When homogeneous turbulence acts upon a non-uniform magnetic field B0(x, t), a mean electromotive force [Escr ](x, t) is in general established owing to the correlation between the velocity field and the fluctuating magnetic field that is generated. The relation between [Escr ] and B0 is linear, and it is generally believed that, when the scale of inhomogeneity of B0 is sufficiently large, it can be expressed as a series involving successively higher spatial derivatives of B0: \[ {\cal E}_i = \alpha_{il} B_{0l}+\beta_{ilm}\partial B_{0l}/\partial x_m+\ldots, \] where the tensors αil, βilm,… are determined (in principle) by the statistical properties of the turbulence and the magnetic diffusivity Λ of the fluid. These tensors are of crucial importance in turbulent dynamo theory. In this paper the question of their asymptotic form in the limit Λ → 0 is considered. By putting Λ = 0 and assuming that the field is non-random at an initial instant t = 0, the developing form of the tensors αl(t) and βm(t) is determined. The expressions involve time integrals of Lagrangian correlation functions associated with the velocity field, which are comparable in structure with the eddy diffusion tensor in the analogous turbulent diffusion problem (Taylor 1921). Some doubts are expressed concerning the convergence of the time integrals as t → ∞, and it is concluded that a satisfactory treatment of the problem will require the inclusion of weak diffusion effects (as recognized by Parker 1955).

The variances of the fluid-particle acceleration and of the pressure-gradient and viscous force are given. The scaling parameters for these variances are velocity statistics measureable with a single-wire anemometer. For both high and low Reynolds numbers, asymptotic scaling formulas are given; these agree quantitatively with DNS data. Thus, the scaling can be presumed known for all Reynolds numbers. Fluid-particle acceleration variance does not obey K41 scaling at any Reynolds number; this is consistent with recent experimental data. The non-dimensional pressure-gradient variance named λT/λP is shown to be obsolete.

In nature, water waves usually propagate in groups and the open question about the characteristics of the highest possible wave in a group is of significant theoretical and practical interest. We examine the problem of the highest non-breaking wave in a wave group by direct numerical simulations of the exact Euler equations. The main aim of the study is twofold: (i) to describe the highest wave in a group in fully nonlinear setting and find its dependence on parameters; (ii) to examine correspondence between the exact breather solutions of weakly nonlinear analytic theory based on the integrable nonlinear Schrödinger (NLS) equation and their strongly nonlinear analogues. In contrast to weakly nonlinear models the very notion of the highest wave is ill-defined: the maximal crest elevation, the maximal trough-to-crest height and the deepest trough all occur at close but different moments; correspondingly, we have to speak about distinctively different extreme waves. In the simulations small initial perturbation of a uniform wave train were prescribed in a way ensuring that the initial perturbation excites a single breather-type modulation. The ensuing growth results in higher wave magnitudes and takes longer time to develop compared with the NLS theory. The maxima of crest elevation noticeably exceed their weakly nonlinear analogues. The wave with the highest crest differs significantly from the unmodulated wave: the local wavelength contracts considerably, the crest becomes noticeably higher; the vicinity of the crest of such an extreme wave is close to that of the limiting Stokes periodic wave. Thus, the shape of the maximal crest wave is almost universal, i.e. it practically does not depend on the way the wave group evolved, or even whether there was initially more than one group. The evolution of a single NLS breather has been shown to have a qualitatively similar but quantitatively quite different analogue in the fully nonlinear setting. The one-to-one mapping of the NLS breather solutions onto fully nonlinear ones has been constructed. The fully nonlinear breathers are found to be robust, which provides grounds for applying the results for developing short-term deterministic forecasting of rogue waves.

In commonly used formulations of the Boussinesq approximation centrifugal buoyancy effects related to differential rotation, as well as strong vortices in the flow, are neglected. However, these may play an important role in rapidly rotating flows, such as in astrophysical and geophysical applications, and also in turbulent convection. Here we provide a straightforward approach resulting in a Boussinesq-type approximation that consistently accounts for centrifugal effects. Its application to the accretion-disc problem is discussed. We numerically compare the new approach to the typical one in fluid flows confined between two differentially heated and rotating cylinders. The results justify the need of using the proposed approximation in rapidly rotating flows.

We study the modulational instability of geophysical Rossby and plasma drift waves within the Charney–Hasegawa–Mima (CHM) model both theoretically, using truncated (four-mode and three-mode) models, and numerically, using direct simulations of CHM equation in the Fourier space. We review the linear theory of Gill (Geophys. Fluid Dyn., vol. 6, 1974, p. 29) and extend it to show that for strong primary waves the most unstable modes are perpendicular to the primary wave, which correspond to generation of a zonal flow if the primary wave is purely meridional. For weak waves, the maximum growth occurs for off-zonal inclined modulations that are close to being in three-wave resonance with the primary wave. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the nonlinear jet pinching predicted by Manin & Nazarenko (Phys. Fluids, vol. 6, 1994, p. 1158). We find that, for strong primary waves, these narrow zonal jets further roll up into Kármán-like vortex streets, and at this moment the truncated models fail. For weak primary waves, the growth of the unstable mode reverses and the system oscillates between a dominant jet and a dominate primary wave, so that the truncated description holds for longer. The two-dimensional vortex streets appear to be more stable than purely one-dimensional zonal jets, and their zonal-averaged speed can reach amplitudes much stronger than is allowed by the Rayleigh–Kuo instability criterion for the one-dimensional case. In the long term, the system transitions to turbulence helped by the vortex-pairing instability (for strong waves) and the resonant wave–wave interactions (for weak waves).

Experiments were performed in dump-stabilized axisymmetric flames to assess turbulent flame speeds (${S}_{T} $) and mean flamelets speeds (stretched laminar flame speeds, ${S}_{L, k} $). Fuels with significantly different thermodiffusive properties have been investigated, ranging from pure methane to syngas (${\mathrm{H} }_{2} \text{{\ndash}} \mathrm{CO} $ blends) and pure hydrogen, while the pressure was varied from 0.1 to 1.25 MPa. Flame front corrugation was measured with planar laser-induced fluorescence (PLIF) of the OH radical, while turbulence quantities were determined with particle image velocimetry (PIV). Two different analyses based on mass balance were performed on the acquired flame images. The first method assessed absolute values of turbulent flame speeds and the second method, by means of an improved fractal methodology, provided normalized turbulent flame speeds (${S}_{T} / {S}_{L, k} $). Deduced average Markstein numbers exhibited a strong dependence on pressure and hydrogen content of the reactive mixture. It was shown that preferential-diffusive-thermal (PDT) effects acted primarily on enhancing the stretched laminar flame speeds rather than on increasing the flame front corrugations. Interaction between flame front and turbulent eddies measured by the fractal dimension was shown to correlate with the eddy temporal activity.

We present experimental results on high-Reynolds-number motion of partially non-wetting liquid drops on inclined plane surfaces using: (i) water on fluoro-alkyl silane (FAS)-coated glass; and (ii) mercury on glass. The former is a high-hysteresis ($3{5}^{\circ } $) surface while the latter is a low-hysteresis one (${6}^{\circ } $). The water drop experiments have been conducted for capillary numbers $0. 0003\lt Ca\lt 0. 0075$ and for Reynolds numbers based on drop diameter $137\lt Re\lt 3142$. The ranges for mercury on glass experiments are $0. 0002\lt Ca\lt 0. 0023$ and $3037\lt Re\lt 20\hspace{0.167em} 069$. It is shown that when $Re\gg 1{0}^{3} $ for water and $Re\gg 10$ for mercury, a boundary layer flow model accounts for the observed velocities. A general expression for the dimensionless velocity of the drop, covering the whole $Re$ range, is derived, which scales with the modified Bond number ($B{o}_{m} $). This expression shows that at low $Re$, $Ca\sim B{o}_{m} $ and at large $Re$, $Ca \sqrt{Re} \sim B{o}_{m} $. The dynamic contact angle (${\theta }_{d} $) variation scales, at least to first-order, with $Ca$; the contact angle variation in water, corrected for the hysteresis, collapses onto the low-$Re$ data of LeGrand, Daerr & Limat (J. Fluid Mech., vol. 541, 2005, pp. 293–315). The receding contact angle variation of mercury has a slope very different from that in water, but the variation is practically linear with $Ca$. We compare our dynamic contact angle data to several models available in the literature. Most models can describe the data of LeGrand et al. (2005) for high-viscosity silicon oil, but often need unexpected values of parameters to describe our water and mercury data. In particular, a purely hydrodynamic description requires unphysically small values of slip length, while the molecular-kinetic model shows asymmetry between the wetting and dewetting, which is quite strong for mercury. The model by Shikhmurzaev (Intl J. Multiphase Flow, vol. 19, 1993, pp. 589–610) is able to group the data for the three fluids around a single curve, thereby restoring a certain symmetry, by using two adjustable parameters that have reasonable values. At larger velocities, the mercury drops undergo a change at the rear from an oval to a corner shape when viewed from above; the corner transition occurs at a finite receding contact angle. Water drops do not show such a clear transition from oval to corner shape. Instead, a direct transition from an oval shape to a rivulet appears to occur.

We consider Lagrangian stochastic modelling of the relative motion of two fluid particles in the inertial range of a turbulent flow. Eulerian analysis of such modelling corresponds to an equation for the Eulerian probability distribution of velocity-vector increments which introduces a hierarchy of constraints for making the model consistent with results from the theory of locally isotropic turbulence. A nonlinear Markov process is presented, which is able to satisfy exactly, in the statistical sense, incompressibility, the exact results on the third-order structure function, and the experimental second-order statistics. The corresponding equation for the Eulerian probability density of velocity-vector increments is solved numerically. Numerical results show non-Gaussian statistics of the one-dimensional Lagrangian probability distributions, and a complex shape of the three-dimensional Eulerian probability density function. The latter is then compared with existing experimental data.

We have performed interface-resolved direct numerical simulations of forced homogeneous-isotropic turbulence in a dilute suspension of spherical particles in the Reynolds number range $Re_{\unicode[STIX]{x1D706}}=115{-}140$. The solid–fluid density ratio was set to $1.5$, gravity was set to zero and two particle diameters were investigated corresponding to approximately $5$ and $11$ Kolmogorov lengths. Note that these particle sizes are clearly outside the range of validity of the point-particle approximation, as has been shown by Homann & Bec (J. Fluid Mech., vol. 651, 2010, pp. 81–91). At the present parameter points the global effect of the particles upon the fluid flow is weak. We observe that the dispersed phase exhibits clustering with moderate intensity. The tendency to cluster, which was quantified in terms of the standard deviation of Voronoï cell volumes, decreases with the particle diameter. We have analysed the relation between particle locations and the location of intense vortical flow structures. The results do not reveal any significant statistical correlation. Contrarily, we have detected a small but statistically significant preferential location of particles with respect to the ‘sticky points’ proposed by Goto & Vassilicos (Phys. Rev. Lett., vol. 100 (5), 2008, 054503), i.e. points where the fluid acceleration field is acting such as to increase the local particle concentration in one-way coupled point-particle models under Stokes drag. The presently found statistical correlation between the ‘sticky points’ and the particle locations further increases when focusing on regions with high local concentration. Our results suggest that small finite-size particles can be brought together along the expansive directions of the fluid acceleration field, as previously observed only for the simplest model for sub-Kolmogorov particles. We further discuss the effect of density ratio and collective particle motion upon the basic Eulerian and Lagrangian statistics.

In many swimming and flying animals, propulsion emerges from the interplay of active muscle contraction, passive body elasticity and fluid–body interaction. Changes in the active and passive body properties can influence performance and cost of transport across a broad range of scales; they specifically affect the vortex generation that is crucial for effective swimming at higher Reynolds numbers. Theoretical models that account for both active contraction and passive elasticity are needed to understand how animals tune both their active and passive properties to move efficiently through fluids. This is particularly significant when one considers the phylogenetic constraints on the jellyfish mechanospace, such as the presence of relatively weak muscles that are only one cell layer thick. In this work, we develop an actively deforming model of a jellyfish immersed in a viscous fluid and use numerical simulations to study the role of active muscle contraction, passive body elasticity and fluid forces in the medusan mechanospace. By varying the strength of contraction and the flexibility of the bell margin, we quantify how these active and passive properties affect swimming speed and cost of transport. We find that for fixed bell elasticity, swimming speed increases with the strength of contraction. For fixed force of contractility, swimming speed increases as margin elasticity decreases. Varying the strength of activation in proportion to the elasticity of the bell margin yields similar swimming speeds, with a cost of transport is substantially reduced for more flexible margins. A scaling study reveals that performance declines as the Reynolds number decreases. Circulation analysis of the starting and stopping vortex rings showed that their strengths were dependent on the relative strength of activation with respect to the bell margin flexibility. This work yields a computational framework for developing a quantitative understanding of the roles of active and passive body properties in swimming.

The statistics of the velocity gradient tensor $\unicode[STIX]{x1D63C}=\boldsymbol{{\rm\nabla}}\boldsymbol{u}$, which embody the fine scales of turbulence, are influenced by turbulent ‘structure’. Whilst velocity gradient statistics and dynamics have been well characterised, the connection between structure and dynamics has largely focused on rotation-dominated flow and relied upon data from numerical simulation alone. Using numerical and spatially resolved experimental datasets of homogeneous turbulence, the role of structure is examined for all local (incompressible) flow topologies characterisable by $\unicode[STIX]{x1D63C}$. Structures are studied through the footprints they leave in conditional averages of the $Q=-\text{Tr}(\unicode[STIX]{x1D63C}^{2})/2$ field, pertinent to non-local strain production, obtained using two complementary conditional averaging techniques. The first, stochastic estimation, approximates the $Q$ field conditioned upon $\unicode[STIX]{x1D63C}$ and educes quantitatively similar structure in both datasets, dissimilar to that of random Gaussian velocity fields. Moreover, it strongly resembles a promising model for velocity gradient dynamics recently proposed by Wilczek & Meneveau (J. Fluid Mech., vol. 756, 2014, pp. 191–225), but is derived under a less restrictive premise, with explicitly determined closure coefficients. The second technique examines true conditional averages of the $Q$ field, which is used to validate the stochastic estimation and provide insights towards the model’s refinement. Jointly, these approaches confirm that vortex tubes are the predominant feature of rotation-dominated regions and additionally show that shear layer structures are active in strain-dominated regions. In both cases, kinematic features of these structures explain alignment statistics of the pressure Hessian eigenvectors and why local and non-local strain production act in opposition to each other.

A commonly observed bedform in wide erodible-bed channels consists of rows of streaks or stripes parallel to the flow. These stripes can be manifested in terms of transverse variation of bed elevation, characteristic grain size (and thus roughness or both. The former case is manifested most strongly in sediment with a nearly uniform size distribution and the latter most strongly in sediment with substantial heterogeneity in size. The amplitude of stripes is rarely larger than one or two grain diameters, and the transverse spacing is invariably of the order of the flow depth. They are closely linked to a pattern of paired cells of secondary flow in the flow cross-section.

An existing theory of streak formation for the case of uniform sediment relies on a second-order turbulence closure which explicitly links the streamwise flow to transverse variations in bed elevation. The theory successfully predicts the formation of streaks, but only at rather high values of the Shields stress, i.e. rather strong sediment transport. Streaks are commonly observed, however, at Shields stresses as low as only slightly above the threshold of motion.

In the present analysis the previous flow model is adapted to the case of transverse variation of roughness as well as elevation, and the constraint of uniform sediment is removed. The theory indicates that allowance for even slight heterogeneity of bed sediment results in the formation of streaks at any Shields stress above the threshold of motion. The resulting streaks are hybrid in the sense that they show transverse variation in both elevation and roughness. The model thus provides a general theory of streak formation.

We investigate viscous fingering instabilities of the double-front system which results when fluid is injected into a porous medium containing fluid of a different composition and temperature. We describe a linear stability analysis based on an eigenfunction expansion method which enables us to investigate the structure of the discrete eigenvalue spectrum. We investigate the extent to which the properties of each front contribute to the tendency of the system to become unstable: we find that instabilities on the compositional front dominate because of the high ratio of thermal to solute diffusion. It is difficult for a stable compositional front to stabilize an unstable thermal front; however, this situation can result in a new fingering phenomenon in which the perturbations undergo coupled oscillations of growing amplitude.

A numerical and experimental study of buoyancy-driven flow in the annulus between two horizontal coaxial cylinders at Rayleigh numbers approaching and exceeding the critical values is presented. The stability of the flow is investigated using linear theory and the energy method. Theoretical predictions of the critical Rayleigh number for onset of secondary flows are obtained for a wide range of radius ratio R and are verified by comparison with results of previous experimental studies. A subcritical Rayleigh number which provides a necessary condition for global flow stability is also determined. The three-dimensional transient equations of fluid flow and heat transfer are solved to study the manifestation of instabilities within annuli having impermeable endwalls, which are encountered in various applications. For the first time, a thorough examination of the development of spiral vortex secondary flow within a moderate gap annulus and its interaction with the primary flow is performed for air. Simulations are conducted to investigate factors influencing the size and number of post-transitional vortex cells. The evolution of stable three-dimensional flow and temperature fields with increasing Rayleigh number in a large gap annulus is also studied. The distinct flow structures which coexist in the large gap annulus at high Rayleigh numbers preceding transition to oscillatory flow, including transverse vortices at the end walls which have not been previously identified, are established numerically and experimentally. The solutions for the large-gap annulus are compared to those for the moderate-gap case to clarify fundamental differences in behaviour. Heat transfer results in the form of local Nusselt number distributions are presented for both the moderate- and large-gap cases. Results from a series of experiments performed with air to obtain data for validation of the numerical scheme and further information on the flow stability are presented. Additionally, the change from a crescent-shaped flow pattern to a unicellular pattern with centre of rotation at the top of the annulus is investigated numerically and experimentally for a Prandtl number of 100. Excellent agreement between the numerical and experimental results is shown for both Prandtl numbers studied. The present work provides, for the first time, quantitative three-dimensional descriptions of spiral convection within a moderate-gap annulus containing air, flow structures preceding oscillation in a large-gap annulus for air, and unicellular flow development in a large-gap annulus for large Prandtl number fluids.

A pattern-recognition procedure designed to extract footprints of organized structures from turbulent signals is developed and used to analyse the large-eddy organization of several turbulent wake flows. The pattern-recognition technique is intended to be a general-purpose analytical tool that makes no use of specific flow characteristics, and that can be implemented as a computer code independent of the types of signals to be processed. The technique is applied to analyse the wake generated by a single cylinder at downstream positions ranging from x/D = 10 to x/D = 220. Also the structural features of the wakes behind a rotating cylinder, two cylinders of unequal diameters and two cylinders of equal diameter, one rotating, are examined at x/D = 140. In the near wake the large-scale motions detected are Kármán vortices, whose periodic activity persists up to 60 diameters. Further downstream the most significant coherent structures detected are single and double rollers with shear-aligned vorticity, whose dimensions and velocity intensities are properly scaled by the half-width of the wake and the local r.m.s. values, respectively. The similarities observed in the organized motions identified in the different wakes at x/D = 140, suggest that the roller organization may be an intrinsic characteristic of fully developed turbulent plane wake flows, irrespective of initial conditions.

Random waves are generated by wind in the first half of a wind-wave flume. The latter half of the flume is kept free from wind to measure the waves unaffected by the wind and wind-generated current. The random waves in the latter area are measured with a linear array of wave gauges, and their phase velocities and coherences are determined by a usual technique of the cross-spectral analysis. The measured results are compared with the nonlinear theory of two-dimensional random waves, which has been presented in part 1 of this paper (Masuda, Kuo & Mitsuyasu 1979). Agreement between the theory and the experiment is satisfactory, and observed characteristics of the phase velocity and coherence of the spectral components can be attributed to the effects of the nonlinearity and angular dispersion of the random waves.

An approximation to the profile of given area with smallest drag in laminar flow is obtained (for Reynolds numbers between 103 and 105). It was shown previously by Pironneau (1974) that the skin friction on such a profile has to satisfy certain optimality conditions; the method used is based on these results. It was found that the optimum profile is long and thin (thickness-to-chord ratio about 10%), the front end being shaped like a wedge of angle 90° and the rear end like a cusp. The drag is very close to the drag on a flat plate of equal length.

Interferometric data were obtained in the 10 cm × 18 cm hypervelocity shock tube of oblique shock-wave reflexions in argon at initial temperatures and pressures of nearly 300 °K and 15 Torr. The shock Mach-number range covered was 2 ⩽ Ms ⩽ 8 over a series of wedge angles 2° ⩽ θw ⩽ 60°. Dual-wavelength laser interferograms were obtained by using a 23 cm diameter field of view Mach-Zehnder interferometer. In addition to our numerous results, the available data for argon and helium obtained over the last two decades were also utilized. It is shown analytically and experimentally that in non-stationary flows six domains exist in the (Ms, θw) plane where regular reflexion (RR), single-Mach reflexion (SMR), complex-Mach reflexion (CMR) and double-Mach reflexion (DMR) can occur. The transition boundaries between these regions were all established analytically. The experimental results from different sources substantiate the present analysis, and areas of disagreement which existed in the literature are now clarified and resolved. It is shown that real-gas effects have a significant influence on the size of the regions and their boundaries. In addition, isopycnics (constant density lines) are given for the four types of reflexion, as well as the density distribution along the wedge surface. This data should provide a solid base for computational fluid dynamicists in comparing numerical techniques with actual experimental results.

Turbulent mass transfer to a wall at high Schmidt numbers is controlled by the velocity field within the viscous sublayer. Measurements have been obtained of the root-mean-square fluctuating mass transfer coefficient and the frequency spectrum of the fluctuating mass transfer coefficient for a Schmidt number of about 2300. From an order-of-magnitude analysis it is concluded that flow fluctuations in the direction of mean flow have little effect on the mass transfer fluctuations. A comparison of the mass transfer spectrum with the spectrum of the component of the velocity gradient in the transverse direction sz reveals that the high-frequency portion of the sz spectrum is not effective in transferring mass. Approximate relations between the mass transfer spectrum and the sz spectrum are developed for high frequencies and for low frequencies.

The instability of a three-dimensional attachment-line boundary layer is considered in the nonlinear regime. Using weakly nonlinear theory, it is found that, apart from a small interval near the (linear) critical Reynolds number, finite-amplitude solutions bifurcate subcritically from the upper branch of the neutral curve. The time-dependent Navier–Stokes equations for the attachment-line flow have been solved using a Fourier–Chebyshev spectral method and the subcritical instability is found at wavenumbers that correspond to the upper branch. Both the theory and the numerical calculations show the existence of supercritical finite-amplitude (equilibrium) states near the lower branch which explains why the observed flow exhibits a preference for the lower branch modes. The effect of blowing and suction on nonlinear stability of the attachment-line boundary layer is also investigated.