The control of separated fluid flow by reduced-order models is studied using the two-dimensional incompressible flow over an open square cavity at Reynolds numbers where instabilities are present. Actuation and measurement locations are taken on the upstream and downstream edge of the cavity. A bi-orthogonal projection is introduced to arrive at reduced-order models for the compensated problem. Global modes, proper orthogonal decomposition (POD) modes and balanced modes are used as expansion bases for the model reduction. The open-loop behaviour of the full and the reduced systems is analysed by comparing the respective transfer functions. This analysis shows that global modes are inadequate to sufficiently represent the input–output behaviour whereas POD and balanced modes are capable of properly approximating the exact transfer function. Balanced modes are far more efficient in this process, but POD modes show superior robustness. The performance of the closed-loop system corroborates this finding: while reduced-order models based on POD are able to render the compensated system stable, balanced modes accomplish the same with far fewer degrees of freedom.

In this paper, I show that a range of environmental flows are inherently dynamically similar. These flows, which are partially obstructed by a permeable medium, are here termed ‘obstructed shear flows’. Examples include aquatic flows over sediment beds, submerged vegetation canopies and coral reefs, as well as atmospheric flows over crop canopies, forests and cities (‘urban canopies’). While the density and geometry of the obstructions may vary, the drag in each system generates a velocity profile with an inflection point. This renders the flow unstable. Consequently, it is expected that (a) the dominant interfacial turbulent structure in obstructed shear flows will be a Kelvin–Helmholtz-type vortex, and (b) that this instability will engender hydrodynamic similarities among obstructed shear flows. Such similarities have been hypothesized but not yet fully explored. An extensive review of existing data confirms these dynamic similarities on scales of O(mm) to O(10 m). The extent of shear penetration into the obstruction, which is a primary determinant of residence time in the obstruction, scales upon the drag length scale. Other relationships that link the strength of turbulence and the ‘slip’ velocity at the top of the obstruction to the friction velocity (u∗) are also evident. The relationships presented here provide predictive capability for flow and transport in obstructed shear flows and suggest the possibility of a single framework to describe such flows on all scales.

The motion of two immiscible liquids in a capillary tube is analysed, theoretically and numerically, for the case in which a residual film confines the displacing liquid to the core of this tube. The theoretical analysis has shown that the three flow regimes predicted by Taylor (J. Fluid Mech., vol. 10, 1961, pp. 161–165), for the case of gas-displacement, can only be achieved when the ratio of the viscosity of the displaced fluid to that of the displacing one is greater than 2. An elliptic mesh generation technique, coupled with the Galerkin finite-element method, is used to compute the velocity field and the configuration of the interface between the two fluids. A map of cases in the Cartesian space defined by the capillary number (Ca) and the viscosity ratio (Nμ) is constructed in order to locate the different flow patterns the problem exhibits. The critical capillary number at which the flow enters the transition range between the bypass regime and the full-recirculating one is given. While a decrease of the fraction of mass attached to the wall is achieved by decreasing Ca or increasing Nμ, bypass flow patterns are formed as a consequence of high values of the capillary number and viscosity ratio.

Spatial and temporal frequency dynamics were experimentally tracked via flow visualization for Newtonian fluids as a function of the inner cylinder Reynolds number (Rei) in the flow between concentric, independently rotating cylinders with a radius ratio of 0.912 and an aspect ratio of 60.7. Eight transitions from laminar to turbulent flow were characterized in detail for a stationary outer cylinder, producing highly resolved space–time and frequency–time plots for wavy, modulated and weakly turbulent states. A previously unreported early-modulated wavy vortex flow was found in our high aspect ratio geometry both with and without the presence of a dislocation. The envelope of stability for this flow state was shown to cross into the co-rotating regime, and is present up to Reo ~ 60, where Reo is the outer cylinder Reynolds number. This early modulation is independent of acceleration in the range 0.18 < dRei/dτ < 2.9, where τ is the time nondimensionalized with a viscous time scale. While many of the flow states have been previously observed in geometries with somewhat different radius ratios, we provide new characterization of transitional structures for Reo = 0 in the range 0 < Re* < 21.4, where Re* = Rei/Rec and Rec is the value of Rei at the primary instability. Special attention has been given to ramp rate. For quasi-static ramps, axisymmetric states are stable over the ranges of Re* = [(0–1.17), > 15.4], states characterized by a single distinct temporal frequency for Re* = [(1.17–1.41), (3.56–5.20), (7.85–15.4)], states with multiple temporal frequencies for Re* = [(1.41–3.56), (5.20–7.85)], and a transition from laminar to weakly turbulent vortices occurs at Re* = 5.49. All flow states are characterized by symmetry/symmetry-breaking features as well as azimuthal and axial wavenumbers.

We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.

Laser Doppler anemometry measurements of the mean velocity and Reynolds stresses are carried out for a rough-surface favourable pressure gradient turbulent boundary layer. The experimental data is compared with smooth favourable pressure gradient and rough zero-pressure gradient data. The velocity and Reynolds stress profiles are normalized using various scalings such as the friction velocity and free stream velocity. In the velocity profiles, the effects of roughness are removed when using the friction velocity. The effects of pressure gradient are not absorbed. When using the free stream velocity, the scaling is more effective absorbing the pressure gradient effects. However, the effects of roughness are almost removed, while the effects of pressure gradient are still observed on the outer flow, when the mean deficit velocity profiles are normalized by the U∞ δ∗/δ scaling. Furthermore, when scaled with U2∞, the 〈u2〉 component of the Reynolds stress augments due to the rough surface despite the imposed favourable pressure gradient; when using the friction velocity scaling u∗2, it is dampened. It becomes ‘flatter’ in the inner region mainly due to the rough surface, which destroys the coherent structures of the flow and promotes isotropy. Similarly, the pressure gradient imposed on the flow decreases the magnitude of the Reynolds stress profiles especially on the 〈v2〉 and -〈uv〉 components for the u∗2 or U∞2 scaling. These effects are reflected in the boundary layer parameter δ∗/δ, which increase due to roughness, but decrease due to the favourable pressure gradient. Additionally, the pressure parameter Λ found not to be in equilibrium, describes the development of the turbulent boundary layer, with no influence of the roughness linked to this parameter. These measurements are the first with an extensive number of downstream locations (11). This makes it possible to compute the required x-dependence for the production term and the wall shear stress from the full integrated boundary layer equation. The finding indicates that the skin friction coefficient depends on the favourable pressure gradient condition and surface roughness.

For the Rayleigh-number range 107 ≲ Ra ≲ 1011 we report measurements of the Nusselt number Nu and of properties of the large-scale circulation (LSC) for cylindrical samples of helium gas (Prandtl number Pr = 0.674) that have aspect ratio Γ ≡ D/L = 0.50 (D and L are the diameter and the height respectively) and are heated from below. The results for Nu are consistent with recent direct numerical simulations. We measured the amplitude δ of the azimuthal temperature variation induced by the LSC at the sidewall, and the LSC circulation-plane orientation θ0, at three vertical positions. For the entire Ra range the LSC involves a convection roll that is coherent over the height of the system. However, this structure frequently collapses completely at irregular time intervals and then reorganizes from the incoherent flow. At small δ the probability distribution p(δ) increases linearly from zero; for Γ = 1 and Pr = 4.38 this increase is exponential. No evidence of a two-roll structure, with one above the other, was observed. This differs from recent direct numerical simulations for Γ = 0.5 and Pr = 0.7, where a one-roll LSC was found to exist only for Ra ≲ 109 to 1010, and from measurements for Γ = 0.5 and Pr ≃ 5, where one- and two-roll structures were observed with transitions between them at random time intervals.

Stirring and sedimentation of solid inertial particles in low-Reynolds-number flows has acquired great relevance in multiple environmental, industrial and microfluidic systems, but few detailed numerical studies have focused on chaotically advected experimentally realizable flows. We carry out one-way coupling simulations to study the dynamics of inertial particles in the steady three-dimensional flow in a cylindrical container with exactly counter-rotating lids, which was recently studied by Lackey & Sotiropoulos (Phys. Fluids, vol. 18, 2006, paper no. 053601). We elucidate the rich Lagrangian dynamics of the flow in the vicinity of toroidal invariant regions and show that depending on the Stokes number inertial particles could get trapped for long times in different equilibrium positions inside integrable islands. In the chaotically advected region of the flow the balance between inertia and gravity forces (represented by the settling velocity) can produce a striking fractal sedimentation regime, characterized by a sequence of discrete deposition events of seemingly random number of particles separated by hiatuses of random duration. The resulting staircase-like distribution of the time series of the number of particles in suspension is shown to be a devil's staircase whose fractal dimension is equal to the 0.87 value found in multiple dissipative dynamical systems in nature. Our work sheds new light on the complex mechanisms governing the stirring and deposition of inertial particles and provides new information about the parameters that are relevant in the characterization of particle dynamics in different regions of chaotically advected flows.

Lighthill's analysis (J. Fluid Mech., vol. 60, 1973, p. 1) of the Weis-Fogh lift mechanism is extended to include the spreading phase of the cycle. Lighthill proposed a two-dimensional inviscid irrotational analytical model to compute the circulation around two flat plates (the wings) as they open out, in opposite directions, about a common centre of rotation taken to be at the point of contact of an edge of each plate (the ‘opening phase’). At a critical opening angle, the plates separate and move apart horizontally (the ‘spreading phase’). During this second phase, the fluid region becomes doubly connected and is not analysed by Lighthill. It can, however, also be studied analytically and the results are presented here. We also extend a similar analysis, in an application to turbomachinery, due to Furber & Ffowcs Williams (J. Fluid Mech., vol. 94, 1979, p. 519).

The oscillations of a class of submerged mass–spring systems are examined. An inviscid fluid model is employed to show that the hydrodynamic effects couple the normal modes of these systems. This coupling of normal modes can excite the displacement mode – yielding passive locomotion of the system – even when starting with zero displacement velocity. This is in contrast with the fact that under similar initial conditions but without the hydrodynamic coupling, such systems cannot achieve a net displacement. These ideas are illustrated via two examples of a two-mass and a three-mass system.

We report the first experimental observation of a bistable dynamo regime. A turbulent flow of liquid sodium is generated between two disks in the von Kármán geometry (VKS experiment). When one disk is kept at rest, bistability is observed between a stationary and an oscillatory magnetic field. The stationary and oscillatory branches occur in the vicinity of a codimension-two bifurcation that results from the coupling between two modes of magnetic field. We present an experimental study of the two regimes and study in detail the region of bistability that we understand in terms of dynamical system theory. Despite the very turbulent nature of the flow, the bifurcations of the magnetic field are correctly described by a low-dimensional model. In addition, the different regimes are robust; i.e. turbulent fluctuations do not drive any transition between the oscillatory and stationary states in the region of bistability.

The topic of density-driven convection in porous media has been the focus of many recent studies due to its relevance as a long-term trapping mechanism during geological sequestration of carbon dioxide. Most of these studies have addressed the problem in homogeneous and anisotropic permeability fields using linear-stability analysis, and relatively little attention has been paid to the analysis for heterogeneous systems. Previous investigators have reduced the governing equations to an initial-value problem and have analysed it either with a quasi-steady-state approximation model or using numerical integration with arbitrary initial perturbations. Recently, Rapaka et al. (J. Fluid Mech., vol. 609, 2008, pp. 285–303) used the idea of non-modal stability analysis to compute the maximum amplification of perturbations in this system, optimized over the entire space of initial perturbations. This technique is a mathematically rigorous extension of the traditional normal-mode analysis to non-normal and time-dependent problems. In this work, we extend this analysis to the important cases of anisotropic and layered porous media with a permeability variation in the vertical direction. The governing equations are linearized and reduced to a set of coupled ordinary differential equations of the initial-value type using the Galerkin technique. Non-modal stability analysis is used to compute the maximum growth of perturbations along with the optimal wavenumber leading to this growth. We show that unlike the solution of the initial-value problem, results obtained using non-modal analysis are insensitive to the choice of bottom boundary condition. For the anisotropic problem, the dependence of critical time and wavenumber on the anisotropy ratio was found to be in good agreement with theoretical scalings proposed by Ennis-King et al. (Phys. Fluids, vol. 17, 2005, paper no. 084107). For heterogeneous systems, we show that uncertainty in the permeability field at low wavenumbers can influence the growth of perturbations. We use a Monte Carlo approach to compute the mean and standard deviation of the critical time for a sample permeability field. The results from theory are also compared with finite-volume simulations of the governing equations using fully heterogeneous porous media with strong layering. We show that the results from non-modal stability analysis match extremely well with those obtained from the simulations as long as the assumption of strong layering remains valid.

Steady dipolar vortices continuously driven by electromagnetic forcing in a shallow layer of an electrolytic fluid are studied experimentally and theoretically. The driving Lorentz force is generated by the interaction of a dc uniform electric current injected in the thin layer and the non-uniform magnetic field produced by a small dipolar permanent magnet (0.33 T). Laminar velocity profiles in the neighbourhood of the zone affected by the magnetic field were obtained with particle image velocimetry in planes parallel and normal to the bottom wall. Flow planes at different depths of the layer were explored for injected currents ranging from 10 to 100 mA. Measurements of the boundary layer attached to the bottom wall reveal that owing to the variation of the field in the normal direction, a slightly flattened developing profile with no shear stresses at the free surface is formed. A quasi-two-dimensional magnetohydrodynamic numerical model that introduces the non-uniformity of the magnetic field, particularly its decay in the normal direction, was developed. Vertical diffusion produced by the bottom friction was modelled through a linear friction term. The model reproduces the main characteristic behaviour of the electromagnetically forced flow.

In the present study we investigate the dynamics of initially spherical capsules (made from elastic membranes obeying the strain-hardening Skalak or the strain-softening neo-Hookean law) in strong planar extensional flows via numerical computations. To achieve this, we develop a three-dimensional spectral boundary element algorithm for membranes with shearing and area-dilatation tensions in Stokes flow. The main attraction of this approach is that it exploits all the benefits of the spectral methods (i.e. high accuracy and numerical stability) but without creating denser systems. To achieve continuity of the interfacial geometry and its derivatives at the edges of the spectral elements during the interfacial deformation, a membrane-based interfacial smoothing is developed, via a Hermitian-like interpolation, for both the interfacial shape and the membrane elastic forces. Our numerical results show that no critical flow rate exists for both Skalak and neo-Hookean capsules in the moderate and strong planar extension flows considered in the present study. As the flow rate increases, both capsules reach elongated ellipsoidal steady-state configurations; the cross-section of the Skalak capsule preserves its elliptical shape, while the neo-Hookean capsule becomes more and more lamellar. The curvature at the pointed edges of these elongated steady-state shapes shows a very fast increase with the flow rate. The large interfacial deformations are accompanied with the development of strong membrane tensions especially for the strain-hardening Skalak capsule; the computed increase of the membrane tensions with the flow rate or the shape extension can be used to predict rupture of a specific membrane (with known lytic tension) due to excessive tensions. The type of the experiment imposed on the capsule as well as the applied flow rate affect dramatically the time evolution of the capsule edges owing to the interaction of the hydrodynamic forces with the membrane tensions; when a spherical Skalak capsule is let to deform in a strong flow, very large edge curvatures (with respect to the steady-state value) are developed during the transient evolution.

The behaviour of an initially planar shock wave propagating into a linearly convergent wedge is investigated experimentally and numerically. In the experiment, a 25° internal wedge is mounted asymmetrically in a pressure-driven shock tube. Shock waves with incident Mach numbers in the ranges of 1.4–1.6 and 2.4–2.6 are generated in nitrogen and carbon dioxide. During each run, the full pressure history is recorded at fourteen locations along the wedge faces and schlieren images are produced. Numerical simulations performed based on the compressible Euler equations are validated against the experiment. The simulations are then used as an additional tool in the investigation.

The linearly convergent geometry strengthens the incoming shock repeatedly, as waves reflected from the wedge faces cross the interior of the wedge. This investigation shows that aspects of this structure persist through multiple reflections and influence the nature of the shock-wave focusing. The shock focusing resulting from the distributed reflected waves of the Mach 1.5 case is distinctly different from the stepwise focusing at the higher incoming shock Mach number. Further experiments using CO2 instead of N2 elucidate some relevant real-gas effects and suggest that the presence or absence of a weak leading shock on the distributed reflections is not a controlling factor for focusing.

We investigate thermally driven convection in a rotating spherical shell subject to inhomogeneous heating on the outer boundary, extending previous results to more rapid rotation rates and larger amplitudes of the boundary heating. The analysis explores the conditions under which steady flows can be obtained, and the stability of these solutions, for two boundary heating modes: first, when the scale of the boundary heating corresponds to the most unstable mode of the homogeneous problem; second, when the scale is larger. In the former case stable steady solutions exhibit a two-layer flow pattern at moderate rotation rates, but at very rapid rotation rates no steady solutions exist. In the latter case, stable steady solutions are always possible, and unstable solutions show convection rolls that cluster into nests that are out of phase with the boundary anomalies and remain trapped for many thermal diffusion times.

We extend a recent theory for steady uniform gravity-driven flow of a highly concentrated granular-fluid mixture over an erodible bed between frictional sidewalls. We first include angles of inclination greater than the angle of repose of the particles; then, we introduce a boundary condition for flow over a rigid bumpy bed. We compare the predictions of the resulting theory with the volume flow rates, depths and angles of inclination measured in the experiments on dry and variously saturated flows over rigid and erodible boundaries. Finally, we employ the resulting theory, with the assumption that the flow is shallow, to solve, in an approximate way, for the variation of height and average velocities along a steady non-uniform inclined flow of a granular-fluid mixture that moves over a rigid bumpy bed. The solutions exhibit features of the flow seen in the experiments – for example, a dry bulbous snout in advance of the fluid, whose length increases with increasing number of the particles and that disappears with increasing velocity – for which satisfactory explanations were lacking.

When an ocean wave breaks against a steep-fronted breakwater, sea wall or a similar marine structure, its impact on the structure can be very violent. This paper describes the theoretical studies that, together with field and laboratory investigations, have been carried out in order to gain a better understanding of the processes involved. The wave's approach towards a structure is modelled with classical irrotational flow to obtain the different types of impact profiles that may or may not lead to air entrapment. The subsequent impact is modelled with a novel compressible-flow model for a homogeneous mixture of incompressible liquid and ideal gas. This enables a numerical description of both trapped air pockets and the propagation of pressure shock waves through the aerated water. An exact Riemann solver is developed to permit a finite-volume solution to the flow model with smallest possible local error.

The high pressures measured during wave impacts on a breakwater are reproduced and it is shown that trapped air can be compressed to a pressure of several atmospheres. Pressure shock waves, reflected off nearby surfaces such as the seabed, can lead to pressures comparable with those of the impact. Typical examples of pressure-time histories, force and impulse are presented and discussed in terms of their practical implications. The numerical model proposed is relevant for a variety of flows where air effects are important. Further applications, including extended studies of wave impacts, are discussed.

In this article, we study the influence of solid-to-fluid density ratio m on the type of vortex-induced oscillation of a square section prism placed inside a two-dimensional channel. We assume that the solid body has neither structural damping nor spring restoring force. Accordingly, the prism equation of motion contains only inertia and aerodynamics forces. The problem is considered in the range of Reynolds numbers Re ∈ [50 200] (based on the prism cross-section height h) and channel widths H = H′/h ∈ [2.5 10]. We found that, for each Re and H, there is a critical mass ratio mc that separates two different oscillation regimes. For m > mc, the prism oscillation is periodical and contains a single harmonic. For m < mc, the prism oscillation changes completely and assumes an irregular pattern that is characterized by multiple harmonics that appear to belong to a uniform spectrum. The change from one regime to the other is abrupt and we were not able to observe a transitional regime in which the number of response harmonics grew by finite steps. The value of the critical mass ratio grows along with the Reynolds number and the channel width.

The dynamics of the air cavity created by vertical water entry of a three-dimensional body is investigated theoretically, computationally and experimentally. The study is focused in the range of relatively low Froude numbers, Fr ≡ V(gD)−1/2 ≤ O(10) (where V is the dropping velocity of the body, D its characteristic dimension and g the gravitational acceleration), when the inertia and gravity effects are comparable. To understand the physical processes involved in the evolution of cavity, we conduct laboratory experiments of water entry of freely dropping spheres. A matched asymptotic theory for the description of the cavity dynamics is developed based on the slender-body theory in the context of potential flow. Direct comparisons with experimental data show that the asymptotic theory properly captures the key physical effects involved in the development of the cavity, and in particular gives a reasonable prediction of the maximum size of the cavity and the time of cavity closure. Due to the inherent assumption in the asymptotic theory, it is incapable of accurately predicting the flow details near the free surface and the body, where nonlinear free surface and body boundary effects are important. To complement the asymptotic theory, a fully nonlinear numerical study using an axisymmetric boundary integral equation is performed. The numerically obtained dependencies of the cavity height and closure time on Froude number and body geometry are in excellent agreement with available experiments.