Multiphase flows in porous media represent fluid dynamics problems of great complexity involving a wide range of physical phenomena. These flows have attracted the attention of an impressive group of renowned researchers and have spawned a number of classic problems in fluid dynamics. These multiphase flows are perhaps best known for their importance in oil recovery from petroleum reservoirs, but they also find application in novel areas such as hydrofracturing for natural gas recovery. In a recent article, Zinchenko & Davis (J. Fluid Mech. 2013, vol. 725, pp. 611–663) present computational simulations that break new ground in the study of emulsions flowing through porous media. These simulations provide sufficient scale to capture the disordered motion and complex break-up patterns of individual droplets while providing sufficient statistical samples for estimating meaningful macroscopic properties of technical interest.

The two-dimensional stationary flow past a rotating cylinder is investigated for both two- and three-dimensional perturbations. The instability mechanisms are analysed using linear stability analysis and the complete neutral curve is presented. It is shown that the first bifurcation in the case of the rotating cylinder occurs for stationary three-dimensional perturbations, confirming recent experiments. Interestingly, the critical Reynolds number at high rotation rates is lower than that for the stationary circular cylinder. The spatial characteristics of the disturbance and a qualitative comparison with the results obtained for linear flows suggest that the stationary unstable three-dimensional mode could be of hyperbolic nature.

Studies on the $\alpha $- and $\beta $-phenomena, terms coined by Bouard & Coutanceau (J. Fluid Mech., vol. 101, 1980, pp. 583–607) for the flow past an impulsively started circular cylinder, have been confined only to the very early stages of the flow. In this paper, besides making a comprehensive in-depth analysis of these phenomena for a much longer period of time, we report the existence of some tertiary vortex phenomena for the first time, which we term the sub-$\alpha $- and sub-$\beta $-phenomena. The mechanism of unsteady flow separation at high Reynolds numbers for the flow past a circular cylinder developed in the last two decades has been used to understand these flow phenomena. The flow is computed using a recently developed compact finite difference method for the biharmonic form of the two-dimensional Navier–Stokes equations for the range of Reynolds number $500\leq \mathit{Re}\leq 10\hspace{0.167em} 000$. We specifically choose $\mathit{Re}= 5000$ to describe the interplay among the primary, secondary and tertiary vortices leading to these interesting vortex dynamics. We also report a $\beta $-like phenomenon which is very similar to the $\beta $-phenomenon, but slightly differs in details. We offer a new perception of the $\alpha $-phenomenon by defining its existence in a strong and weak sense along with a clearer characterization of the $\beta $-phenomenon. Apart from numerical computation, a detailed theoretical characterization using topological aspects of the boundary layer separation leading to the secondary and tertiary vortex phenomena has also been carried out. We compare our numerical results with established experimental and numerical results wherever available and an excellent match with the experimental results is obtained in all cases.

Wing deformation is observed during the flight of some insect species; however, the effect of these distorted wing shapes on the leading edge vortex (LEV) is not well understood. In this study, we investigate the effect of one of these deformation parameters, (rigid) wing camber, on the flow structures and aerodynamic forces for insect-like wings, using a numerical model of an altered fruit fly wing revolving at a constant angular velocity. Both positive and negative camber was investigated at Reynolds numbers of 120 and 1500, along with the chordwise location of maximum camber. It was found that negatively cambered wings produce very similar LEV structures to non-cambered wings at both Reynolds numbers, but high positive camber resulted in the formation of multiple streamwise vortices at the higher Reynolds number, which disrupt the development of the main LEV. Despite this, positively cambered wings were found to produce higher lift to drag ratios than flat or negatively cambered wings. It was determined that a region of low pressure near the wing’s leading edge, combined with the curvature of the wing’s upper surface in this region, resulted in a vertical tilting of the net force vector for positively cambered wings, which explains how insects can benefit from wing camber.

Grid turbulence of polyethylene oxide (PEO) solutions (Polyox WSR-301 in ${\mathrm{H} }_{2} \mathrm{O} $) has been investigated experimentally for three concentrations of 25, 50 and 100 weight ppm, at a turbulence Reynolds number based on a Taylor microscale of ${\mathit{Re}}_{\lambda } \approx 100$. For the first time, time sequences of turbulence spectra have been acquired at a rate of 0.005 Hz to reveal the spectral evolution due to mechanical degradation of the polymers. In contrast to spectra averaged over the entire degradation process, the sequence of spectra reveals a clear but time-dependent Lumley scale at which the energy spectrum changes abruptly from the Kolmogorov ${\kappa }^{- 5/ 3} $ inertial range to a ${\kappa }^{- 3} $ elastic range, in which the rate of strain is maintained constant by the polymers. The scaling of the initial Lumley length with Kolmogorov dissipation rate ${\epsilon }_{0} $ and molecular weight is determined, and a cascade model for the temporal decrease of molecular weight, i.e. for the breaking of polymer chains is presented. Finally, a heuristic model spectrum is developed which covers the cases of both maximum and partial turbulence reduction by polymers.

Experiments are performed on an elliptic circulation control aerofoil in an open-jet wind tunnel facility. The influence of blowing from a single trailing-edge slot on the external flow is assessed using two-component particle image velocimetry (PIV) and steady surface pressure measurements. The test section configuration (open jet or closed wall) significantly affects the leading-edge region of the flow field. PIV is also used to measure the curved wall jet and its interaction with the external flow near the trailing edge. PIV measurements of the curved wall jet reveal mean tangential velocity similarity in the outer region of the flow above the location where the tangential velocity reaches a local maximum. The length and velocity parameters required for similarity scale with the product of the chord Reynolds number and the momentum coefficient in accordance with the recent publication by Stalnov, Kribus & Seifert (J. Renew. Sustain. Energy, vol. 2, 2010, p. 063101). The separation location is also a function of the product of these parameters. The dataset is used to assemble equations to predict the similarity length scales, velocity scales and separation location. These equations compare well with the present measurements.

This paper analyses the adjoint solution of the Navier–Stokes equation. We focus on flow across a circular cylinder at three Reynolds numbers, ${\mathit{Re}}_{D} = 20, 100$ and $500$. The quantity of interest in the adjoint formulation is the drag on the cylinder. We use classical fluid mechanics approaches to analyse the adjoint solution, which is a vector field similar to a flow field. Production and dissipation of kinetic energy of the adjoint field is discussed. We also derive the evolution of circulation of the adjoint field along a closed material contour. These analytical results are used to explain three numerical solutions of the adjoint equations presented in this paper. The adjoint solution at ${\mathit{Re}}_{D} = 20$, a viscous steady state flow, exhibits a downstream suction and an upstream jet, the opposite of the expected behaviour of a flow field. The adjoint solution at ${\mathit{Re}}_{D} = 100$, a periodic two-dimensional unsteady flow, exhibits periodic, bean-shaped circulation in the near-wake region. The adjoint solution at ${\mathit{Re}}_{D} = 500$, a turbulent three-dimensional unsteady flow, has complex dynamics created by the shear layer in the near wake. The magnitude of the adjoint solution increases exponentially at the rate of the first Lyapunov exponent. These numerical results correlate well with the theoretical analysis presented in this paper.

This paper presents the results of an in-depth study of the flow past a streamwise oscillating cylinder, examining the impact of varying the amplitude and frequency of the oscillation, and the Reynolds number of the incoming flow. These findings are presented in a framework that shows that the relationship between the frequency of vortex shedding ${f}_{s} $ and the amplitude of oscillation ${A}^{\ast } $ is governed by two primary factors: the first is a reduction of ${f}_{s} $ proportional to a series in ${A}^{\ast 2} $ over a wide range of driving frequencies and Reynolds numbers; the second is nonlinear synchronization when this adjusted ${f}_{s} $ is in the vicinity of $N= {(1- {f}_{s} / {f}_{d} )}^{- 1} $, where $N$ is an integer. Typically, the influence of higher-order terms is small, and truncation to the first term of the series (${A}^{\ast 2} $) well represents the overall trend of vortex shedding frequency as a function of amplitude. However, discontinuous steps are overlaid on this trend due to the nonlinear synchronization. When ${f}_{s} $ is normalized by the Strouhal frequency ${f}_{St} $ (the frequency of vortex shedding from an unperturbed cylinder), the rate at which ${f}_{s} / {f}_{St} $ decreases with amplitude, at least for ${f}_{d} / {f}_{St} = 1$, shows a linear dependence on the Reynolds number. For a fixed $\mathit{Re}= 175$, the truncated series shows that the rate of decrease of ${f}_{s} / {f}_{St} $ with amplitude varies as ${(2- {f}_{d} / {f}_{St} )}^{- 1/ 2} $ for $1\leqslant {f}_{d} / {f}_{St} \leqslant 2$, but is essentially independent of ${f}_{d} / {f}_{St} $ for ${f}_{d} / {f}_{St} \lt 1$. These trends of the rate of decrease of ${f}_{s} $ with respect to amplitude are also used to predict the amplitudes of oscillation around which synchronization occurs. These predicted amplitudes are shown to fall in regions of the parameter space where synchronized modes occur. Further, for the case of varying ${f}_{d} / {f}_{St} $, a very reasonable prediction of the amplitude of oscillation required for the onset of synchronization to the mode where ${f}_{s} = 0. 5{f}_{d} $ is given. In a similar manner, amplitudes at which ${f}_{s} = 0$ are calculated, predicting where the natural vortex shedding is completely supplanted by the forcing. These amplitudes are found to coincide approximately with those at which the onset of a symmetric vortex shedding mode is observed. This result is interpreted as meaning that the symmetric shedding mode occurs when the dynamics crosses over from being dominated by the vortex shedding to being dominated by the forcing.

Fully developed plane channel flow rotating in the spanwise direction has been studied analytically and numerically. Linear stability analysis reveals that all cross-flow modes are stable for supercritical rotation numbers, $Ro\gt R{o}_{c} $, where $R{o}_{c} $ will approach 3 from below for increasing Reynolds number. Plane Tollmien–Schlichting (TS) waves are independent of rotation and always linearly unstable for supercritical Reynolds numbers. Direct numerical simulation (DNS) of the laminarization process reveals that the turbulence is damped when $Ro$ approaches $R{o}_{c} $. Hence, the laminarization is dominated by linear mechanisms. The flow becomes periodic for supercritical Reynolds numbers and rotation rates, i.e. when $Ro\gt R{o}_{c} $ and $Re\gt R{e}_{c} $. At such rotation rates, all oblique (cross-flow) modes are damped and when the disturbance amplitude becomes small enough, the TS modes start to grow exponentially. Secondary instabilities are initially blocked by the rotation since all cross-flow modes are linearly stable and the breakdown to turbulence will be strongly delayed. Hence, the TS waves will reach extremely high amplitudes, much higher than for typical turbulent fluctuations. Eventually, the extreme-amplitude state with TS-like waves will break down to turbulence and the flow will laminarize due to the influence of the rapid rotation, thus completing the cycle that will then be repeated. This flow is strongly dominated by linear mechanisms, which is remarkable considering the extremely high amplitudes involved in the processes of laminarization of the turbulence at $Ro\geq R{o}_{c} $ and the growth of the unstable TS waves.

The characteristics of flow past a partial cross-stream array of (idealized) tidal turbines are investigated both analytically and computationally to understand the mechanisms that determine the limiting performance of partial tidal fences. A two-scale analytical partial tidal fence model reported earlier is further extended by better accounting for the effect of array-scale flow expansion on device-scale dynamics, so that the new model is applicable to short fences (consisting of a small number of devices) as well as to long fences. The new model explains theoretically general trends of the limiting performance of partial tidal fences. The new model is then compared to three-dimensional Reynolds-averaged Navier–Stokes (RANS) computations of flow past an array of various numbers (up to 40) of actuator disks. On the whole, the analytical model agrees well with the RANS computations, suggesting that the two-scale dynamics described in the analytical model predominantly determines the fence performance in the RANS computations as well. The comparison also suggests that the limiting performance of short partial fences depends on how much of device far-wake mixing takes place within the array near-wake region. This factor, however, depends on the structures of the wake and therefore on the type/design of devices to be arrayed.

In this paper we examine the dynamics of an initially stable bubble due to ultrasonic forcing by an acoustic wave. A tissue layer is modelled as a density interface acted upon by surface tension to mimic membrane effects. The effect of a rigid backing to the thin tissue layer is investigated. We are interested in ultrasound contrast agent type bubbles which have immediate biomedical applications such as the delivery of drugs and the instigation of sonoporation. We use the axisymmetric boundary integral technique detailed in Curtiss et al. (J. Comput. Phys., 2013, submitted) to model the interaction between a single bubble and the tissue layer. We have identified a new peeling mechanism whereby the re-expansion of a toroidal bubble can peel away tissue from a rigid backing. We explore the problem over a large range of parameters including tissue layer depth, interfacial tension and ultrasonic forcing.

Direct numerical simulations have been performed to study the drag reduction resulting from spatial oscillations of a segment of the wall under a turbulent boundary layer. The oscillating motion is imposed by utilizing a streamwise modulated spanwise wall forcing. The results are compared with earlier simulations using temporal oscillations with an identical segment and forcing amplitudes, and with a frequency related to the wavelength through a convective velocity. Two different oscillation amplitudes with equal oscillation wavelength have been used, which allows for a direct comparison between a relatively weak and strong forcing of the flow. The weaker forcing results in 25 % drag reduction while the stronger forcing, with twice the amplitude, yields 41 % drag reduction. Comparison with the temporal cases reveals drastically improved energy savings for the spatial oscillation technique, in accordance with earlier channel flow investigations. The streamwise variation of spanwise shear is shown to follow the analytical solution to the laminar Navier–Stokes equations derived under the assumption of constant friction velocity. Furthermore, the spanwise velocity profiles at various phases are compared with the analytical solution, and show very good agreement. The downstream development of the spatial Stokes layer thickness is theoretically estimated to be ${\sim }{x}^{1/ 15} $, in general agreement with the simulation data. The spatial variation of the spanwise Reynolds stress is investigated and compared with the variation in time for the temporal wall forcing cases. The controversy regarding a zero or non-zero production of spanwise Reynolds stress in the temporal case is elucidated. In addition, comparison with the spatial case reveals that a second production term originating from the downstream variation of the spanwise wall velocity has a negative contribution to the production, and hence relates to the larger drag reduction in the case of spatial forcing.

The unifying theory of scaling in thermal convection (Grossmann & Lohse, J. Fluid. Mech., vol. 407, 2000, pp. 27–56; henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (Phys. Rev. Lett., vol. 86, 2001, pp. 3320–3323) in the regime $3\times 1{0}^{7} \leq \mathit{Ra}\leq 3\times 1{0}^{9} $ and $4\leq \mathit{Pr}\leq 34$ and Grossmann & Lohse (Phys. Rev. E, vol. 66, 2002, p. 016305) used the experimental data point from Qiu & Tong (Phys. Rev. E, vol. 64, 2001, p. 036304) and the fact that $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ is independent of the parameter $a$, which relates the dimensionless kinetic boundary thickness with the square root of the wind Reynolds number, to fix the Reynolds number dependence. Meanwhile the theory is, on the one hand, well-confirmed through various new experiments and numerical simulations; on the other hand, these new data points provide the basis for an updated fit in a much larger parameter space. Here we pick four well-established (and sufficiently distant) $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ data points and show that the resulting $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ function is in agreement with almost all established experimental and numerical data up to the ultimate regime of thermal convection, whose onset also follows from the theory. One extra $\mathit{Re}(\mathit{Ra}, \mathit{Pr})$ data point is used to fix $\mathit{Re}(\mathit{Ra}, \mathit{Pr})$. As $\mathit{Re}$ can depend on the definition and the aspect ratio, the transformation properties of the GL equations are discussed in order to show how the GL coefficients can easily be adapted to new Reynolds number data while keeping $\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$ unchanged.

We investigate the transfer properties of energy and helicity fluctuations in fully developed homogeneous and isotropic turbulence by changing the nature of the nonlinear Navier–Stokes terms. We perform a surgery of all possible interactions, by keeping only those triads that have sign-definite helicity content. In order to do this, we apply an exact decomposition of the velocity field in a helical Fourier basis, as first proposed by Constantin & Majda (Commun. Math. Phys, vol. 115, 1988, p. 435) and exploited in great detail by Waleffe (Phys. Fluids A, vol. 4, 1992, p. 350), and we evolve the Navier–Stokes dynamics keeping only those velocity components carrying a well-defined (positive or negative) helicity. The resulting dynamics preserves translational and rotational symmetries but not mirror invariance. We give clear evidence that this three-dimensional homogeneous and isotropic chiral turbulence is characterized by a stationary inverse energy cascade with a spectrum ${E}_{back} (k)\sim {k}^{- 5/ 3} $ and by a direct helicity cascade with a spectrum ${E}_{forw} (k)\sim {k}^{- 7/ 3} $. Our results are important to highlight the dynamics and statistics of those subsets of all possible Navier–Stokes interactions responsible for reversal events in the energy-flux properties, and demonstrate that the presence of an inverse energy cascade is not necessarily connected to a two-dimensionalization of the flow. We further comment on the possible relevance of such findings to flows of geophysical interest under rotations and in thin layers. Finally we propose other innovative numerical experiments that can be achieved by using a similar decimation of degrees of freedom.

Extreme wetting properties of solids, either superhydrophobic or superhydrophilic, provide versatile methods to achieve unusual liquid deposit morphologies, such as liquid pearls or polygonal films. Here we report the dynamics of liquid drops that impact on solid surfaces where the extreme wetting properties are coupled in such a way that a superhydrophilic annulus is patterned on a superhydrophobic background. The drop that initially spreads on the inner superhydrophobic region is arrested by the hydrophilic annulus. The liquid deposit gets destabilized because of the strong water repellence of the inner region, exhibiting the burst and disengagement of the liquid film. This process leads to the formation of a liquid ring defined by the annulus pattern, which has practical implications in rapid printing of functional liquids. We visualize such drop dynamics with a high-speed camera and characterize their salient features by combining experimental measurements and theoretical considerations.

We consider the three-dimensional, cylindrical equivalent to the problem of instability between two inviscid fluids due to a velocity shear between them, known as Kelvin–Helmholtz instability. We begin by developing the solution to the linearized equations for a rotating fluid. While this solution is not in itself new, we carry the analysis further than previous treatments by including non-zero modes and considering the effect of the surface tension, particularly on the dispersion relation. We then consider a system of two fluids rotating at different rates and derive its dispersion relation, which is rather more complicated than that for a single rotating fluid. While a general analytic solution is unattainable, by investigating some special cases we show that the fundamental mode is always stable, and that Kelvin–Helmholtz instability also exists for the system. We compare our results with experiments and conclude by suggesting some hypothetical links between the theory and nature.

The stability of a two-dimensional inviscid flow in an annulus between two permeable cylinders is examined. The basic flow is irrotational, and both radial and azimuthal components of the velocity are non-zero. The direction of the radial flow can be from the inner cylinder to the outer one (the diverging flow) or from the outer cylinder to the inner one (the converging flow). It is shown that, independent of the direction of the radial flow, the basic flow is unstable to small two-dimensional perturbations provided that the ratio of the azimuthal component of the velocity to the radial one is sufficiently large. The instability is oscillatory and persists if the viscosity of the fluid is taken into consideration.

The flow around an isolated cylinder spinning at high rotation rates in free stream is investigated. The existence of two steady two-dimensional states is confirmed, as is the existence of a secondary mode of vortex shedding. The stability of the two steady states to three-dimensional perturbations is established using linear stability analysis. At lower rotation rates on the first steady state, two three-dimensional modes are confirmed, and their structure and curves of marginal stability as a function of rotation rate and Reynolds number are determined. One mode (named mode $E$) appears consistent with a hyperbolic instability in the wake, while the second (named mode $F$) appears to be a centrifugal instability of the flow very close to the cylinder surface. At higher rotation rates on the second steady state, a single three-dimensional mode due to centrifugal instability (named mode ${F}^{\prime } $) is found. This mode becomes increasingly difficult to excite as the rotation rate is increased.

The velocity potential is derived for a transient source of arbitrary strength undergoing arbitrary three-dimensional motion. The initially quiescent fluid of infinite depth is assumed to be inviscid, incompressible and homogeneous. The upper surface of the fluid is covered by a thin layer of elastic material of uniform density with lateral stress. The linearized initial boundary-value problem is formulated within the framework of the potential-flow theory, and the Laplace transform technique is employed to obtain the solution. The potential of a time-harmonic source with forward speed is obtained as a particular case. The far-field wave motion at long time is determined via the method of stationary phase. The problems of radiation (surge, sway and heave) of the flexural–gravity waves by a submerged sphere advancing at constant forward speed are investigated. The method of multipole expansions is used. Numerical results are obtained for the wave-making resistance and lift, added-mass and damping coefficients. The effects of an ice sheet and broken ice on the hydrodynamic loads are discussed in detail.

Instabilities and long-term evolution of two-dimensional circular flows around a rigid circular cylinder (island) are studied analytically and numerically. For that we consider a base flow consisting of two concentric neighbouring rings of uniform but different vorticity, with the inner ring touching the cylinder. We first study the inviscid linear stability of such flows to perturbations of the free edges of the rings. For a given ratio of the vorticity in the rings, the governing parameters of the problem are the radii of the inner and outer rings scaled on the cylinder radius. In this two-dimensional parameter space, we determine analytically the regions of linear stability/instability of each azimuthal mode $m= 1, 2, \ldots . $ In the physically most meaningful case of zero net circulation, for each mode $m\gt 1$, two regions are identified: a regular instability region where mode $m$ is unstable along with some other modes, and a unique instability region where only mode $m$ is unstable. After the conditions of linear instability are established, inviscid contour-dynamics and high-Reynolds-number finite-element simulations are conducted. In the regular instability regions, simulations of both kinds typically result in the formation of vortical dipoles or multipoles. In the unique instability regions, where the inner vorticity ring is much thinner than the outer ring, the inviscid contour-dynamics simulations do not reveal dipole emission. In the viscous simulation, because viscosity has time to widen the inner ring, the instability develops in the same manner as in the regular instability regions.