, and filter size, and this ratio allows us to connect measurements of SFS variables with LES applications. We find that the SFS fluxes and variances collapse reasonably well for atmospheric conditions and filter widths in the range $\Lambda_w/\Delta_f = [0.2,15]$. The SFS variances are anisotropic and the SFS energy is non-inertial, exhibiting a strong dependence on the stratification, large-scale shear, and proximity to the surface. SFS flux decomposition into modified-Leonard, cross-, and Reynolds terms illustrates that these terms are of comparable magnitude and scale content at large $\Lambda_w/\Delta_f$. As $\Lambda_w/\Delta_f \rightarrow 0$, the SFS flux approaches the-ensemble-average flux and is dominated by the Reynolds term. Backscatter of energy from the SFS motions to the resolved fields is small in the bulk of the surface layer, less than 20% for $\Lambda_w/\Delta_{f} < 2$.

A priori testing of typical SFS models using the HATS dataset shows that the turbulent kinetic energy and Smagorinsky model coefficients $C_k$ and $C_s$ depend on $\Lambda_w/\Delta_f$ and are smaller than theoretical estimates based on the assumption of a sharp spectral cutoff filter in the inertial range. $C_k$ and $C_s$ approach zero for small $\Lambda_w/\Delta_f$. Much higher correlations between measured and modelled SFS fluxes are obtained with a mixed SFS model that explicitly includes the modified-Leonard term. The eddy-viscosity model coefficients still retain a significant dependence on $\Lambda_w/\Delta_f$ with the mixed model. A dissipation model of the form $\epsilon = C_\epsilon E_s^{3/2}/\Delta_f$ is not universal across the range of $\Lambda_w/\Delta_f$ typical of atmospheric LES applications. The inclusion of a shear-stability-dependent length scale (Canuto & Cheng 1997) captures a large fraction of the variation in the eddy-viscosity and dissipation model coefficients.

Based on linear inviscid theory, a two-dimensional source oscillating with frequency $\omega_{0}$ in a uniformly stratified (constant Brunt–Väisälä frequency $N_{0}$) Boussinesq fluid induces a steady-state wave pattern, also known as St Andrew's Cross, that features four straight wave beams stretching radially outwards from the source at angles $\pm\cos^{-1}(\omega_{0}/N_{0})$ relative to the vertical. Similar wave beams are generated by oscillatory stratified flow over topography and also appear in simulations of thunderstorm-generated gravity waves in the atmosphere. Uniform plane-wave beams of infinite extent are in fact exact solutions of the nonlinear inviscid equations of motion, and this property is used here to study the propagation of finite-amplitude wave beams taking into account weak viscous and refraction effects. Oblique beams ($\omega_{0}\,{<}\,N_{0}$) are considered first and an amplitude-evolution equation is derived assuming slow modulations along the beam direction. Remarkably, the leading-order nonlinear terms cancel out in this evolution equation and, as a result, the steady-state similarity solution of Thomas & Stevenson (1972) for linear viscous beams is also valid in the nonlinear régime. Moreover, for the same reason, nonlinear effects are found to be relatively unimportant for two-dimensional and axisymmetric beams that propagate nearly vertically ($\omega_{0}\,{\approx}\,N_{0}$) in a Boussinesq fluid. Owing to the fact that the group velocity vanishes when $\omega_{0}\,{=}\,N_{0}$, however, the transient evolution of nearly vertical beams takes place on a slower time scale than that of oblique beams; this is shown to account for the discrepancies between the steady-state similarity solution of Gordon & Stevenson (1972) and their experimental observations. Finally, the present asymptotic theory is used to study the refraction of nearly vertical nonlinear beams in the presence of background shear and variations in the Brunt–Väisälä frequency.

We report the results of an experimental study of magnetohydrodynamic damping of sidewall convection in a rectangular enclosure filled with gallium. In particular we investigate the suppression of convection when a steady magnetic field is applied separately in each of the three principal directions of the flow. The strongest damping of the steady flow is found for a vertical magnetic field, which is in agreement with theory. However, we observe that the application of a field transverse to the flow provides greater damping than a longitudinal one, which seems to contradict available theory. We provide a possible resolution of this apparent dichotomy in terms of the length scale of the experiment.

The spectrum of linear modes governing the multidimensional instabilities of gaseous detonations is revisited by combining a numerical analysis with new analytical results. In view of recent develop`ments in nonlinear analyses for describing the cellular structure of weakly unstable detonation fronts, particular attention is paid to the neighbourhood of the instability threshold. A first objective is to check the validity domain of the analytical results and to investigate to what extent they are useful when approaching the self-sustained regime (Chapman–Jouguet conditions). A second objective is to study how the multidimensional instabilities are influenced by multiple-step chemistry. The roles of the induction period and of the stiffness of the exothermic runaway will be investigated separately.

The equilibrium states of homogeneous turbulent buoyant flows are investigated through a fixed-point analysis of the evolution equations for the Reynolds stress anisotropy tensor and the scaled heat flux vector. The mean velocity and thermal fields are assumed to be two-dimensional. Scalar invariants formed from the Reynolds stress anisotropy tensor, the scaled heat flux vector, and the strain rate and rotation rate tensors are governed by a closed set of algebraic equations derived for the stress anisotropy and scaled heat flux under a (weak) equilibrium assumption. Six equilibrium state variables are identified for the buoyant case and contrasted with the corresponding two state variables obtained for the non-buoyant homogeneous turbulence case. These results, while dependent on the functional forms of the models for the pressure–strain rate correlation tensor and the pressure–scalar-gradient correlation and viscous dissipation vector, can be used as in the non-buoyant case to either calibrate new closure models or validate the performance of existing models. In addition, since the analysis only involves the turbulent time scales (both velocity and thermal) and their ratio, the results of the analysis are independent of the specific models for the dissipation rates of the turbulent kinetic energy and the temperature variance. The analytical results are compared with model predictions as well as recent direct numerical simulation (DNS) data for buoyant shear flows. Good agreement with DNS data is obtained.

It is shown that there exist bodies such that in two-dimensional steady inviscid incompressible flow the pressure gradient is favourable over the entire surface of the body, and the lift is non-zero, if the body is immersed in a uniform stream and there are also two trapped point vortices.

A two-dimensional wake-like compound flow, formed by a low-speed stream embedded within a high-speed flow, is examined in this article. It is shown that the range of absolutely unstable flow in parameter space greatly increases when such a flow is confined within a duct. Parameters studied here are: the density ratio, which is from 0.1 to 1000; the velocity ratio, which varies from co-flow to counter-flow; and the ratio of the duct width to the width of the central jet. Absolutely unstable flows permit perturbations to propagate upstream, and can lead to self-sustained global oscillations similar to the vortex shedding process which takes place in the wake of a bluff body. This theoretical situation models the wake-like behaviour of a two-fluid coaxial injector with a recessed central tube. The aerodynamic destabilizing mechanism is of primary importance whereas the stabilizing mechanisms, which are not considered here, are of secondary importance. The conclusions from this analysis of a ducted compound flow can explain why one observes self-sustained oscillations in recessed coaxial injectors. The presence of a recirculation bubble in the central flow, which is the basis of other proposed explanations, is not required.

The added mass interaction of a body of variable volume translating in a fluid depends not only on the relative acceleration, but also on the rate of change of the volume. In the present study this prediction is put to an experimental test by observing the motion of a bubble rising in a pressurized tube. When the pressure is brought back to ambient by a fast-opening valve, the bubble expands and, from an analysis of its acceleration, it is possible to deduce the effect of the volume change on the added mass interaction. The results support the validity of the conventional theory for this effect. Some other observations on the transition from a straight to a spiral or zig-zag trajectory – a regime which may well be called ‘Leonardo's paradox’–are reported.

The air cushioning effect in the gap between an almost inviscid body of water and a nearby solid wall (or another body of water) is studied theoretically and is found to depend on predominantly lubricating forces in the air, in certain applications. The situation in which the density and viscosity in air are taken as small compared with those in water is investigated. In this situation potential-flow dynamics in the water couples with lubrication behaviour in the air, leading to a nonlinear integro-differential system for the evolution of the interface. The numerical values of the main parameters are investigated and indicate a wide range of practical applications. Specifically, the lubrication/inviscid balance holds for typical global Reynolds numbers below the order of the viscosity ratio divided by the cube of the density ratio, i.e. below about 10$^{7}$ in the case of air and water; for Reynolds numbers of that order the lubrication behaviour is replaced by an unsteady boundary-layer response, whereas above that order formally the response is totally inviscid. A variety of spatio-temporal flow solutions are presented for the lubrication/inviscid system and these all indicate a relatively rapid closure of the gap, in a common form which is analysed.

Direct numerical simulation has been performed to explore the turbulence near a freely deformable interface in a countercurrent air–water flow, at a shear Reynolds number $\Re_{\star}=171$. The deformations of the interface fall in the range of capillary waves of waveslope $ak=0.01$, and very small phase speed-to-friction velocity ratio, $c/u_{\star}$. The results for the gas side are compared to open-channel flow data at the same shear Reynolds number, placing emphasis upon the influence of the waves in the interfacial viscosity-affected region, and away from it in the outer core flow. Comparison shows a similarity in the distribution of the turbulence intensities near the interface, confirming that for the range of flow conditions considered, the lighter phase perceives the interface like a flexible solid surface, at least in the limit of non-breaking waves. Overall, in a time-averaged sense, the interfacial motion affects the turbulence in the near-interface region; the most pertinent effect is a general dampening of the turbulent fluctuating field which, in turn, leads to a reduction in the interfacial dissipation. Furthermore, the turbulence is found to be less anisotropic at the interface than at the wall. This is confirmed by the analysis of the pressure–rate-of-strain tensor, where the effect of interfacial motion is shown to decrease the pressure strain correlation in the direction normal to the interface and in the spanwise direction. The analysis of the turbulent kinetic energy and Reynolds stress budgets reveals that the interface deformations mainly affect the so-called boundary term involving the redistribution of energy, i.e. by the action of pressure, turbulent fluctuations and molecular viscosity, and the dissipation terms, leaving the production terms almost unchanged. The non-zero value of the turbulent kinetic energy at the interface, together with the reduced dissipation, implies that the turbulent activity persists near the interface and contributes to accelerating the turbulent transfer mechanisms. Away from the interface, the decomposition of the fluctuating velocity gradient tensor demonstrates that the fluctuating rate-of-strain and rate-of-rotation at the interface influence the flow throughout the boundary layer more vigorously. The study also reveals the streaky structure over the deformable interface to be less organized than over a rigid wall. However, the elongation of the streaks does not seem to be much affected by the interfacial motion. A simple qualitative analysis of the quasi–streamwise vortices using different eduction techniques shows that the interfacial turbulent structures do not change with a change of boundary conditions.