In this article we present direct numerical simulations of stratified flow at resolutions of up to $204{8}^{2} \times 513$ , to explore scalings for the dynamics of stably stratified turbulence. Recent work suggests that for strong enough stratification, the vertical integral scale of the turbulence adjusts to yield a vertical Froude number, ${F}_{v} $ , of order unity at high enough Reynolds number, whilst the horizontal Froude number, ${F}_{h} $ , decreases as stratification is increased. Our numerical simulations are consistent with predictions by Lindborg (J. Fluid Mech., vol. 550, 2006, pp, 207–242), and with numerical simulations at lower resolution, in that the horizontal kinetic energy spectrum follows a Kolmogorov spectrum (after replacing the wavenumber with the horizontal wavenumber) and that the horizontal potential energy spectrum similarly follows the Corrsin–Obukhov spectrum for a passive scalar. Most importantly, we build upon these previous results by thoroughly exploring the dependence of the horizontal spectrum of horizontal kinetic energy on both the stratification and the relative size of the vertical dissipation terms, as quantified by the buoyancy Reynolds number. Our most important result is that variations in the power-law exponent scale entirely with the buoyancy Reynolds number and not with the stratification itself, lending considerable support to the Lindborg (2006) hypothesis that horizontal spectra are independent of stratification at large Reynolds numbers. We further demonstrate that even at the large numerical resolution of this study, the spectrum and hence the dynamics are affected by the buoyancy Reynolds number unless it is larger than $O(10)$ , indicating that extreme care must be taken when assessing claims made from previous numerical simulations of stratified flow at low or moderate resolution and extrapolating the results to geophysical or astrophysical Reynolds numbers.

The effects of small-amplitude, two-dimensional grooves on pressure losses in a laminar channel flow have been analysed. Grooves with an arbitrary shape and an arbitrary orientation with respect to the flow direction have been considered. It has been demonstrated that losses can be expressed as a superposition of two parts, one associated with change in the mean positions of the walls and one induced by flow modulations associated with the geometry of the grooves. The former effect can be determined analytically, while the latter has to be determined numerically and can be captured with an acceptable accuracy using reduced-order geometry models. Projection of the wall shape onto a Fourier space has been used to generate such a model. It has been found that in most cases replacement of the actual wall geometry with the leading mode of the relevant Fourier expansion permits determination of pressure losses with an error of less than 10 %. Detailed results are given for sinusoidal grooves for the range of parameters of practical interest. These results describe the performance of arbitrary grooves with the accuracy set by the properties of the reduced-order geometry model and are exact for sinusoidal grooves. The results show a strong dependence of the pressure losses on the groove orientation. Longitudinal grooves produce the smallest drag, and oblique grooves with an inclination angle of ${\sim }42\textdegree $ exhibit the largest flow turning potential. Detailed analyses of the extreme cases, i.e. transverse and longitudinal grooves, have been carried out. For transverse grooves with small wavenumbers, the dominant part of the drag is produced by shear, while the pressure form drag and the pressure interaction drag provide minor contributions. For the same grooves with large wavenumbers, the stream lifts up above the grooves due to their blocking effect, resulting in a change in the mechanics of drag formation: the contributions of shear decrease while the contributions of the pressure interaction drag increase, leading to an overall drag increase. In the case of longitudinal grooves, drag is produced by shear, and its rearrangement results in a drag decrease for long-wavelength grooves in spite of an increase of the wetted surface area. An increase of the wavenumber leads to the fluid being squeezed from the troughs and the stream being forced to lift up above the grooves. The shear is nearly eliminated from a large fraction of the wall but the overall drag increases due to reduction of the effective channel opening. It is shown that properly structured grooves are able to eliminate wall shear from the majority of the wetted surface area regardless of the groove orientation, thus exhibiting the potential for the creation of drag-reducing surfaces. Such surfaces can become practicable if a method for elimination of the undesired pressure and shear peaks through proper groove shaping can be found.

Geophysical fluid models often support both fast and slow motions. As the dynamics are often dominated by the slow motions, it is desirable to filter out the fast motions by constructing balance models. An example is the quasi-geostrophic (QG) model, which is used widely in meteorology and oceanography for theoretical studies, in addition to practical applications such as model initialization and data assimilation. Although the QG model works quite well in the mid-latitudes, its usefulness diminishes as one approaches the equator. Thus far, attempts to derive similar balance models for the tropics have not been entirely successful as the models generally filter out Kelvin waves, which contribute significantly to tropical low-frequency variability. There is much theoretical interest in the dynamics of planetary-scale Kelvin waves, especially for atmospheric and oceanic data assimilation where observations are generally only of the mass field and thus do not constrain the wind field without some kind of diagnostic balance relation. As a result, estimates of Kelvin wave amplitudes can be poor. Our goal is to find a balance model that includes Kelvin waves for planetary-scale motions. Using asymptotic methods, we derive a balance model for the weakly nonlinear equatorial shallow-water equations. Specifically we adopt the ‘slaving’ method proposed by Warn et al. (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 723–739), which avoids secular terms in the expansion and thus can in principle be carried out to any order. Different from previous approaches, our expansion is based on a long-wave scaling and the slow dynamics is described using the height field instead of potential vorticity. The leading-order model is equivalent to the truncated long-wave model considered previously (e.g. Heckley & Gill, Q. J. R. Meteorol. Soc., vol. 110, 1984, pp. 203–217), which retains Kelvin waves in addition to equatorial Rossby waves. Our method allows for the derivation of higher-order models which significantly improve the representation of Rossby waves in the isotropic limit. In addition, the ‘slaving’ method is applicable even when the weakly nonlinear assumption is relaxed, and the resulting nonlinear model encompasses the weakly nonlinear model. We also demonstrate that the method can be applied to more realistic stratified models, such as the Boussinesq model.

A new vortex model of inviscid propeller and wind-turbine wakes is proposed based on an asymptotic expansion of the Biot–Savart induction law to account for the finite vortex core size. The circulation along the blade is assumed to be constant from the blade root to the tip approximating a turbine with maximum power production for given operating conditions. The model iteratively calculates the tip-vortex path, allowing the wake to expand/contract freely, and is afterward able to evaluate the velocity field in the whole domain. The ‘roller-bearing analogy’, proposed by Okulov and Sørensen (J. Fluid Mech., vol. 649, 2010, pp. 497–508), is used to determine the vortex core size. A comparison of the main outcomes of the present model with the general momentum theory is performed in terms of the operating parameters (namely the number of blades, the tip-speed ratio, the blade circulation and the vortex core size), demonstrating good agreement between the two. Furthermore, experimental data have been compared with the model outputs to validate the model under real operating conditions.

A current in a homogeneous rotating fluid is subject to simultaneous inertial and barotropic instabilities. Inertial instability causes rapid mixing of streamwise absolute linear momentum and alters the vertically averaged velocity profile of the current. The resulting profile can be predicted by a construction based on absolute-momentum conservation. The alteration of the mean velocity profile strongly affects how barotropic instability will subsequently change the flow. If a current with a symmetric distribution of cyclonic and anticyclonic vorticity undergoes only barotropic instability, the result will be cyclones and anticyclones of the same shape and amplitude. Inertial instability breaks this symmetry. The combined effect of inertial and barotropic instability produces anticyclones that are broader and weaker than the cyclones. A two-step scheme for predicting the result of the combined inertial and barotropic instabilities is proposed and tested. This scheme uses the construction for the redistribution of streamwise absolute linear momentum to predict the mean current that results from inertial instability and then uses this equilibrated current as the initial condition for a two-dimensional simulation that predicts the result of the subsequent barotropic instability. Predictions are made for the evolution of a Gaussian jet and are compared with three-dimensional simulations for a range of Rossby numbers. It is demonstrated that the actual redistribution of absolute momentum in the three-dimensional simulations is well predicted by the construction used here. Predictions are also made for the final number and size of vortices that result from the combined inertial and barotropic instabilities.

Numerical simulations of acoustic radiation from a jet engine inlet are performed using advanced computational aeroacoustics algorithms and high-quality numerical boundary treatments. As a model of modern commercial jet engine inlets, the inlet geometry of the NASA Source Diagnostic Test is used. Fan noise consists of tones and broadband sound. This investigation considers the radiation of tones associated with upstream-propagating duct modes. The primary objective is to identify the dominant physical processes that determine the directivity of the radiated sound. Two such processes have been identified. They are acoustic diffraction and refraction. Diffraction is the natural tendency for an acoustic duct mode to follow a curved solid surface as it propagates. Refraction is the turning of the direction of propagation of a duct mode by mean flow gradients. Parametric studies on the changes in the directivity of radiated sound due to variations in forward flight Mach number, duct mode frequency, azimuthal mode number and radial mode number are carried out. It is found there is a significant difference in directivity for the radiation of the same duct mode from an engine inlet when operating in static condition versus one in forward flight. It will be shown that the large change in directivity is the result of the combined effects of diffraction and refraction.

We study both theoretically and numerically two-dimensional magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl number $\mathit{Pm}$ (and the limits thereof), with an emphasis on solution regularity. For $\mathit{Pm}= 0$ , both $\Vert \omega \Vert ^{2} $ and $\Vert j\Vert ^{2} $ , where $\omega $ and $j$ are, respectively, the vorticity and current, are uniformly bounded. Furthermore, $\Vert \boldsymbol{\nabla} j\Vert ^{2} $ is integrable over $[0, \infty )$ . The uniform boundedness of $\Vert \omega \Vert ^{2} $ implies that in the presence of vanishingly small viscosity $\nu $ (i.e. in the limit $\mathit{Pm}\rightarrow 0$ ), the kinetic energy dissipation rate $\nu \Vert \omega \Vert ^{2} $ vanishes for all times $t$ , including $t= \infty $ . Furthermore, for sufficiently small $\mathit{Pm}$ , this rate decreases linearly with $\mathit{Pm}$ . This linear behaviour of $\nu \Vert \omega \Vert ^{2} $ is investigated and confirmed by high-resolution simulations with $\mathit{Pm}$ in the range $[1/ 64, 1] $ . Several criteria for solution regularity are established and numerically tested. As $\mathit{Pm}$ is decreased from unity, the ratio $\Vert \omega \Vert _{\infty } / \Vert \omega \Vert $ is observed to increase relatively slowly. This, together with the integrability of $\Vert \boldsymbol{\nabla} j\Vert ^{2} $ , suggests global regularity for $\mathit{Pm}= 0$ . When $\mathit{Pm}= \infty $ , global regularity is secured when either $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $ , where $\boldsymbol{u}$ is the fluid velocity, or $\Vert j\Vert _{\infty } / \Vert j\Vert $ is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range $\mathit{Pm}\in [1, 64] $ show that $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $ varies slightly (with similar behaviour for $\Vert j\Vert _{\infty } / \Vert j\Vert $ ), thereby lending strong support for the possibility $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert \lt \infty $ in the limit $\mathit{Pm}\rightarrow \infty $ . The peak of the magnetic energy dissipation rate $\mu \Vert j\Vert ^{2} $ is observed to decrease rapidly as $\mathit{Pm}$ is increased. This result suggests the possibility $\Vert j\Vert ^{2} \lt \infty $ in the limit $\mathit{Pm}\rightarrow \infty $ . We discuss further evidence for the boundedness of the ratios $\Vert \omega \Vert _{\infty } / \Vert \omega \Vert $ , $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $ and $\Vert j\Vert _{\infty } / \Vert j\Vert $ in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields.

Motivated by the importance of diapycnal mixing parameterizations in large-scale ocean general circulation models, we provide a detailed analysis of high-Reynolds-number mixing in density stratified shear flows which constitute an archetypical example of the small-scale physical processes occurring in the oceanic interior that control turbulent diffusion. Our focus is upon the issue as to whether the route to fully developed turbulence in the stratified mixing layer is in any significant way determinant of diapycnal mixing efficiency as represented by an effective turbulent diffusivity. We characterize different routes to fully developed turbulence by the nature of the secondary instabilities through which a primary Kelvin–Helmholtz billow executes the transition to this state. We then demonstrate that different mechanisms of turbulence transition characterized in these different transition mechanisms lead to considerably different values for the efficiency of diapycnal mixing and also for the effective vertical flux of buoyancy. We show that the widely employed value of 0.15–0.2 for the efficiency of mixing in shear-induced stratified turbulence based upon both laboratory measurements and similarly low-Reynolds-number numerical simulations may be too low for the high-Reynolds-number regime characteristic of geophysical flows. Our results show that the mixing efficiency tends to a value of approximately $1/ 3$ for sufficiently large Reynolds number at an intermediate value of 0.12 for the Richardson number. This is in agreement with a theoretical predictions of Caulfield, Tang and Plasting (J. Fluid Mech., vol. 498, 2004, pp. 315–332) for the asymptotic value of mixing efficiency in stratified Couette flows. In the high-Reynolds-number regime, mixing efficiency is shown to vary over a considerable range during the course of a particular shear-induced mixing event. We explain this variation on the basis of a detailed examination of the underlying dynamics. Since values in the range 0.15–0.2 for mixing efficiency have been extensively employed to infer an effective diffusivity from ocean microstructure measurements and also in energy balance analyses of the requirements of the global ocean circulation, our findings have potentially important implications for large-scale ocean modelling. We also quantify the errors introduced by employing the Osborn (J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89) formula along with an efficiency of 0.15 to infer values for effective diffusivity, and explain the logical underpinnings of this conclusion. One of the more important aspects of this work from the perspective of our theoretical understanding of stratified turbulence is the demonstration that the inverse cascade of energy, which is facilitated by the vortex-merging process that is typical of laboratory experiments and of the low-Reynolds-number simulations of shear flow evolution, is strongly suppressed by increase of the Reynolds number to values typical of geophysical flows. Based on this finding, the application of results based on low-Reynolds-number (numerical or laboratory) experiments to high-Reynolds-number geophysical shear flows needs to be reconsidered.

The stability of the flow between two concentric cylinders is studied numerically and analytically when the fluid is stably stratified. We show that such flow is unstable when the angular velocity $\Omega (r)$ increases along the radial direction, a regime never explored before. The instability is highly non-axisymmetric and involves the resonance of two families of inertia–gravity waves like for the strato-rotational instability. The growth rate is maximum when only the outer cylinder is rotating and goes to zero when $\Omega (r)$ is constant. The sufficient condition for linear, inviscid instability derived previously, $\mathrm{d} {\Omega }^{2} / \mathrm{d} r\lt 0$ , is therefore extended to $\mathrm{d} {\Omega }^{2} / \mathrm{d} r\not = 0$ , meaning that only the regime of solid-body rotation is stable in stratified fluids. A Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) analysis in the inviscid limit, confirmed by the numerical results, shows that the instability occurs only when the Froude number is below a critical value and only for a particular band of azimuthal wavenumbers. It is also demonstrated that the instability originates from a reversal of the radial group velocity of the waves, or equivalently from a wave over-reflection phenomenon. The instability persists in the presence of viscous effects.

Using filter-space techniques (FSTs), we study the spatial structure of the scale-to-scale flux of energy in two-dimensional flow. Analysing data from a weakly turbulent, experimental quasi-two-dimensional flow, we find rotationally symmetric patterns consisting of lobes of spectral flux of alternating sign that are associated with vortical motion in the flow field. Such patterns also occur in a simple analytical model, even though the single-scale model flow should have no scale-to-scale energy transfer. Thus, the interpretation of these alternating patterns must be handled with care. By decomposing the spectral flux into three distinct components, we show that these lobe patterns are entirely associated with the Leonard and, to a lesser extent, cross terms. In addition, we show that the contributions from these two terms are localized around the energy injection scale, and that the bulk of the inverse energy transfer in our flow is carried by the subgrid term alone.

We analyse the creeping flow generated by a spherical particle moving through a viscous fluid with nematic directional order, in which momentum diffusivity is anisotropic and which opposes resistance to bending. Specifically, we provide closed-form analytical expressions for the response function, i.e. the equivalent to Stokes’s drag formula for nematic fluids. Particular attention is given to the rotationally pseudo-isotropic condition defined by zero resistance to bending, and to the strain pseudo-isotropic condition defined by isotropic momentum diffusivity. We find the former to be consistent with the rheology of biopolymer networks and the latter to be closer to the rheology of nematic liquid crystals. These ‘pure’ anisotropic conditions are used to benchmark existing particle tracking microrheology methods that provide effective directional viscosities by applying Stokes’s drag law separately in different directions. We find that the effective viscosity approach is phenomenologically justified in rotationally isotropic fluids, although it leads to significant errors in the estimated viscosity coefficients. On the other hand, the mere concept of directional effective viscosities is found to be misleading in fluids that oppose an appreciable resistance to bending. Finally, we observe that anisotropic momentum diffusivity leads to asymmetric streamline patterns displaying enhanced (reduced) streamline deflection in the directions of lower (higher) diffusivity. The bending resistance of the fluid is found to modulate the asymmetry of streamline deflection. In some cases, the combined effects of both anisotropy mechanisms leads to streamline patterns that converge towards the sphere.

We study the degree to which Kraichnan–Leith–Batchelor (KLB) phenomenology describes two-dimensional energy cascades in $\alpha $ turbulence, governed by $\partial \theta / \partial t+ J(\psi , \theta )= \nu {\nabla }^{2} \theta + f$ , where $\theta = {(- \Delta )}^{\alpha / 2} \psi $ is generalized vorticity, and $\hat {\psi } (\boldsymbol{k})= {k}^{- \alpha } \hat {\theta } (\boldsymbol{k})$ in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow ( $\alpha = 1$ ), regular two-dimensional flow ( $\alpha = 2$ ) and rotating shallow flow ( $\alpha = 3$ ), which is the isotropic limit of a mantle convection model. We re-examine arguments for dual inverse energy and direct enstrophy cascades, including Fjørtoft analysis, which we extend to general $\alpha $ , and point out their limitations. Using an $\alpha $ -dependent eddy-damped quasinormal Markovian (EDQNM) closure, we seek self-similar inertial range solutions and study their characteristics. Our present focus is not on coherent structures, which the EDQNM filters out, but on any self-similar and approximately Gaussian turbulent component that may exist in the flow and be described by KLB phenomenology. For this, the EDQNM is an appropriate tool. Non-local triads contribute increasingly to the energy flux as $\alpha $ increases. More importantly, the energy cascade is downscale in the self-similar inertial range for $2. 5\lt \alpha \lt 10$ . At $\alpha = 2. 5$ and $\alpha = 10$ , the KLB spectra correspond, respectively, to enstrophy and energy equipartition, and the triad energy transfers and flux vanish identically. Eddy turnover time and strain rate arguments suggest the inverse energy cascade should obey KLB phenomenology and be self-similar for $\alpha \lt 4$ . However, downscale energy flux in the EDQNM self-similar inertial range for $\alpha \gt 2. 5$ leads us to predict that any inverse cascade for $\alpha \geq 2. 5$ will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for $\alpha \geq 2. 5$ is significantly steeper than the KLB prediction, while for $\alpha \lt 2. 5$ we obtain the KLB spectrum.

We derive a growth-rate model for the Richtmyer–Meshkov mixing layer, given arbitrary but known initial conditions. The initial growth rate is determined by the net mass flux through the centre plane of the perturbed interface immediately after shock passage. The net mass flux is determined by the correlation between the post-shock density and streamwise velocity. The post-shock density field is computed from the known initial perturbations and the shock jump conditions. The streamwise velocity is computed via Biot–Savart integration of the vorticity field. The vorticity deposited by the shock is obtained from the baroclinic torque with an impulsive acceleration. Using the initial growth rate and characteristic perturbation wavelength as scaling factors, the model collapses the growth-rate curves and, in most cases, predicts the peak growth rate over a range of Mach numbers ( $1. 1\leq {M}_{i} \leq 1. 9$ ), Atwood numbers ( $- 0. 73\leq A\leq - 0. 35$ and $0. 22\leq A\leq 0. 73$ ), adiabatic indices ( $1. 40/ 1. 67\leq {\gamma }_{1} / {\gamma }_{2} \leq 1. 67/ 1. 09$ ) and narrow-band perturbation spectra. The mixing layer at late times exhibits a power-law growth with an average exponent of $\theta = 0. 24$ .

Fluid dynamics instabilities are usually investigated in two types of situations, either confined in cells with fixed boundaries, or free to grow in open space. In this article we study the Faraday instability triggered in a floating liquid lens. This is an intermediate situation in which a hydrodynamical instability develops in a domain with flexible boundaries. The instability is observed to be initially disordered with fluctuations of both the wave field and the lens boundaries. However, a slow dynamics takes place, leading to a mutual adaptation so that a steady regime is reached with a stable wave field in a stable lens contour. The most recurrent equilibrium lens shape is elongated with the Faraday wave vector along the main axis. In this self-organized situation an equilibrium is reached between the radiation pressure exerted by Faraday waves on the borders and their capillary response. The elongated shape is obtained theoretically as the exact solution of a Riccati equation with a unique control parameter and compared with the experiment.

We consider the flow of an electrically conducting fluid confined in a rotating spherical shell. The flow is driven by a directly imposed electromagnetic body force, created by the combination of an electric current flowing from the inner sphere to a ring-shaped electrode around the equator of the outer sphere and a separately imposed predominantly axial magnetic field. We begin by numerically computing the axisymmetric basic states, which consist of a strong zonal flow. We next compute the linear onset of non-axisymmetric instabilities, and fully three-dimensional solutions up to ten times supercritical. We demonstrate that an experimental liquid-sodium device 50 cm in diameter could achieve and exceed these parameter values.

The three-dimensional analogue of Moffatt eddies is derived for a corner formed by the intersection of three orthogonal planes. The complex exponents of the first few modes are determined and the flows resulting from the primary modes (those which decay least rapidly as the apex is approached and, hence, should dominate the near-apex flow) examined in detail. There are two independent primary modes, one symmetric, the other antisymmetric, with respect to reflection in one of the symmetry planes of the cone. Any linear combination of these modes yields a possible primary flow. Thus, there is not one, but a two-parameter family of such flows. The particle-trajectory equations are integrated numerically to determine the streamlines of primary flows. Three special cases in which the flow is antisymmetric under reflection lead to closed streamlines. However, for all other cases, the streamlines are not closed and quasi-periodic limiting trajectories are approached when the trajectory equations are integrated either forwards or backwards in time. A generic streamline follows the backward-time trajectory in from infinity, undergoes a transient phase in which particle motion is no longer quasi-periodic, before being thrown back out to infinity along the forward-time trajectory.

We consider the static and dynamic behaviour of two-dimensional droplets on inclined heterogeneous substrates. We utilize an evolution equation for the droplet thickness based on the long-wave approximation of the Stokes equations in the presence of slip. Through a singular perturbation procedure, evolution equations for the location of the two moving fronts are obtained under the assumption of quasi-static dynamics. The deduced equations, which are verified by direct comparisons with numerical solutions to the governing equation, are scrutinized in a variety of dynamic and equilibrium settings. For example, we demonstrate the possibility for stick–slip dynamics, substrate-induced hysteresis, the uphill motion of the droplet for sufficiently strong chemical gradients and the existence of a critical inclination angle beyond which the droplet can no longer be supported at equilibrium. Where possible, analytical expressions are obtained for various quantities of interest, which are also verified by appropriate numerical experiments.

A conservative hyperbolic two-parameter model of shear shallow-water flows is used to study the classical turbulent hydraulic jump. The parameters of the model, which are the wall enstrophy and the roller dissipation coefficient, are determined from measurements of the roller length and the deviation from the Bélanger equation of the sequent depth ratio (experimental data by Hager & Bremen, J. Hydraul. Res., vol. 27, 1989, pp. 565–585; and Hager, Bremen & Kawagoshi, J. Hydraul. Res., vol. 28, 1990, pp. 591–608). Stationary solutions to the model describe with a good accuracy the free-surface profile of the hydraulic jump. The model is also capable of predicting the oscillations of the jump toe. We show that if the upstream Froude number is larger than ${\sim }1. 5$ , the jump toe oscillates with a particular frequency, while for the Froude number smaller than 1.5 the solution becomes stationary. In particular, we show that for a given flow discharge, the oscillation frequency is a decreasing function of the Froude number.

A realistic, efficient and robust technique for the control of amplifier flows has been investigated. Since this type of fluid system is extremely sensitive to upstream environmental noise, an accurate model capturing the influence of these perturbations is needed. A subspace identification algorithm is not only a convenient and effective way of constructing this model, it is also realistic in the sense that it is based on input and output data measurements only and does not require other information from the detailed dynamics of the fluid system. This data-based control design has been tested on an amplifier model derived from the Ginzburg–Landau equation, and no significant loss of efficiency has been observed when using the identified instead of the exact model. Even though system identification leads to a realistic control design, other issues such as state estimation, have to be addressed to achieve full control efficiency. In particular, placing a sensor too far downstream is detrimental, since it does not provide an estimate of incoming perturbations. This has been made clear and quantitative by considering the relative estimation error and, more appropriately, the concept of a visibility length, a measure of how far upstream a sensor is able to accurately estimate the flow state. It has been demonstrated that a strongly convective system is characterized by a correspondingly small visibility length. In fact, in the latter case the optimal sensor placement has been found upstream of the actuators, and only this configuration was found to yield an efficient control performance. This upstream sensor placement suggests the use of a feed-forward approach for fluid systems with strong convection. Furthermore, treating upstream sensors as inputs in the identification procedure results in a very efficient and robust control. When validated on the Ginzburg–Landau model this technique is effective, and it is comparable to the optimal upper bound, given by full-state control, when the amplifier behaviour becomes convection-dominated. These concepts and findings have been extended and verified for flow over a backward-facing step at a Reynolds number $\mathit{Re}= 350$ . Environmental noise has been introduced by three independent, localized sources. A very satisfactory control of the Kelvin–Helmholtz instability has been obtained with a one-order-of-magnitude reduction in the averaged perturbation norm. The above observations have been further confirmed by examining a low-order model problem that mimics a convection-dominated flow but allows the explicit computation of control-relevant measures such as observability. This study casts doubts on the usefulness of the asymptotic notion of observability for convection-dominated flows, since such flows are governed by transient effects. Finally, it is shown that the feed-forward approach is equivalent to an optimal linear–quadratic–Gaussian control for spy sensors placed sufficiently far upstream or for sufficiently convective flows. The control design procedure presented in this paper, consisting of data-based subspace identification and feed-forward control, was found to be effective and robust. Its implementation in a real physical experiment may confidently be carried out.

An axisymmetric drop spreads on a radially heated, partially wetting solid substrate in a rotating geometry. The lubrication approximation is applied to the field equations for this thin viscous drop to yield an evolution equation that captures the dependence of viscosity, surface tension, gravity, centrifugal forces and thermocapillarity. We study the quasi-static spreading regime, whereby droplet motion is controlled by a constitutive law that relates the contact angle to the contact-line speed. Non-uniform heating of the substrate can generate both vertical and radial temperature gradients along the drop interface, which produce distinct thermocapillary forces and equivalently flows that affect the spreading process. For the non-rotating system, competition between surface chemistry (wetting) and thermocapillary flows induced by the thermal gradients gives rise to bistability in certain regions of parameter space in which the droplets converge to an equilibrium shape. The centrifugal forces that develop in a rotating geometry enlarge the bistability regions. Parameter regimes in which the droplet spreads indefinitely are identified and spreading laws are computed to compare with experimental results from the literature.