In this paper, we describe a theoretical asymptotic model for large-amplitude travelling solitary waves in an axially symmetric rotating flow of an inviscid incompressible fluid confined in an infinitely long circular tube. By considering the special, but important, case when the upstream flow is close to that of uniform axial flow and uniform rotation, we are able to construct analytical solutions which describe solitary waves with ‘bubbles’, that is, recirculation zones with reversed flow, located on the axis of the tube. Such waves have amplitudes which slightly exceed the critical amplitude, where there is incipient flow reversal. The effect of the recirculation zone is to introduce into the governing amplitude equation an extra nonlinear term, which is proportional to the square of the difference between the wave amplitude and the critical amplitude. We consider in detail a special, but representative, class of upstream inflow conditions. We find that although the structure of the recirculation zone is universal, the presence of such solitary waves is quite sensitive to the actual upstream axial and rotational velocity shear configurations. Our results are compared with previous theories and observations, and related to the well-known phenomenon of vortex breakdown.

Incompressible slip flow past a semi-infinite flat plat at zero incidence is treated in terms of the linearized viscous flow equations. A formal solution is obtained using Fourier transforms and the Wiener-Hopf technique. Explicit inversion of the transform is not possible, but asymptotic expansions are discussed. These reveal the inadequacy of boundary-layer theory in predicting the nature of the solution, even at the plate surface. For example, the local shear forces on the plate are significantly different from boundary-layer values, even far downstream, where slip effects are small. The boundary-layer limit is approached as the Reynolds number based on the mean free path or, equivalently, the free-stream Mach number tends to infinity.

The stretched-vortex subgrid model is used to run large-eddy simulations of temporal mixing layers at various Reynolds and Schmidt numbers, with different initial and boundary conditions. A self-similar flow is obtained, during which the growth rate, mean velocity and Reynolds stresses are in accord with experimental results. However, predictions of the amount of mixed fluid, and of the variation in its composition across the layer, are excessive, especially at high Schmidt number. More favourable comparisons between experiment and simulation are obtained when the large-scale flow is quasi-two-dimensional; however, such states are not self-similar and not sustainable. Present model assumptions lead to predictions of the continued subgrid spectrum with a viscous cutoff that is dependent on grid resolution.

Direct numerical simulations (DNS) of two inviscid flows, the Taylor–Green flow and two orthogonal interacting Lamb dipoles, together with the DNS of forced isotropic turbulence, were performed to generate data for a comparative study. The isotropic turbulent field was considered after the transient and, in particular, when the velocity derivative skewness oscillates around −0.5. At this time, Rλ ≈ 257 and a one decade wide k−5/3 range was present in the energy spectrum. For the inviscid flows the fields were considered when a wide k−3 range was achieved. This power law spectral decay corresponds to infinite enstrophy and is considered one of the requirements to demonstrate that the Euler equations lead to a finite time singularity (FTS). Flow visualizations and statistics of the strain rate tensor and vorticity components in the principal axes of the strain rate tensor (λ, λ) were used to classify structures. The key role of the intermediate component 2 is demonstrated by its good correlation with enstrophy production. Filtering of the fields shows that the slope of the power law is directly connected to self-similar structures, whose radius of curvature is smaller the steeper the spectrum.

The asymptotic solutions of second-order accuracy taking account of the velocity boundary conditions imposed on the sidewalls are derived for wall-localized boundary-layer convection in a rotating fluid layer in the presence of stress-free or no-slip vertical sidewalls. The second-order asymptotic solutions give a satisfactory quantitative agreement with the fully numerical solutions obtained in a rotating channel with either stress-free or no-slip sidewalls. Furthermore, we show, through fully three-dimensional numerical simulations, that the structure of the boundary-layer convection is highly robust in strongly nonlinear regimes.

Steady, supersonic, dissipative, three-dimensional, axisymmetric flow is considered. A system of Burgers-type equations is shown to govern the flow field. In inviscid regions the Whitham theory gives the limiting form. Dissipative effects ultimately engulf the inviscid zone and at sufficiently large distances from the body the flow is governed by linear dissipative theory. The flow field is divided into zones based on the presence or absence of nonlinearity and dissipation. Estimates and criteria which describe the extent of these zones are given.

We consider the drainage of fluid trapped beneath a two-dimensional drop that sediments towards a horizontal plane. The governing equation is closely related to that for capillary drainage (without gravity) of an annular film discussed in a companion paper (Lister et al., J. Fluid Mech. vol. 552, 2006, p. 311). When drainage starts, dynamical structures rapidly appear that are usually called dimples in the context of sedimentation. Dimples are constant-pressure regions to which most of the fluid in the film is confined, which are analogous to the collars and lobes that appear in annular capillary drainage.

The process of drainage is controlled by a Bond number, $B$, that measures the relative importance of gravity and surface tension for the sedimenting drop. When $B$ is sufficiently small, all the fluid ultimately drains from a single small dimple and the drop takes a static sessile equilibrium shape. The dimple-drainage process is the same as that of a lobe. When $B$ is sufficiently large, several permanent dimples are formed under the drop, and these exhibit complex dynamics of collision and interaction analogous to that of collars and lobes. No static drop shape is reached, even for long times. For critical values of $B$, fluid may be permanently trapped in one or more stationary dimples (analogous to collars), and families of equilibrium drop shapes are found that depend upon the quantity of trapped fluid.

The process of mixing at the top of a turbulent cloud layer contains a new feature which has not been considered in previous studies of mixing; evaporation of liquid water can cause density changes which may affect the dynamics. A model experiment has been devised to study this problem, using liquids whose density behaviour is non-linear to simulate evaporation.

The existence of a moist, stable, turbulent region above cloudtop can be explained using the laboratory results, which suggest that this region can be regarded dynamically as part of the cloud. Comparison of the rates of mixing in the model experiments with and without ‘evaporation’ suggests that evaporation could cause a small decrease in the mixing rate for a given density difference, but the change would be negligible in practice. This result also sheds some light on the mechanism of mixing, in both the linear and non-linear cases.

Fluid dynamics induced by periodically forced flow around a cylinder is analysed computationally for the case when the forcing frequency is much lower than the von Kármán vortex shedding frequency corresponding to the constant flow velocity condition. By using the Koopman mode decomposition approach, we find a new normal form equation that extends the classical Hopf bifurcation normal form by a time-dependent term for Reynolds numbers close to the Hopf bifurcation value. The normal form describes the dynamics of an observable and features a forcing (control) term that multiplies the state, and is thus a parametric – i.e. not an additive – forcing effect. We find that the dynamics of the flow in this regime is characterized by alternating instances of quiescent and strong oscillatory behaviour and that this pattern persists indefinitely. Furthermore, the spectrum of the associated Koopman operator is shown to possess quasi-periodic features. We establish the theoretical underpinnings of this phenomenon – that we name quasi-periodic intermittency – using the new normal form model and show that the dynamics is caused by the tendency of the flow to oscillate between the unstable fixed point and the stable limit cycle of the unforced flow. The quasi-periodic intermittency phenomenon is also characterized by positive finite-time Lyapunov exponents that, over a long period of time, asymptotically approach zero.

Turbulent miscible fountains discharged vertically from a round source into quiescent uniform unbounded environments of density $\unicode[STIX]{x1D70C}_{0}$ are investigated numerically using large-eddy simulations. Both upward and downward fountains are considered. The numerical simulations cover a wide range of the density ratio $\unicode[STIX]{x1D70C}_{i}/\unicode[STIX]{x1D70C}_{0}$, where $\unicode[STIX]{x1D70C}_{i}$ is the source density of the released fluid. These simulations are used to evaluate how the initial maximum height $H_{i}$ and the steady state height $H_{ss}$ of the fountains are affected by large density contrasts, i.e. in the general non-Boussinesq case. For both upward and downward non-Boussinesq fountains, the ratio $\unicode[STIX]{x1D706}=H_{i}/H_{ss}$ remains close to $1.45$, as usually observed for Boussinesq fountains. However the Froude (linear) scaling originally introduced by Turner (J. Fluid Mech., vol. 26 (4), 1966, pp. 779–792) for Boussinesq fountains is no longer valid to determine the steady fountain height. The ratio between $H_{ss}$ and the height predicted by the Turner’s relation turns out to be proportional to $(\unicode[STIX]{x1D70C}_{i}/\unicode[STIX]{x1D70C}_{0})^{n}$. Remarkably, it is found that the power exponent $n$ differs following the direction in which the buoyant fluid is released ($n=1/2$ for downward fountains and $n=3/4$ for upward fountains). This new result demonstrates that for non-Boussinesq turbulent fountains the configurations heavy/light and light/heavy are not equivalent. Finally, scalings are proposed for fountains, regardless of the direction (upwards and downwards) and of the density difference (Boussinesq and non-Boussinesq).

This paper reports on the experimental testing of oxygen-enriched porous fuel injection in a scramjet engine. Fuel was injected via inlet mounted, oxide-based ceramic matrix composite (CMC) injectors on both flow path surfaces that covered a total of 9.2 % of the intake surface area. All experiments were performed at an enthalpy of $3.93{-}4.25\pm 3.2\,\%~\text{MJ}~\text{kg}^{-1}$, flight Mach number 9.2–9.6 and an equivalence ratio of $0.493\pm 3\,\%$. At this condition, the engine was shown to be on the verge of achieving appreciable combustion. Oxygen was then added to the fuel prior to injection such that two distinct enrichment levels were achieved. Combustion was found to increase, by as much as 40 % in terms of combustion-induced pressure rise, over the fuel-only case with increasing oxygen enrichment. Further, the onset of combustion was found to move upstream with increasing levels of oxygen enrichment. Thrust, both uninstalled and specific, and specific impulse were found to be improved with oxygen enrichment. Enhanced fuel–air mixing due to the pre-mixing of oxygen with the fuel together with the porous fuel injection are believed to be the main contributors to the observed enhanced performance of the tested engine.

We consider the dynamics of an axisymmetric, partially wetting droplet of a one-component volatile liquid. The droplet is supported on a smooth superheated substrate and evaporates into a pure vapour atmosphere. In this process, we take the liquid properties to be constant and assume that the vapour phase has poor thermal conductivity and small dynamic viscosity so that we may decouple its dynamics from the dynamics of the liquid phase. This leads to a so-called ‘one-sided’ lubrication-type model for the evolution of the droplet thickness, which accounts for the effects of evaporation, capillarity, gravity, slip and kinetic resistance to evaporation. By asymptotically matching the flow near the contact line region and the bulk of the droplet in the limit of a small slip length and commensurably small evaporation and kinetic resistance effects, we obtain coupled evolution equations for the droplet radius and volume. The predictions of our asymptotic analysis, which also include an estimate of the evaporation time, are found to be in excellent agreement with numerical simulations of the governing lubrication model for a broad range of parameter regimes.

In this paper separation-induced shock reflection is studied theoretically and experimentally. An analytical model is proposed to establish the connections among upstream conditions, downstream conditions and shock configurations. Furthermore, the minimum entropy production principle is employed to determine the incident shock angles as well as the criterion for the transition from regular reflection to Mach reflection, which agrees well with experimental results. Additionally, a solution path for a reflected shock that fulfills the minimum entropy production principle is found in the overall regular reflection domain, based on which the steadiest shock configuration may be determined according to upstream and downstream conditions.

This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of $5000$ and $15\,000$ and an extensive range of Hartmann numbers $0\leqslant \mathit{Ha}\leqslant 800$ were investigated. Three broad regimes are identified in which optimal mode topology and non-modal growth mechanisms are distinct. These regimes, corresponding to low, moderate and high magnetic field strengths, are found to be governed by the independent parameters; Hartmann number, Reynolds number based on the Hartmann layer thickness $R_{H}$ and Reynolds number built upon the Shercliff layer thickness $R_{S}$, respectively. Transition between regimes respectively occurs at $\mathit{Ha}\approx 2$ and no lower than $R_{H}\approx 33.\dot{3}$. Notably for the high Hartmann number regime, quasi-two-dimensional magnetohydrodynamic models are shown to be excellent predictors of not only transient growth magnitudes, but also the fundamental growth mechanisms of linear disturbances. This paves the way for a precise analysis of transition to quasi-two-dimensional turbulence at much higher Hartmann numbers than is currently achievable.

A small water channel network is proposed for designing shallow water fields, and the network is applied to attain water wave cloaking. The design formula is derived from an analogy between waves in a water channel and in an electric circuit; an approach of a transmission line metamaterial is extended to water waves and the water channel is used as an alternative of the transmission line. The size of the water channel is sufficiently smaller than the wavelength and a number of the channels are periodically connected as a network. This small water channel network makes artificial wave fields, and it works for a wide band of frequencies. First, we make an isotropic network equivalent to a shallow water space with constant depth in order to validate the proposed design method. It shows no wave reflection at the interface due to impedance matching. After that, the proposed theory is applied to designing an anisotropic small water channel network for demonstrating shallow water cloaking. A cylinder is cloaked from waves by the network surrounding the cylinder. Both cases are confirmed with numerical computations by solving the boundary-value problem based on linear potential theory.

In this study, direct numerical simulation of the flow around a rotating sphere at high Mach and low Reynolds numbers is conducted to investigate the effects of rotation rate and Mach number upon aerodynamic force coefficients and wake structures. The simulation is carried out by solving the three-dimensional compressible Navier–Stokes equations. A free-stream Reynolds number (based on the free-stream velocity, density and viscosity coefficient and the diameter of the sphere) is set to be between 100 and 300, the free-stream Mach number is set to be between 0.2 and 2.0, and the dimensionless rotation rate defined by the ratio of the free-stream and surface velocities above the equator is set between 0.0 and 1.0. Thus, we have clarified the following points: (1) as free-stream Mach number increased, the increment of the lift coefficient due to rotation was reduced; (2) under subsonic conditions, the drag coefficient increased with increase of the rotation rate, whereas under supersonic conditions, the increment of the drag coefficient was reduced with increasing Mach number; and (3) the mode of the wake structure becomes low-Reynolds-number-like as the Mach number is increased.