An analytical vorticity-based model is introduced for steady-state inviscid Boussinesq gravity currents in sheared ambients. The model enforces the conservation of mass and horizontal and vertical momentum, and it does not require any empirical closure assumptions. As a function of the given gravity current height, upstream ambient shear and upstream ambient layer thicknesses, the model predicts the current velocity as well as the downstream ambient layer thicknesses and velocities. In particular, it predicts the existence of gravity currents with a thickness greater than half the channel height, which is confirmed by direct numerical simulation (DNS) results and by an analysis of the energy loss in the flow. For high-Reynolds-number gravity currents exhibiting Kelvin–Helmholtz instabilities along the current/ambient interface, the DNS simulations suggest that for a given shear magnitude, the current height adjusts itself such as to allow for maximum energy dissipation.

The influence of high levels of wall cooling on the stability of hypervelocity boundary layers is investigated. Such conditions are relevant to experiments in high-enthalpy impulse facilities, where the wall temperature is much smaller than the free-stream temperature, as well as to some real flight scenarios. Some effects of wall cooling are well known, for instance, the stabilization of the first mode and destabilization of the second mode. In this paper, several new instability phenomena are investigated that arise only for high Mach numbers and high levels of wall cooling. In particular, certain unstable modes can travel supersonically with respect to the free stream, which changes the nature of the dispersion curve and leads to instability over a much wider band of frequencies. The cause of this phenomenon, the range of parameters for which it occurs and its implications for boundary layer stability are examined. Additionally, growth rates are systematically reported for a wide range of conditions relevant to high-enthalpy impulse facilities, and the stability trends in terms of Mach number and wall temperature are mapped out. Thermal non-equilibrium is included in the analysis and its influence on the stability characteristics of flows in impulse facilities is assessed.

The nonlinear temporal instability of gas-surrounded planar liquid sheets, whose linear instability contains both sinuous and varicose modes, is studied. Both the weakly nonlinear analysis using a second-order perturbation expansion and the numerical simulation using a boundary integral method have been applied. Their comparison shows that the weakly nonlinear analysis can precisely predict the shapes of sheets for most of the time of disturbance evolution and qualitatively explain the instability mechanism when sheets break up. Both the first harmonics of the linear sinuous mode and linear varicose mode are varicose; they contribute to the breakup of sheets, but the first harmonic generated by the coupling between the linear sinuous and varicose modes is sinuous; it plays an important role in modulating the wave profile. The instability with various initial phase differences between the upper and lower interfaces is examined. Except for the varicose initial disturbance, the linear sinuous mode dominates in the shapes of sheets when their amplitudes grow large. Within the second-order analysis, the major modes that can cause the breakup include the linear varicose mode, the first harmonic of the linear sinuous mode and the first harmonic of the linear varicose mode. The effects of various flow parameters have been investigated. At relatively large wavenumbers where approximate analytical and numerical results agree well when sheets break up, increasing the wavenumber reduces the wave amplitude. Reducing the initial disturbance amplitude makes the first harmonic of the linear sinuous mode the dominant mode in causing the breakup. Increasing the Weber number or gas-to-liquid density ratio significantly reduces breakup time and enhances instability.

We provide a detailed physical description of the bubble formation processes taking place in a type of flow where the liquid pressure gradient can be straightforwardly controlled. The analysis, which is supported by an exhaustive experimental study in which the liquid viscosity is varied by three orders of magnitude, provides closed expressions for both the bubbling frequencies and the bubble diameters. Different equations are obtained depending on the values of the three dimensionless parameters characterizing this physical situation, namely the Weber and Reynolds numbers and the gas to liquid flow rate ratio. Since both the inertia dominated and viscous dominated bubbling regimes are simply described in terms of the local pressure gradient and the flow rate ratio, the same types of ideas can be applied in the design of bubble makers in which the pressure gradients are controlled in completely different ways.

We study the propagation of viscous gravity currents along a thin permeable substrate where slow vertical drainage is allowed from the boundary. In particular, we report the effect of this vertical fluid drainage on the second-kind self-similar solutions for the shape of the fluid–fluid interface in three contexts: (i) viscous axisymmetric gravity currents converging towards the centre of a cylindrical container; (ii) viscous gravity currents moving towards the origin in a horizontal Hele-Shaw channel with a power-law varying gap thickness in the horizontal direction; and (iii) viscous gravity currents propagating towards the origin of a porous medium with horizontal permeability and porosity gradients in power-law forms. For each of these cases with vertical leakage, we identify a regime diagram that characterizes whether the front reaches the origin or not; in particular, when the front does not reach the origin, we calculate the final location of the front. We have also conducted laboratory experiments with a cylindrical lock gate to generate a converging viscous gravity current where vertical fluid drainage is allowed from various perforated horizontal substrates. The time-dependent position of the propagating front is captured from the experiments, and the front position is found to agree well with the theoretical and numerical predictions when surface tension effects can be neglected.

It is known that longitudinal grooves may stabilize or destabilize the travelling wave instability in a channel flow depending on the groove wavenumber. These waves reduce to the classical Tollmien–Schlichting waves in the absence of grooves. It is shown that another class of travelling wave instability exists if grooves with sufficiently high amplitude and proper wavelengths are used. It is demonstrated that the new instability mode is driven by the inviscid mechanism, with the disturbance motion having the form of a wave propagating in the streamwise direction with phase speed approximately four times larger than the Tollmien–Schlichting wave speed and with its streamwise wavelength being approximately twice the spanwise groove wavelength. The instability motion is concentrated mostly in the middle of the channel and has a planar character, i.e. the dominant velocity components are parallel to the walls. A significant reduction of the corresponding critical Reynolds number can be achieved by increasing the groove amplitude. Conditions that guarantee the flow stability in a grooved channel, i.e. the grooved surface behaves as a hydraulically smooth surface, have been identified.

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a Prandtl number of unity and for Rayleigh numbers up to $10^{9}$. In this geometry, curvature and radial variations of the gravitational acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from ${\it\eta}=r_{i}/r_{o}=0.2$ to ${\it\eta}=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl–Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with ${\it\eta}=0.6$ and with gravity proportional to $1/r^{2}$. We show that our spherical models are consistent with the predictions of Grossmann & Lohse’s (J. Fluid Mech., vol. 407, 2000, pp. 27–56) theory and that $\mathit{Nu}(\mathit{Ra})$ and $\mathit{Re}(\mathit{Ra})$ scalings are in good agreement with plane layer results.

Direct numerical simulations of flows in cylinders subjected to both rapid rotation and axial precession are presented and analysed in the context of a stability theory based on the triadic resonance of Kelvin modes. For a case that was chosen to provide a finely tuned resonant instability with a small nutation angle, the simulations are in good agreement with the theory and previous experiments in terms of mode shapes and dynamics, including long-time-scale regularization of the flow and recurrent collapses. Cases not tuned to the most unstable triad, but with the nutation angle still small, are also in quite good agreement with theoretical predictions, showing that the presence of viscosity makes the physics of the triadic-resonance model robust to detuning. Finally, for a case with $45^{\circ }$ nutation angle for which it has been suggested that resonance does not occur, the simulations show that a slowly growing triadic resonance predicted by theory is in fact observed if sufficient evolution time is allowed.

A modelling methodology to reproduce the experimental measurements of a turbulent flow in the presence of symmetry is presented. The flow is a three-dimensional wake generated by an axisymmetric body. We show that the dynamics of the turbulent wake flow can be assimilated by a nonlinear two-dimensional Langevin equation, the deterministic part of which accounts for the broken symmetries that occur in the laminar and transitional regimes at low Reynolds numbers and the stochastic part of which accounts for the turbulent fluctuations. Comparison between theoretical and experimental results allows the extraction of the model parameters.

Here, we study noisy transitional flows in imperfect millimetre-scale channels. For probing the flows, we use microcantilever sensors embedded in the channel walls. We perform experiments in two nominally identical channels. The different sets of imperfections in the two channels result in two random flows in which the high-order moments of the near-wall fluctuations differ by orders of magnitude. Surprisingly, however, the lowest-order statistics in both cases appear to be qualitatively similar and can be described by a proposed noisy Landau equation for a slow mode. The noise, regardless of its origin, regularizes the Landau singularity of the relaxation time and makes transitions driven by different noise sources appear similar.

We present measurements of the azimuthal rotation velocity $\dot{{\it\theta}}(t)$ and thermal amplitude ${\it\delta}(t)$ of the large-scale circulation in turbulent Rayleigh–Bénard convection with modulated rotation. Both $\dot{{\it\theta}}(t)$ and ${\it\delta}(t)$ exhibit clear oscillations at the modulation frequency ${\it\omega}$. Fluid acceleration driven by oscillating Coriolis force causes an increasing phase lag in $\dot{{\it\theta}}(t)$ when ${\it\omega}$ increases. The applied modulation produces oscillatory boundary layers and the resulting time-varying viscous drag modifies ${\it\delta}(t)$ periodically. Oscillation of $\dot{{\it\theta}}(t)$ with maximum amplitude occurs at a finite modulation frequency ${\it\omega}^{\ast }$. Such a resonance-like phenomenon is interpreted as a result of optimal coupling of ${\it\delta}(t)$ to the modulated rotation velocity. We show that an extended large-scale circulation model with a relaxation time for ${\it\delta}(t)$ in response to the modulated rotation provides predictions in close agreement with the experimental results.

We report an experimental study of the large-scale circulation (LSC) reversal in quasi-2D turbulent thermal convection, in which the aspect ratio ${\it\Gamma}$ ($=\text{height}/\text{length}$ of a rectangular box) is used as a parameter to perturb the stability of the LSC. It is found that the mean time interval $\langle {\it\tau}\rangle$ between two successive reversals increases strongly with increasing ${\it\Gamma}$. A stochastic model is proposed to incorporate the effect of the corner rolls. In the model, the aspect ratio serves as a tuning parameter for the relative weight of the corner rolls that damp the LSC. The model predictions for the shape of the bistable states of the system and $\langle {\it\tau}\rangle$ agree excellently with the experimental results, with $\langle {\it\tau}\rangle$ having an unexpected stretched exponential Rayleigh number dependence, ${\sim}\!\exp (Ra^{{\it\alpha}})$. We further show quantitatively that the main damping force of the LSC in a quasi-2D system is from the corner rolls rather than the viscous drag from the sidewalls, which bridges the difference found in quasi-2D and 3D systems.

We study the behaviour of elongated puddles deposited on non-wetting substrates. Such liquid strips retract and adopt circular shapes after a few oscillations. Their thickness and horizontal surface area remain constant during this reorganization, so that the energy of the system is only lowered by minimizing the length of the contour (and the corresponding surface energy); despite the large scale of the experiments (several centimetres), motion is driven by surface tension. We focus on the retraction stage, and show that its velocity results from a balance between the capillary driving force and inertia, due to the frictionless motion on non-wetting substrates. As a consequence, the retraction velocity has a special Taylor–Culick structure, where the puddle width replaces the usual thickness.