A novel method for performing spectral analysis of a fluid flow solely based on snapshot sequences from numerical simulations or experimental data is presented by Schmid (J. Fluid Mech., 2010, this issue, vol. 656, pp. 5–28). Dominant frequencies and wavenumbers are extracted together with dynamic modes which represent the associated flow structures. The mathematics underlying this decomposition is related to the Koopman operator which provides a linear representation of a nonlinear dynamical system. The procedure to calculate the spectra and dynamic modes is based on Krylov subspace methods; the dynamic modes reduce to global linear eigenmodes for linearized problems or to Fourier modes for (nonlinear) periodic problems. Schmid (2010) also generalizes the analysis to the propagation of flow variables in space which produces spatial growth rates with associated dynamic modes, and an application of the decomposition to subdomains of the flow region allows the extraction of localized stability information. For finite-amplitude flows this spectral analysis identifies relevant frequencies more effectively than global eigenvalue analysis and decouples frequency information more clearly than proper orthogonal decomposition.

The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders.

In this paper we study the waves generated over the slipline and their interactions with other waves for Mach reflection in steady two-dimensional supersonic flow. We find that a series of expansion and compression waves exist over the slip line, even in the region immediately behind the leading part of the reflected shock wave, previously regarded as a uniform flow. These waves make the leading part of the slipline, previously regarded as straight, deviate nonlinearly towards the reflecting surface. When the transmitted expansion waves from the upper corner first intersect the slipline, an inflexion point is produced. Downstream of this inflexion point, compression waves are produced over the slipline. By considering the interaction between the various expansion or compression waves, we obtain a Mach stem height, the shape and position of the slipline and reflected shock wave, compared well to computational fluid dynamics (CFD) results. We also briefly consider the case with a subsonic portion behind the reflected shock wave. The global flow pattern is obtained through CFD and the starting point of the sonic line is identified through a simple analysis. The sonic line appears to coincide with the first Mach wave from the upper corner expansion fan after transmitted from the reflected shock wave.

The effects of small-scale dissipative fluidic actuation on the evolution of large- and small-scale motions in a turbulent shear layer downstream of a backward-facing step are investigated experimentally. Actuation is applied by modulation of the vorticity flux into the shear layer at frequencies that are substantially higher than the frequencies that are typically amplified in the near field, and has a profound effect on the evolution of the vortical structures within the layer. Specifically, there is a strong broadband increase in the energy of the small-scale motions and a nearly uniform decrease in the energy of the large-scale motions which correspond to the most amplified unstable modes of the base flow. The near field of the forced shear layer has three distinct domains. The first domain (x/θ0 < 50) is dominated by significant concomitant increases in the production and dissipation of turbulent kinetic energy and in the shear layer cross-stream width. In the second domain (50 < x/θ0 < 300), the streamwise rates of change of these quantities become similar to the corresponding rates in the unforced flow although their magnitudes are substantially different. Finally, in the third domain (x/θ0 > 350) the inviscid instability of the shear layer re-emerges in what might be described as a ‘new’ baseline flow.

The main aim of this paper is to relate instability modes with modes obtained from proper orthogonal decomposition (POD) in the study of global spatio-temporal nonlinear instabilities for flow past a cylinder. This is a new development in studying nonlinear instabilities rather than spatial and/or temporal linearized analysis. We highlight the importance of multi-modal interactions among instability modes using dynamical system and bifurcation theory approaches. These have been made possible because of accurate numerical simulations. In validating computations with unexplained past experimental results, we noted that (i) the primary instability depends upon background disturbances and (ii) the equilibrium amplitude obtained after the nonlinear saturation of primary growth of disturbances does not exhibit parabolic variation with Reynolds number, as predicted by the classical Stuart–Landau equation. These are due to the receptivity of the flow to background disturbances for post-critical Reynolds numbers (Re) and multi-modal interactions, those produce variation in equilibrium amplitude for the disturbances that can be identified as multiple Hopf bifurcations. Here, we concentrate on Re = 60, which is close to the observed second bifurcation. It is also shown that the classical Stuart–Landau equation is not adequate, as it does not incorporate multi-modal interactions. To circumvent this, we have used the eigenfunction expansion approach due to Eckhaus and the resultant differential equations for the complex amplitudes of disturbance field have been called here the Landau–Stuart–Eckhaus (LSE) equations. This approach has not been attempted before and here it is made possible by POD of time-accurate numerical simulations. Here, various modes have been classified either as a regular mode or as anomalous modes of the first or the second kind. Here, the word anomalous connotes non-compliance with the Stuart–Landau equation, although the modes originate from the solution of the Navier–Stokes equation. One of the consequences of multi-modal interactions in the LSE equations is that the amplitudes of the instability modes are governed by stiff differential equations. This is not present in the traditional Stuart–Landau equation, as it retains only the nonlinear self-interaction. The stiffness problem of the LSE equations has been resolved using the compound matrix method.

We consider a viscous, incompressible fluid confined in a narrow annular channel rotating rapidly about its axis of symmetry with angular velocity Ω that itself precesses slowly about an axis fixed in an inertial frame. The precessional problem is characterized by three parameters: the Ekman number E, the Poincaré number ε and the aspect ratio of the channel Γ. Dependent upon the size of Γ, precessionally driven flows can be either resonant or non-resonant with the Poincaré forcing. By assuming that it is the viscous effect, rather than the nonlinear effect, that plays an essential role at exact resonance, two asymptotic expressions for ε ≪ 1 and E ≪ 1 describing the single and double inertial-mode resonance are derived under the non-slip boundary condition. An asymptotic expression describing non-resonant precessing flows is also derived. Further studies based on numerical integrations, including two-dimensional linear analysis and direct three-dimensional nonlinear simulation, show a satisfactory quantitative agreement between the three asymptotic expressions and the fuller numerics for small and moderate Reynolds numbers at an asymptotically small E. The transition from two-dimensional precessing flow to three-dimensional small-scale turbulence for large Reynolds numbers is also investigated.

Cloud cavitation occurrence about a sphere is investigated in a variable-pressure water tunnel using low- and high-speed photography. The model sphere, 0.15 m in diameter, was sting-mounted within a 0.6 m square test section and tested at a constant Reynolds number of 1.5 × 106 with cavitation numbers varying between 0.36 and 1.0. High-speed photographic recordings were made at 6 kHz for several cavitation numbers providing insight into cavity shedding and nucleation physics. Shedding phenomena and frequency content were investigated by means of pixel intensity time series data using wavelet analysis. Instantaneous cavity leading edge location was investigated using image processing and edge detection.

The boundary layer at cavity separation is shown to be laminar for all cavitation numbers, with Kelvin–Helmholtz instability and transition to turbulence in the separated shear layer the main mechanism for cavity breakup and cloud formation at high cavitation numbers. At intermediate cavitation numbers, cavity lengths allow the development of re-entrant jet phenomena, providing a mechanism for shedding of large-scale Kármán-type vortices similar to those for low-mode shedding in single-phase subcritical flow. This shedding mode, which exists at supercritical Reynolds numbers for single-phase flow, is eliminated at low cavitation numbers with the onset of supercavitation.

Complex interactions between the separating laminar boundary layer and the cavity were observed. In all cases the cavity leading edge was structured in laminar cells separated by well-known ‘divots’. The initial laminar length and divot density were modulated by the unsteady cavity shedding process. At cavitation numbers where shedding was most energetic, with large portions of leading edge extinction, re-nucleation was seen to be circumferentially periodic and to consist of stretched streak-like bubbles that subsequently became fleck-like. This process appeared to be associated with laminar–turbulent transition of the attached boundary layer. Nucleation occurred periodically in time at these preferred sites and formed the characteristic cavity leading edge structure after sufficient accumulation of vapour had occurred. These observations suggest that three-dimensional instability of the decelerating boundary layer flow may have significantly influenced the developing structure of the cavity leading edge.

We use direct numerical simulations to study a steady bathtub vortex in a cylindrical tank with a central drain-hole, a flat stress-free surface and velocity prescribed at the inlet. We find that the qualitative structure of the meridional flow does not depend on the radial Reynolds number, whereas we observe a weak overall rotation at a low radial Reynolds number and a concentrated vortex above the drain-hole at a high radial Reynolds number. We introduce a simple analytically integrable model that shows the same qualitative dependence on the radial Reynolds number as the simulations and compares favourably with the results for the radial velocity and the azimuthal velocity at the surface. Finally, we describe the height dependence of the radius of the vortex core and the maximum of the azimuthal velocity at a high radial Reynolds number, and we show that the data on the radius of the vortex core and the maximum of the azimuthal velocity as functions of height collapse on single curves by appropriate scaling.

This theoretical study is devoted to description of fluid flow and heat transfer in a spreading viscous drop with phase transition. A similarity solution for the combined full Navier–Stokes equations and energy equation for the expanding lamella generated by drop impact is obtained for a general case of oblique drop impact with high Weber and Reynolds numbers. The theory is applicable to the analysis of the phenomena of drop solidification, target melting and film boiling. The theoretical predictions for the contact temperature at the substrate surface agree well with the existing experimental data.

Modifications have been made to an analytical model proposed by Shusser & Gharib (J. Fluid Mech., vol. 416, 2000b, pp. 173–185) for the vortex ring formation and pinch-off process in a starting jet. Compared with previous models, the present investigation distinguishes the leading vortex ring from its trailing jet so as to consider the details of the pinch-off process in terms of the properties of the leading vortex ring, which are determined by considering the flux of kinetic energy, impulse and circulation from the trailing jet into the leading vortex ring by convective transportation. A two-stage process has been identified before the complete separation of the leading vortex ring from its trailing jet. The first stage involves the growth of the leading vortex ring by absorbing all the ejected fluid from the nozzle until certain optimum size is achieved. The second stage is characterized by the appreciable translational velocity of the leading vortex ring followed by a trailing jet. The leading vortex ring is approximated as a Norbury vortex ring with growing characteristic core radius ϵ such that dimensionless energy α, as well as its translational velocity and penetration depth, can be estimated. By applying the Kelvin–Benjamin variational principle, the pinch-off process is signified by two time scales, i.e. the formation number, which indicates the onset of the pinch-off process, and the separation time, which corresponds to the time when the leading vortex ring becomes physically separated from the trailing jet and is therefore referred to as the end of the pinch-off process. The effect of nozzle geometry, i.e. a straight nozzle or a converging nozzle, has also been taken into account by using different descriptions of the growth of the trailing jet. The prediction of the formation number and the characteristics of the vortex ring are found to be in good agreement with previous experimental results on starting jets with straight nozzles and converging nozzles, respectively.

The dynamics of a viscous compressible fluid, confined between two parallel plane walls and excited by a sudden impulse transverse to the walls, is studied on the basis of the linearized Navier–Stokes equations. It is shown that the time-dependent flow depends strongly on the sound velocity and on the shear and volume viscosity. Under favourable conditions an echoing effect can be observed, with a sound pulse bouncing many times between the two plates. The velocity correlation function of a Brownian particle immersed in the fluid is calculated in point approximation. It shows a similar strong dependence on fluid properties.

The three-dimensional global optimal dynamics of a flat-plate boundary layer is studied by means of an adjoint-based optimization in a spatial domain of long – but finite – streamwise dimension. The localized optimal initial perturbation is characterized by a pair of streamwise-modulated counter-rotating vortices, tilted upstream, yielding at the optimal time elongated streaks of alternating sign in the streamwise direction. This indicates that perturbations with non-zero streamwise wavenumber have a role in the transient dynamics of a boundary layer. A scaling law is provided, describing the variation of the streamwise modulation of the optimal initial perturbation with respect to the streamwise domain length and to the Reynolds number. For spanwise-extended domains, a near-optimal three-dimensional perturbation is extracted during the optimization process; it is localized also in the spanwise direction, resulting in a wave packet of elongated disturbances modulated in the spanwise and streamwise directions. The nonlinear evolution of the optimal and near-optimal perturbations is investigated by means of direct numerical simulations. Both perturbations are found to induce transition at lower levels of the initial energy than local optimal and suboptimal perturbations. Moreover, it is observed that transition occurs in a well-defined region of the convected wave packet, close to its centre, via a mechanism including at the same time oscillations of the streaks of both quasi-sinuous and quasi-varicose nature. Hairpin vortices are observed before transition; they have an active role in the breakdown of the streaks and result in a turbulent spot which spreads out in the boundary layer.

A wind-forced weakly nonlinear weakly dispersive evolution model is derived for a continuously stratified circular lake of slowly varying depth under the effect of the Earth's rotation. The model was numerically integrated to investigate the evolution of long internal waves of vertical mode one for various sets of environmental parameters. It is demonstrated that the Kelvin wave steepens as it propagates, and the steepened front subsequently generates a train of oscillatory waves. It is demonstrated that Poincaré waves do not steepen, but their amplitude is modulated in an oscillatory manner with time, exhibiting a pseudo recurrence character. The model was applied to the wind forced problem, confirming that Kelvin and Poincaré waves are the dominant response. Energy partition among Kelvin and Poincaré wave modes is estimated as a function of wind-forcing parameters. For large lakes, the most significant wave amplitude is found in the Kelvin wave mode, but the gross field energy is most significantly contained in Poincaré wave modes.

We present three-dimensional direct numerical simulations (DNS) of the Kida vortex flow, a prototypical turbulent flow, using a novel high-order lattice Boltzmann (LB) model. Extensive comparisons of various global and local statistical quantities obtained with an incompressible-flow spectral element solver are reported. It is demonstrated that the LB method is a promising alternative for DNS as it quantitatively captures all the computed statistics of fluid turbulence.

This theoretical study examines confined viscous planar jet/wake flows with continuous velocity profiles. These flows are characterized by the shear, confinement, Reynolds number and shear-layer thickness. The primary aim of this paper is to determine the effect of confinement on viscous jets and wakes and to compare these results with corresponding inviscid results. The secondary aim is to consider the effect of viscosity and shear-layer thickness. A spatio-temporal analysis is performed in order to determine absolute/convective instability criteria. This analysis is carried out numerically by solving the Orr–Sommerfeld equation using a Chebyshev collocation method. Results are produced over a large range of parameter space, including both co-flow and counter-flow domains and confinements corresponding to 0.1 < h2/h1 < 10, where the subscripts 1 and 2 refer to the inner and outer streams, respectively. The Reynolds number, which is defined using the channel width, takes values between 10 and 1000. Different velocity profiles are used so that the shear layers occupy between 1/2 and 1/24 of the channel width. Results indicate that confinement has a destabilizing effect on both inviscid and viscous flows. Viscosity is found always to be stabilizing, although its effect can safely be neglected above Re = 1000. Thick shear layers are found to have a stabilizing effect on the flow, but infinitely thin shear layers are not the most unstable; having shear layers of a small, but finite, thickness gives rise to the strongest instability.

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.

A complete solution is obtained for the wave field generated by the time-harmonic edgewise oscillations of a horizontal circular disk in an incompressible stratified viscous fluid. The linearized equations of viscous internal waves and the no-slip condition on the rigid disk are used to derive sets of dual integral equations for the fluid velocity and vorticity. The dual integral equations are solved by analytic reduction to sets of linear algebraic equations. Asymptotic results confirm that this edgewise motion no longer excites waves in the small-viscosity limit. Broadside oscillations and the effect of density diffusion are also considered.

When a charged particle moves along a charged wall in a polar fluid, it experiences an electroviscous lift force normal to the surface and an electroviscous drag, superimposed on the viscous drag, parallel to the surface. Here a theoretical analysis is presented to determine the electroviscous drag on a charged spherical particle surrounded by a thin electrical double layer near a charged plane wall, when the particle translates parallel to the wall without rotation, in a symmetric electrolyte solution at rest. The electroviscous (electro-hydrodynamic) forces, arising from the coupling between the electrical and hydrodynamic equations, are determined as a solution of three partial differential equations, for electroviscous ion concentration (perturbed ion clouds), electroviscous potential (perturbed electric potential) and electroviscous or electro-hydrodynamic flow field (perturbed flow field). The problem was previously solved for small gap widths and low Peclet numbers in the inner region around the gap between the sphere and the wall, using lubrication theory. Here the restriction on the particle–wall distances is removed, and an analytical and numerical solution is obtained valid for the whole domain of interest. For large sphere–wall separations the solution approaches that for the electroviscous drag on an isolated sphere in an unbounded fluid. For small particle–wall distances it differs from that obtained by the use of lubrication theory, showing that lubrication theory is inadequate for electroviscous problems. The analytical results are in complete agreement with the full numerical calculations. For small particle–wall distances a model is given which provides both physical insight and an easy way to calculate the force with high precision.

Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. Among these equations, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact two-dimensional linear theory compared to other MS-type equations. Nevertheless, it has a disadvantage of being a vector equation, i.e. it requires solving a system of two coupled partial differential equations. In addition, for three-dimensional problems, there is a difficulty in constructing the additional boundary condition needed for the solution. In the present work, it is shown how the vector CMSE can be transformed into an equivalent scalar equation using a pseudo-potential formulation. The pseudo-potential mild-slope equation (PMSE) preserves the accuracy of the CMSE while avoiding the need of an additional boundary condition. Furthermore, the PMSE significantly reduces the computational effort relative to the CMSE, since it is a scalar equation. The accuracy of the new model was tested numerically by comparing it to laboratory data and analytical solutions.

We present a detailed study of the linear stability of the plane Couette–Poiseuille flow in the presence of a crossflow. The base flow is characterized by the crossflow Reynolds number Rinj and the dimensionless wall velocity k. Squire's transformation may be applied to the linear stability equations and we therefore consider two-dimensional (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, k ∈ [0, 1], two ranges of Rinj exist where unconditional stability is observed. In the lower range of Rinj, for modest k we have a stabilization of long wavelengths leading to a cutoff Rinj. This lower cutoff results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Crossflow stabilization and Couette stabilization appear to act via very similar mechanisms in this range, leading to the potential for a robust compensatory design of flow stabilization using either mechanism. As Rinj is increased, we see first destabilization and then stabilization at very large Rinj. The instability is again a long-wavelength mechanism. An analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien–Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilization at very large Rinj appears to be due to decay in energy production, which diminishes like Rinj−1. Our study is limited to two-dimensional, spanwise-independent perturbations.