Wing or fin flexibility can dramatically affect the performance of flying and swimming animals. Both laboratory experiments and numerical simulations have been used to study these effects, but analytical results are notably lacking. Here, we develop small-amplitude theory to model a flapping wing that pitches passively due to a combination of wing compliance, inertia and fluid forces. Remarkably, we obtain a class of exact solutions describing the wing’s emergent pitching motions, along with expressions for how thrust and efficiency are modified by compliance. The solutions recover a range of realistic behaviours and shed new light on how flexibility can aid performance, the importance of resonance, and the separate roles played by wing and fluid inertia. The simple robust estimates afforded by our theory may prove valuable even in situations where details of the flapping motion and wing geometry differ.

The effect of initial conditions on transition to turbulence is studied in a variable-density shock-driven flow. Richtmyer–Meshkov instability (RMI) evolution of fluid interfaces with two different imposed initial perturbations is observed before and after interaction with a second shock reflected from the end wall of a shock tube (reshock). The first perturbation is a predominantly single-mode long-wavelength interface which is formed by inclining the entire tube to 80$^{\circ }$ relative to the horizontal, yielding an amplitude-to-wavelength ratio, $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D706}=0.088$, and thus can be considered as half the wavelength of a triangular wave. The second interface is multi-mode, and contains additional shorter-wavelength perturbations due to the imposition of shear and buoyancy on the inclined perturbation of the first case. In both cases, the interface consists of a nitrogen-acetone mixture as the light gas over carbon dioxide as the heavy gas (Atwood number, $A\sim 0.22$) and the shock Mach number is $M\approx 1.55$. The initial condition was characterized through Proper Orthogonal Decomposition and density energy spectra from a large set of initial condition images. The evolving density and velocity fields are measured simultaneously using planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) techniques. Density, velocity, and density–velocity cross-statistics are calculated using ensemble averaging to investigate the effects of additional modes on the mixing and turbulence quantities. The density and velocity data show that a distinct memory of the initial conditions is maintained in the flow before interaction with reshock. After reshock, the influence of the long-wavelength inclined perturbation present in both initial conditions is still apparent, but the distinction between the two cases becomes less evident as smaller scales are present even in the single-mode case. Several methods are used to calculate the Reynolds number and turbulence length scales, which indicate a transition to a more turbulent state after reshock. Further evidence of transition to turbulence after reshock is observed in the velocity and density fluctuation spectra, where a scaling close to $k^{-5/3}$ is observed for almost one decade, and in the enstrophy fluctuation spectra, where a scaling close to $k^{1/3}$ is observed for a similar range. Also, based on normalized cross correlation spectra, local isotropy is reached at lower wave numbers in the multi-mode case compared with the single-mode case before reshock. By breakdown of large scales to small scales after reshock, rapid decay can be observed in cross-correlation spectra in both cases.

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

In this study we investigate the Kolmogorov flow (a shear flow with a sinusoidal velocity profile) in a weakly stratified, two-dimensional fluid. We derive amplitude equations for this system in the neighbourhood of the initial bifurcation to instability for both low and high Péclet numbers (strong and weak thermal diffusion, respectively). We solve amplitude equations numerically and find that, for low Péclet number, the stratification halts the cascade of energy from small to large scales at an intermediate wavenumber. For high Péclet number, we discover diffusively spreading, thermal boundary layers in which the stratification temporarily impedes, but does not saturate, the growth of the instability; the instability eventually mixes the temperature inside the boundary layers, so releasing itself from the stabilizing stratification there, and thereby grows more quickly. We solve the governing fluid equations numerically to compare with the asymptotic results, and to extend the exploration well beyond onset. We find that the arrest of the inverse cascade by stratification is a robust feature of the system, occurring at higher Reynolds, Richards and Péclet numbers – the flow patterns are invariably smaller than the domain size. At higher Péclet number, though the system creates slender regions in which the temperature gradient is concentrated within a more homogeneous background, there are no signs of the horizontally mixed layers separated by diffusive interfaces familiar from doubly diffusive systems.

The gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. The limiting behaviour of the film profile as the steepness of the wall geometry is increased is discussed. Using substantial numerical evidence, it is established that as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. The effect of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates is also considered.

A model of vortex breakdown is presented and its predictions compared with the experiments of Sarpkaya (1971). The model is cntred about a theory of long, weakly nonlinear waves propagating on critical flows in tubes of variable cross-section. Although the weakly nonlinear theory must be extended beyond its domain of formal validity, many of the experimentally observed features of vortex breakdown are reproduced by the model. The description of the time evolution of the flow field that is presented requires numerical calculations that are not simple, but some important conclusions may be determined by easy computations. In particular, the axial position of a breakdown may be found from a very simple equation (10).

A model fractal forcing for direct numerical simulations (DNS) of stationary, homogeneous and isotropic turbulence is proposed. The power spectrum of this forcing is a power-law function of wavenumber with a positive exponent that is an increasing function of $D_{f}$, the fractal dimension of the fractal stirrer. The following results are obtained for DNS turbulence subjected to fully self-similar fractal forcing of fractal dimension $D_f$. (i) The Taylor and Kolmogorov microscales are proportional to each other and to the smallest length scale of the fractal forcing. (ii) The integral length scale is much smaller than the size $L_b$ of the DNS box and a decreasing function of the extent of the fractal forcing range because fractal forcing generates very irregular velocity fields. (iii) In qualitative (but not quantitative) agreement with renormalization group (RG) theories of turbulence, higher values of $D_f$ lead to increased energy at the highest wavenumbers. (iv) The energy and energy input rate spectra and the inter-scale energy transfer $T(k)$ all scale with the turbulence r.m.s. velocity $u'$ and the Taylor microscale $\lambda$. (v) More than 80% of the total dissipation occurs in the fractal forcing range of scales which extends from $L_b$ to about one to two times the Taylor microscale. In that range, $T(k)$ is negligible. (vi) $\lambda \,{\sim}\, \nu/u'$ (where $\nu$ is the kinematic viscosity), the kinetic energy dissipation rate per unit mass $\epsilon \,{\sim}\, u'^{3}/\lambda \,{\sim}\,u'^{4}/\nu$ and the velocity derivative skewness $S$ is independent of Reynolds number in the limit where the Reynolds number and the fractal forcing range are increased together. (vii) The intermediate eigenvalue of the strain rate tensor is on average positive, and the negligible values of $T(k)$ in the fractal forcing range are accompanied by lower values of $S$ and a significant reduction in local compression by comparison to turbulence forced only at the large scales. (viii) The geometrical alignments between vorticity, strain rate tensor eigenvectors and vortex stretching vector are qualitatively as in turbulence forced only at the large scales but significantly weakened. (ix) The p.d.f.s of velocity increments are approximately, though not exactly, Gaussian at all scales between the Kolmogorov and integral length scales. A few preliminary DNS results are also given for the case of turbulence generated by a fractal forcing that is discretely, as opposed to fully, self-similar.

The unsteady drag and lift generated by the interaction of a gravity current with a bottom-mounted square cylinder are investigated by means of high-resolution Navier–Stokes simulations. Two-dimensional simulations for Reynolds numbers (Re) O(1000) and three-dimensional simulations for Re = O(10000) demonstrate that the drag coefficient increases exponentially towards a maximum as the current meets the cylinder, then undergoes strong fluctuations and eventually approaches a quasi-steady value. The simulation results show that the maximum drag coefficient can reach a value of 3, with the quasi-steady value being O(1), which should aid in selecting a design drag coefficient for submarine structures under the potential impact of gravity currents. The transient drag and lift fluctuations after impact are associated with the Kelvin–Helmholtz vortices in the mixing layer between the gravity current and the ambient fluid. As these vortices pass over the cylinder, they cause the convection of separated flow regions along the bottom wall towards the cylinder. In two-dimensional simulations at Re = O(10000), these flow structures are seen to be unrealistically coherent and to persist throughout the interaction, thus resulting in a noticeable overprediction of the drag and lift fluctuations. On the other hand, the impact of the current on the cylinder is seen to be very well captured by two-dimensional simulations at all Re values. Three-dimensional simulations lead to excellent agreement with available experimental data throughout the flow/structure interaction. They show that the spanwise variation of the drag is determined by the gravity current's lobe-and-cleft structure at impact and by an unsteady cellular flow structure similar to that found in constant-density flows at later times. A comparison between gravity-current flows and corresponding constant-density flows shows the hydrostatic drag component to be important for gravity currents.

In this paper we describe the restrictions that the probability density function (p.d.f.) of the size of particles resulting from the rupture of a drop or bubble must satisfy. Using conservation of volume, we show that when a particle of diameter, D0, breaks into exactly two fragments of sizes D and D2 = (D30−D3)1/3 respectively, the resulting p.d.f., f(D; D0), must satisfy a symmetry relation given by D22f(D; D0) = D2f(D2; D0), which does not depend on the nature of the underlying fragmentation process. In general, for an arbitrary number of resulting particles, m(D0), we determine that the daughter p.d.f. should satisfy the conservation of volume condition given by m(D0) ∫0D0 (D/D0)3f(D; D0) dD = 1. A detailed analysis of some contemporary fragmentation models shows that they may not exhibit the required conservation of volume condition if they are not adequately formulated. Furthermore, we also analyse several models proposed in the literature for the breakup frequency of drops or bubbles based on different principles, g(ϵ, D0). Although, most of the models are formulated in terms of the particle size D0 and the dissipation rate of turbulent kinetic energy, ϵ, and apparently provide different results, we show here that they are nearly identical when expressed in dimensionless form in terms of the Weber number, g*(Wet) = g(ϵ, D0) D2/30 ϵ−1/3, with Wet ~ ρ ϵ2/3D05/3/σ, where ρ is the density of the continuous phase and σ the surface tension.

This paper presents experimental results on the nonlinear phase locking present in the resonant growth of unstable modes in the shear layer of an axisymmetric jet. The initial instability modes scale with the exiting shear layer and grow convectively with downstream distance. Because of the special condition at the exit lip of the jet, the initial growth of modes is very sensitive to local unsteady pressure fields. A part of the unsteady field is stochastic in nature. To a larger extent, the pressure field at the lip of the jet contains the imprint of the downstream-developing instability modes, in particular the first unstable axisymmetric mode and its subharmonic. These are felt at the lip of the jet as a result of the energetic processes of the first vortex rollup and vortex pairing. As a result, a resonant feedback exists which under special conditions makes the initial region of this flow in some sense absolutely unstable. The features of this process are brought out by the normalized crossbispectrum or cross-bicoherence between the instantaneous unsteady pressure at the lip of the jet and velocity time series measured at the same azimuthal position for different downstream locations. These give a measure of the nonlinear phase locking between the principle modes and their sum and difference modes. Analysis of these show a perfect nonlinear phase locking at the fundamental axisymmetric and subharmonic frequencies between the pressure field at the lip and the velocity field at the downstream locations corresponding to the energy saturations of the fundamental and subharmonic modes. This resonance process can be suppressed or enhanced by low-amplitude axisymmetric mode forcing at the natural preferred frequency of slightly detuned cases. Contrasted to this is the behaviour of the fundamental m = ± 1 helical mode. This mode was found to have the same spatial growth rate as the axisymmetric mode and a streamwise frequency approximately 20 % higher, in agreement with theoretical predictions. However, short-time spectral estimates showed that these two fundamental modes do not exist at the same time or space. This suggests that each is a basin of attraction which suppresses the existence of the other. The apparent non-deterministic switching observed between these modes is probably the result of the response of the jet to stochastic input of axisymmetric or non-axisymmetric disturbances. This scenario may lead to a low-dimensional temporal model based on the interaction between these two modes which captures most of the early random nature seen in our experiments.

The laminar viscous flow in the gap between two concentric spheres is investigated for a rotating inner sphere. The solution is obtained by solving the Navier-Stokes equations by means of finite-difference techniques, where the equations are restricted to axially symmetric flows. The flow field is hydrodynamically unstable above a critical Reynolds number. This investigation indicates that the critical Reynolds number beyond which Taylor vortices appear is slightly higher in a spherical gap than for the flow between concentric cylinders. The formation of Taylor vortices could be observed only for small gap widths s ≤ 0·17. The final state of the flow field depends on the initial conditions and the acceleration of the inner sphere. Steady and unsteady flow modes are predicted for various Reynolds numbers and gap widths. The results are in agreement with experiment if certain accuracy conditions of the finite-difference methods are satisfied. It is seen that the equatorial symmetry is of great importance for the development of the Taylor vortices in the gap.

Two-dimensional simulations of gas cavity responses to both weak shocks (p ≤ 30 MPa) and strong shocks (p ranging from 500 to 2000 MPa) are performed using a finite volume method. An artificial viscosity to capture the shock and a simple, stable, and adaptive mesh generation technique have been developed for the computations. The details of the shock propagation, rarefaction, transmission and bubble wall motions are obtained from the numerical computations. A weak shock is defined in the present context as one that does not cause liquid jet formation upon impact with the bubble. For this case, a large pressure is created within the gas upon collapse due to rapid compression of the gas, ultimately causing the re-expansion of the bubble. The bubble collapse and re-expansion time predicted by this model agree well with spherically symmetric computations. When impacted by strong shock waves, the bubble will collapse and a liquid jet is formed that propagates through the bubble to the opposite bubble wall. Jet speeds as high as 2000 m s−1 are predicted by this model.

This paper presents experimental data obtained for the refraction of a plane shock wave at a carbon dioxide–helium interface. The gases were separated initially by a delicate polymer membrane. Both regular and irregular wave systems were studied, and a feature of the latter system was the appearance of bound and free precursor shocks. Agreement between theory and experiment is good for regular systems, but for irregular ones it is sometimes necessary to take into account the effect of the membrane inertia to obtain good agreement. The basis for the analysis of irregular systems is one-dimensional piston theory and Snell's law.

A thin ‘rope’ of viscous fluid falling from a sufficient height onto a surface forms a series of regular coils. Here we investigate theoretically and experimentally a curious feature of this instability: the existence of multiple states with different frequencies at a fixed value of the fall height. Using a numerical model based on asymptotic ‘thin rope’ theory, we determine curves of coiling frequency $\Omega$vs. fall height $H$ as functions of the fluid viscosity $\nu$, the diameter $d$ of the injection hole, the volumetric injection rate $Q$, and the acceleration due to gravity $g$. In addition to the three coiling modes previously identified (viscous, gravitational and inertial), we find a new multivalued ‘inertio-gravitational’ mode that occurs at heights intermediate between gravitational and inertial coiling. In the limit when the rope is strongly stretched by gravity and $\Pi_1\,{\equiv}\, (\nu^5/g Q^3)^{1/5}\,{\gg}\, 1$, inertio-gravititational coiling occurs in the height range $O(\Pi_1^{-1/6})\,{\leq}\, H(g/\nu^2)^{1/3}\,{\leq}\, O(\Pi_1^{-5/48})$. The frequencies of the individual branches are proportional to $(g/H)^{1/2}$, and agree closely with the eigenfrequencies of a whirling liquid string with negligible resistance to bending and twisting. The number of coexisting branches scales as $\Pi_1^{5/32}$. The predictions of the numerical model are in excellent agreement with laboratory experiments performed by two independent groups using different apparatus and working fluids. The experiments further show that interbranch transitions in the inertio-gravitational regime occur via an intermediate state with a ‘figure of eight’ geometry that changes the sense of rotation of the coiling.

The steady motion of a Newtonian fluid in a rectangular enclosure open on its upper side is considered under the action of thermocapillary forces due to surface-tension gradients along the free surface. An asymptotic solution, in the limiting case of the aspect ratio A → 0, is found and discussed for the cases where the surface deformation may be neglected, that is for contact angles at the lateral walls equal to ½π and very small values of the crispation number. The flow field has also been analysed in a wide range of the governing parameters A, Mg, Cr, by a computational model particularly appropriate to simulate free-surface flows. For A [Lt ] 1 the numerical results confirm the behaviour predicted by the asymptotic theory, while for A [ges ] 1 several characteristic features of the flow-field structure are emphasized. For increasing Mg, the surface layer under the free surface maintains in the mid-section a constant value, dependent only on A, and decreases together with the thermal boundary-layer thickness near the lateral walls. For increasing A, the motion remains confined in a region near the free surface; hence the overall Nu, starting from the pure conduction value (Nu → 1 as A → 0) inccreases with A, reaching a maximum, to tend again to unity as A → ∞. The surface deformation, at least for very small values of the crispation number, seems to have a negligible influence on the qualitative aspects of the flow-field structure.

Similarity solutions in which the vorticity decays algebraically towards the outer limit of the viscous layer are shown to be possible limit solutions of the full boundary-layer equations with exponential decay. The discussion is illustrated by consideration of a mainstream applicable to viscous flow in a cone.

Based on a similarity between piston-mode sloshing in a two-dimensional rectangular moonpool and a small-amplitude oscillatory flow through a slot in a slatted screen, a pressure discharge in the moonpool opening is theoretically quantified. This makes it possible, by changing the dynamic free-surface condition inside the moonpool, to account for the vortex-induced damping and thereby to modify the inviscid potential flow statement. The present paper is an important addition to Faltinsen et al. (J. Fluid Mech., vol. 575, 2007, pp. 359–397), where a discrepancy between the potential flow theory and model tests was reported. New calculations are supported by the earlier model tests of the authors as well as numerical data obtained by means of viscous solvers.

The effect of torsion on fully developed laminar flow in a twisted pipe is considered in the low-Reynolds-number regime. The flow in a pipe of elliptical cross-section whose axis is straight but which is twisted about that axis is examined and the secondary flow pattern found. The helical pipe of circular cross-section is also revisited; in this case, the first-order torsional effect on the streamtubes is verified, and previous conclusions that such an effect would be of higher order are explained.

The purpose of this paper is to provide secondary instability analysis of stationary crossflow vortices on a hypersonic yawed straight circular cone with a $7^{\circ }$ half-angle at $6^{\circ }$ angle of attack, free-stream Mach number 6 and unit Reynolds number $10.09\times 10^{6}~\text{m}^{-1}$. At an angle of attack, a three-dimensional boundary layer is developed between the windward and leeward symmetry planes. Under the action of azimuthal pressure gradients, the flow near the surface is deflected more than the flow near the edge of the boundary layer. This results in an inflectional velocity profile that can sustain the growth of crossflow vortices. The stationary crossflow instability is computed by means of the nonlinear parabolized stability equations, including a methodology to predict the stationary-crossflow marching path and variation of the spanwise number of waves in the marching direction solely from the basic state. Secondary instability analysis is performed using spatial BiGlobal equations based on two-dimensional partial differential equations. The secondary instabilities are calculated at different axial locations along two crossflow vortex trajectories selected to complement experiments conducted in the Mach 6 Quiet Tunnel at Texas A&M University and in the Boeing/AFOSR Mach 6 Quiet Tunnel at Purdue University. The secondary instability analysis captures various instability modes. Similar to observations in the low-speed regime for an infinite swept wing, secondary shear-layer instabilities are amplified as a consequence of the three-dimensional shear layer formed by crossflow vortices. Also, low-frequency travelling crossflow and high-frequency second modes coexist with the shear-layer instabilities. These results are shown to be in good agreement with the two sets of hypersonic yawed cone experiments (one with natural surface roughness and one with artificial discrete roughness) and compare well with experimental measurements of an incompressible swept wing.

A reactive fluid dissolving a uniform porous material triggers an instability in the dissolution front, leading to spontaneous formation of pronounced well-spaced channels in the surrounding rock matrix. The concentration field within the dissolving region contains two different length scales, upstream (no reaction) and downstream of the front position. Previous investigations of the reactive-infiltration instability have considered one or other of the scales to be dominant, leading to rather different conclusions. Here we describe a more general linear stability analysis which includes both length scales simultaneously. We show how previous work corresponds to special cases of our more general analysis and obtain closed-form solutions for small permeability gradients.