We study the axisymmetric stretching of a thin sheet of viscous fluid driven by a centrifugal body force. Time-dependent simulations show that the sheet radius R(t) tends to infinity in finite time. As time t approaches the critical time t*, the sheet becomes partitioned into a very thin central region and a relatively thick rim. A net momentum and mass balance in the rim leads to a prediction for the sheet radius near the singularity that agrees with the numerical simulations. By asymptotically matching the dynamics of the sheet with the rim, we find that the thickness h in the central region is described by a similarity solution of the second kind, with h ∝ (t* − t)α where the exponent α satisfies a nonlinear eigenvalue problem. Finally, for non-zero surface tension, we find that the exponent increases rapidly to infinity at a critical value of the rotational Bond number B = 1/4. For B > 1/4, surface tension defeats the centrifugal force, causing the sheet to retract rather than to stretch, with the limiting behaviour described by a similarity solution of the first kind.

We present the results from numerical and theoretical investigations of rotating Rayleigh–Bénard convection for relatively large dimensionless rotation rates, 170 < Ω < 274, and a Prandtl number of 6.4. Unexpected square patterns were found experimentally by Bajaj et al. (Phys. Rev. Lett., vol. 81, 1998, p. 806) in this parameter regime and near threshold for instability in the bulk. These square patterns have not yet been understood theoretically. Sánchez-Álvarez et al. (Phys. Rev. E, vol. 72, 2005, p. 036307) have found square patterns in numerical simulations for similar parameters when only the Coriolis force is included. We performed detailed numerical studies of rotating Rayleigh–Bénard convection for the same parameters as the experiments and simulations. To better understand these patterns, we compared the effects of the Coriolis force as well as the centrifugal force. We also computed the coefficients of the amplitude equation describing one-, two- and three-mode bulk solutions to rotating Rayleigh–Bénard convection. We find that squares are unstable, but we do find stable limit cycles consisting of three coupled oscillating amplitudes, which can superficially resemble squares, since one of the three amplitudes is rather small.

Locomotion of a passively flapping flat plate has been studied numerically by means of a multiblock lattice Boltzmann method. A flexible plate is modelled by a rigid plate with a torsion spring acting about the pivot at the leading edge of the plate. A dynamic model of this kind is called a lumped-torsional-flexibility model. When the leading edge is forced to heave sinusoidally, the plate pitches passively and propels itself in the horizontal direction as a result of the fluid–plate interaction. We have investigated various aspects of the mechanics behind the behaviour of the flapping plate, including the periodic- and non-periodic-flow states, the spontaneous motion of the plate, vortical structure and how they compare to similar propulsion systems in animals. In the periodic-flow regime, two dynamical responses of the passively pitching plate (forward and backward movements) are observed. Which movement will occur depends only on the frequency ratio F of the natural frequency of the system and the heaving frequency associated with the lumped torsional flexibility. It is found that the plate will select the forward movement when F > 1 and the backward movement when F ≤ 1. In the forward-movement regime, analysis of the dynamical behaviours and propulsive properties of the passively pitching plate indicates that the torsional flexibility can remarkably improve the propulsive performance. In addition, four kinds of vortex structures in the near wake are identified, which mainly depend on the forward speed of the plate. Finally the forward movement is compared to the flapping-based locomotion of swimming and flying animals. The results obtained in this study are consistent with the observations and measurements of swimming and flying animals; thus, they may provide physical insights into understanding of the propulsive mechanisms of the flapping wings and fins of animals.

Buoyancy-driven coastal currents, which are bounded by a coast and a surface density front, are ubiquitous and play essential role in the mesoscale variability of the ocean. Their highly unstable nature is well known from observations, laboratory and numerical experiments. In this paper, we revisit the linear stability problem for such currents in the simplest reduced-gravity model and study nonlinear evolution of the instability by direct numerical simulations. By using the collocation method, we benchmark the classical linear stability results on zero-potential-vorticity (PV) fronts, and generalize them to non-zero-PV fronts. In both cases, we find that the instabilities are due to the resonance of frontal and coastal waves trapped in the current, and identify the most unstable long-wave modes. We then study the nonlinear evolution of the unstable modes with the help of a new high-resolution well-balanced finite-volume numerical scheme for shallow-water equations. The simulations are initialized with the unstable modes obtained from the linear stability analysis. We found that the principal instability saturates in two stages. At the first stage, the Kelvin component of the unstable mode breaks, forming a Kelvin front and leading to the reorganization of the mean flow through dissipative and wave–mean flow interaction effects. At the second stage, a new, secondary unstable mode of the Rossby type develops on the background of the reorganized mean flow, and then breaks, forming coherent vortex structures. We investigate the sensitivity of this scenario to the along-current boundary and initial conditions. A study of the same problem in the framework of the fully baroclinic two-layer model will be presented in the companion paper.

The aerodynamic performance of a flapping two-dimensional wing section with simplified chord-wise flexibility is studied computationally. Bending stiffness is modelled by a torsion spring connecting two or three rigid components. The leading portion of the wing is prescribed with kinematics that are characteristic of biological hovering, and the aft portion responds passively. Coupled simulations of the Navier–Stokes equations and the wing dynamics are conducted for a wide variety of spring stiffnesses and kinematic parameters. Performance is assessed by comparison of the mean lift, power consumption and lift per unit power, with those from an equivalent rigid wing, and two cases are explored in greater detail through force histories and vorticity snapshots. From the parametric survey, four notable mechanisms are identified through which flexible wings behave differently from rigid counterparts. Rigid wings consistently require more power than their flexible counterparts to generate the same kinematics, as passive deflection leads to smaller drag and torque penalties. Aerodynamic performance is degraded in very flexible wings undergoing large heaving excursions, caused by a premature detachment of the leading-edge vortex. However, a mildly flexible wing has consistently good performance over a wide range of phase differences between pitching and heaving – in contrast to the relative sensitivity of a rigid wing to this parameter – due to better accommodation of the shed leading-edge vortex into the wake during the return stroke, and less tendency to interact with previously shed trailing-edge vortices. Furthermore, a flexible wing permits lift generation even when the leading portion remains nearly vertical, as the wing passively deflects to create an effectively smaller angle of attack, similar to the passive pitching mechanism recently identified for rigid wings. It is found that an effective pitch angle can be defined that accounts for wing deflection to align the results with those of the equivalent rigid wing.

Statistics obtained from seven different direct numerical simulations (DNSs) pertaining to a canonical turbulent boundary layer (TBL) under zero pressure gradient are compiled and compared. The considered data sets include a recent DNS of a TBL with the extended range of Reynolds numbers Reθ = 500–4300. Although all the simulations relate to the same physical flow case, the approaches differ in the applied numerical method, grid resolution and distribution, inflow generation method, boundary conditions and box dimensions. The resulting comparison shows surprisingly large differences not only in both basic integral quantities such as the friction coefficient cf or the shape factor H12, but also in their predictions of mean and fluctuation profiles far into the sublayer. It is thus shown that the numerical simulation of TBLs is, mainly due to the spatial development of the flow, very sensitive to, e.g. proper inflow condition, sufficient settling length and appropriate box dimensions. Thus, a DNS has to be considered as a numerical experiment and should be the subject of the same scrutiny as experimental data. However, if a DNS is set up with the necessary care, it can provide a faithful tool to predict even such notoriously difficult flow cases with great accuracy.

The self-similar evolution to turbulence of a multi-mode miscible Rayleigh–Taylor (RT) mixing layer has been investigated for Atwood numbers 0.03–0.6, using an air–helium gas channel experiment. Two co-flowing gas streams, one containing air (on top) and the other a helium–air mixture (at the bottom), initially flowed parallel to each other at the same velocity separated by a thin splitter plate. The streams met at the end of the splitter plate, with the downstream formation of a buoyancy unstable interface, and thereafter buoyancy-driven mixing. This buoyancy-driven mixing layer experiment permitted long data collection times, short transients and was statistically steady. Several significant designs and operating characteristics of the gas channel experiment are described that enabled the facility to be successfully run for At ~ 0.6. We report, and discuss, statistically converged measurements using digital image analysis and hot-wire anemometry. In particular, two hot-wire techniques were developed for measuring the various turbulence and mixing statistics in this air–helium RT experiment. Data collected and discussed include: mean density profiles, growth rate parameters, various turbulence and mixing statistics, and spectra of velocity, density and mass flux over a wide range of Atwood numbers (0.03 ≤ At ≤ 0.6). In particular, the measured data at the small Atwood number (0.03–0.04) were used to evaluate several turbulence-model constants. Measurements of the root mean square (r.m.s.) velocity and density fluctuations at the mixing layer centreline for the large At case showed a strong similarity to lower At behaviours when properly normalized. A novel conditional averaging technique provided new statistics for RT mixing layers by separating the bubble (light fluid) and spike (heavy fluid) dynamics. The conditional sampling highlighted differences in the vertical turbulent mass flux, and vertical velocity fluctuations, for the bubbles and spikes, which were not otherwise observable. Larger values of the vertical turbulent mass flux and vertical velocity fluctuations were found in the downward-falling spikes, consistent with larger growth rates and momentum of spikes compared with the bubbles.

Micro-cavitation bubbles generated by ultrasound have wide and important applications in medical ultrasonics and sonochemistry. An approximate theory is developed for nonlinear and non-spherical bubbles in a compressible liquid by using the method of matched asymptotic expansions. The perturbation is performed to the second order in terms of a small parameter, the bubble-wall Mach number. The inner flow near the bubble can be approximated as incompressible at the first and second orders, leading to the use of Laplace's equation, whereas the outer flow far away from the bubble can be described by the linear wave equation, also for the first and second orders. Matching between the two expansions provides the model for the non-spherical bubble behaviour in a compressible fluid. A numerical model using the mixed Eulerian–Lagrangian method and a modified boundary integral method is used to obtain the evolving bubble shapes. The primary advantage of this method is its computational efficiency over using the wave equation throughout the fluid domain. The numerical model is validated against the Keller–Herring equation for spherical bubbles in weakly compressible liquids with excellent agreement being obtained for the bubble radius evolution up to the fourth oscillation. Numerical analyses are further performed for non-spherical oscillating acoustic bubbles. Bubble evolution and jet formation are simulated. Outputs also include the bubble volume, bubble displacement, Kelvin impulse and liquid jet tip velocity. Bubble behaviour is studied in terms of the wave frequency and amplitude. Particular attention is paid to the conditions if/when the bubble jet is formed and when the bubble becomes multiply connected, often forming a toroidal bubble. When subjected to a weak acoustic wave, bubble jets may develop at the two poles of the bubble surface after several cycles of oscillations. A resonant phenomenon occurs when the wave frequency is equal to the natural oscillation frequency of the bubble. When subjected to a strong acoustic wave, a vigorous liquid jet develops along the direction of wave propagation in only a few cycles of the acoustic wave.

Existence of edge waves travelling along three-dimensional periodic structures is considered within the linear water-wave theory. A condition ensuring the existence is derived by analysing the spectrum of a suitably defined trace operator. The sufficient condition is a simple inequality comparing a weighted volume integral, taken over the submerged part of an element in the infinite array of identical obstacles, to the area of the free surface pierced by the obstacle. Various examples are given, and the results are extended to edge waves along periodic coastlines and over a periodically varying ocean floor.

To date, analytical models of internal tide generation by two-dimensional ridges have considered only idealized shapes. Here, we advance the Green function approach to address the generation of internal tides by two-dimensional topography of arbitrary shape, employing the Wentzel-Kramers-Brillouin (WKB) approximation to consider the impact of non-uniform stratifications. This allows for a more accurate analytical estimation of tidal conversion rates. Studies of single and double ridges reveal that the conversion rate and the nature of the radiated internal tide can be sensitive to the topographic shape, particularly around criticality and when there is interference between wave fields generated by neighbouring ridges. The method is then applied to the study of two important internal tide generation sites, the Hawaiian and Luzon Ridges, where it captures key features of the generation process.

An analysis is made of the noise generated during the passage of quiescent temperature/entropy inhomogeneities through regions of rapidly accelerated mean flow. This is an important source of jet engine core noise. Bake et al. (J. Sound Vib., vol. 326, 2009, pp. 574–598) have used an ‘entropy wave generator’ coupled with a converging–diverging nozzle to perform a series of canonical measurements of the sound produced when the inhomogeneity consists of a nominally uniform slug of hot gas. When flow separation and jet formation occur in the diffuser section of the nozzle, it is shown in this paper that the vortex sound generated by the jet is strongly correlated with the entropy noise produced by the slug and that the overall noise level is significantly reduced. Streamwise ‘stretching’ of the hot slug during high subsonic acceleration into the nozzle and the consequent attenuation of the entropy gradient in the nozzle are shown to significantly decrease the effective rate at which indirect combustion noise increases with the Mach number. Numerical predictions indicate that this is responsible for the peak observed by Bake et al. in the entropy-generated sound pressure at a nozzle Mach number near 0.6.

The breaking of fully nonlinear internal solitary waves of depression shoaling upon a uniformly sloping boundary in a smoothed two-layer density field was investigated using high-resolution two-dimensional simulations. Our simulations were limited to narrow-crested waves, which are more common than broad-crested waves in geophysical flows. The simulations were performed for a wide range of boundary slopes S ∈ [0.01, 0.3] and wave slopes extending the parameter range to weaker slopes than considered in previous laboratory and numerical studies. Over steep slopes (S ≥ 0.1), three distinct breaking processes were observed: surging, plunging and collapsing breakers which are associated with reflection, convective instability and boundary-layer separation, respectively. Over mild slopes (S ≤ 0.05), nonlinearity varies gradually and the wave fissions into a train of waves of elevation as it passes through the turning point where solitary waves reverse polarity. The dynamics of each breaker type were investigated and the predominance of a particular mechanism was associated with a relative developmental time scale. The breaking location was modelled as a function of wave amplitude (a), characteristic wave length and the isopycnal length along the slope. The breaker type was characterized in wave slope (Sw = a/Lw, where Lw is a measure of half of the wavelength) versus S space, and the reflection coefficient (R), modelled as a function of the internal Iribarren number, was in agreement with other studies. The effects of grid resolution and wave Reynolds number (Rew) on R, boundary-layer separation and the evolution of global instability were studied. High Reynolds numbers (Rew ~ 104) were found to trigger a global instability, which modifies the breaking process relative to the lower Rew case, but not necessarily the breaking location, and results in a ~ 10 % increase in R, relative to the Rew ~ 103 case.

The stability of pressure-driven flow in a rectangular channel with deformable neo-Hookean viscoelastic solid walls is analysed for a wide range of Reynolds numbers (from Re ≪ 1 to Re ≫ 1) by considering both sinuous and varicose modes for the perturbations. Pseudospectral numerical and asymptotic methods are employed to uncover the various unstable modes, and their stability boundaries are determined in terms of the solid elasticity parameter Γ = Vη/(ER) and the Reynolds number Re = RV ρ/η; here V is the maximum velocity of the laminar flow, R is the channel half-width, η and ρ are respectively the viscosity and density of the fluid and E is the shear modulus of the solid layer. We show that for small departures from a rigid solid, wall deformability could have a destabilizing or stabilizing effect on the Tollmien–Schlichting (TS) instability (a sinuous mode) depending on the solid-layer thickness. Upon further increase in solid deformability, the TS mode coalesces with another unstable mode (absent in rigid channels) giving rise to a single unstable mode which extends to very low Reynolds number (<1) for highly deformable walls. There are other types of instabilities that exist only due to wall deformability. In the absence of inertia (Re = 0), there is a short-wave instability of both sinuous and varicose modes arising due to the discontinuity of the first normal stress difference across the fluid–solid interface. For both sinuous and varicose modes, it is shown that inclusion of inertia is important even for Re ≪ 1, wherein a new class of long-wavelength unstable modes are predicted which are absent at Re = 0. These unstable modes are a type of shear waves in an elastic solid which are destabilized by the flow. These long- and short-wave instabilities are absent if a simple linear elastic model is used for the solid. At intermediate and high Re, upstream and downstream travelling waves of both sinuous and varicose modes become unstable. We show that sinuous and varicose modes become critical in different parameter regimes, thereby demonstrating the importance of capturing all the unstable modes. Inclusion of dissipative effects in the neo-Hookean model is generally shown to play a stabilizing role on the instabilities due to both sinuous and varicose modes. The predicted instabilities will be important for the flow of liquids (with viscosity ≥ 10−3 Pa s) in deformable channels of width ≤1 mm, and with shear modulus ≤ 105 Pa.

High-resolution direct numerical simulations are used to investigate freely decaying two-dimensional turbulence. We focus on the interplay between coherent vortices and vortex filaments, the second of which give rise to an inertial range. We find that Batchelor's prediction for the inertial-range enstrophy spectrum Eω(k, t) ~ β2/3k−1, where β is the enstrophy dissipation rate, is reasonably well satisfied once the turbulence is fully developed, but that the assumptions which underpin the usual interpretation of his theory are not valid. For example, the lack of a quasi-equilibrium cascade means the enstrophy flux Πω(k) is highly non-uniform throughout the inertial range, thus the common assumption that β can act as a surrogate for Πω(k) becomes questionable. We present a variant of Batchelor's theory which accounts for the wavenumber-dependence of Πω; in particular we propose Eω(k, t) ~ Πω(k1)2/3k−1, where k1 is the wavenumber marking the start of the observed k−1 region of the enstrophy spectrum. This provides a better collapse of the data and, unlike Batchelor's original theory, can be justified on theoretical grounds. The basis for our proposal is the observation that the straining of the vortex filaments, which fuels the enstrophy flux through the inertial range, comes almost exclusively from the strain field of the coherent vortices, and this can be characterized by Πω(k1)1/3. Thus Eω(k) is a function of only k and Πω(k1) in the inertial range, and dimensional analysis then yields Eω ~ Πω(k1)2/3k−1. We also confirm the prediction by Davidson (Phys. Fluids, vol. 20, 2008, 025106) that in the inertial range Πω varies as Πω(k)/Πω(k1) = 1 − a−1 ln(k/k1), where a is a constant of order 1. This corresponds to ∂Eω/∂t ~ k−1. Surprisingly, the measured enstrophy fluxes imply that the dynamics of the inertial range as defined by the behaviour of Πω extend to wavenumbers much smaller than k1, but this is masked in Eω(k, t) by the presence of coherent vortices which also contribute to Eω in this region. In fact, we find that kEω(k, t) ≈ H(k) + A(t), or ∂Eω/∂t ~ k−1 in this extended low-k region, where H(k) is almost independent of time and represents the signature of the coherent vortices. In short, the inertial range defined by ∂Eω/∂t ~ k−1 or Πω(k) ~ ln(k) is much broader than the observed Eω ~ k−1 region.

The flow field around a 6:1 prolate spheroid has been investigated by means of direct numerical simulations. Contrary to earlier studies the major axis of the spheroid was oriented perpendicular to the oncoming flow. At the subcritical Reynolds number 10 000 the laminar boundary layer separated from the frontal side of the spheroid and formed an elliptical vortex sheet. The detached shear layer was unstable from its very inception and even the near-wake turned out to be turbulent. The Strouhal number associated with the large-scale shedding was 0.156, significantly below that of the wake of a sphere. A higher-frequency mode was associated with Kelvin–Helmholtz instabilities in the detached shear layer. The shape of the near-wake mirrored the shape of the spheroid. Some 10 minor diameters downstream, the major axis of the wake became aligned with the minor axis of the spheroid.

Cylinder wakes have been studied extensively over several decades to better understand the basic flow phenomena encountered in such flows. The physics of the very near wake of the cylinder is perhaps the most challenging of them all. This region comprises the two detached shear layers, the recirculation region and wake flow. A study of the instability of the detached shear layers is important because these shear layers have a considerable impact on the dynamics of the very near wake. It has been observed experimentally that during certain periods of time that are randomly distributed, the measured fluctuating velocity component near the shear layers shows considerable amplification and it subsequently returns to its normal level (intermittency). Here, direct numerical simulations are used to accomplish a number of objectives such as confirming the presence of intermittency (computationally) and shedding light on processes that contribute significantly to intermittency and shear-layer transition/breakdown. Velocity time traces together with corresponding instantaneous vorticity contours are used in deciphering the fundamental processes underlying intermittency and shear-layer transition. The computed velocity spectra at three locations along the shear layer are provided. The computed shear-layer frequency agrees well with a power-law fit to experimental data.

The behaviour of fluid-particle acceleration in near-wall turbulent flows is investigated in numerically simulated turbulent channel flows at low to moderate Reynolds numbers, Reτ = 180~600). The acceleration is decomposed into pressure-gradient (irrotational) and viscous contributions (solenoidal acceleration) and the statistics of each component are analysed. In near-wall turbulent flows, the probability density function of acceleration is strongly dependent on the distance from the wall. Unexpectedly, the intermittency of acceleration is strongest in the viscous sublayer, where the acceleration flatness factor of O(100) is observed. It is shown that the centripetal acceleration around coherent vortical structures is an important source of the acceleration intermittency. We found sheet-like structures of strong solenoidal accelerations near the wall, which are associated with the background shear modified by the interaction between a streamwise vortex and the wall. We found that the acceleration Kolmogorov constant is a linear function of y+ in the log layer. The Reynolds number dependence of the acceleration statistics is investigated.

The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.

The addition of elastic polymers to turbulent liquids is known to produce significant drag reduction. In this study, we prove that the drag in pipe and channel flows of an unforced laminar fluid constitutes a lower bound for the drag of a fluid containing dilute elastic polymers. Further, the addition of elastic polymers to laminar fluids invariably increases drag. This proof does not rely on the adoption of a particular constitutive equation for the polymer force, and would also be applicable to other similar methods of drag reduction, which are also achieved by the addition of certain particles to a flow. Examples of such methods include the addition of surfactants to a flowing liquid and the presence of sand particles in sandstorms and water droplets in cyclones.

The exact theory of linearized water waves in a channel of indefinite length with bottom corrugations of finite amplitude (Howard & Yu, J. Fluid Mech., vol. 593, 2007, pp. 209–234) is extended to study the higher order Bragg resonances of water waves occurring when the corrugation wavelength is close to an integer multiple of half a water wavelength. The resonance tongues (ranges of water-wave frequencies) are given for these higher order cases. Within a resonance tongue, the wave amplitude exhibits slow exponential modulation over the corrugations, and slow sinusoidal modulation occurs outside it. The spatial rate of wave amplitude modulation is analysed, showing its quantitative dependence on the corrugation height, water-wave frequency and water depth. The effects of these higher order Bragg resonances are illustrated using the normal modes of a rectangular tank.