The three-dimensional structure of a moderate-Reynolds-number (≈ 100) plane wake behind a flat plate subjected to periodic spanwise perturbations has been studied both experimentally and numerically. Comparisons between experimental interface visualizations and numerical calculations demonstrate that important features of the development of the three-dimensional evolution can be reproduced by numerical inviscid vortex dynamics.

It is shown that the redistribution, reorientation and stretching of vorticity leads to the formation of counter-rotating pairs of streamwise vortices which superimpose onto the spanwise Kármán-like vortices. These streamwise vortices exhibit lambdashaped structures and are located in the braids connecting consecutive Kármán vortices of opposite sign.

The interaction of the evolving streamwise structure with the spanwise Kármán vortices results in the formation of closed vortex loops. Depending on the orientation of the initial perturbation, the three-dimensional vorticity field of the wake acquires either a symmetric or a non-symmetric configuration. Under the effect of a periodic vertical perturbation, the wake develops a non-symmetric vorticity mode exhibiting a staggered array of closed vortex loops of alternating sign. In contrast, under the effect of a periodic horizontal perturbation, the wake acquires a symmetric vorticity mode with the closed vortex loops of alternating sign aligned in the flow direction.

We numerically investigate the dynamics of thin disks falling under gravity in a viscous fluid medium at rest at infinity. Varying independently the density and thickness of the disk reveals the influence of the disk aspect ratio which, contrary to previous belief, is found to be highly significant as it may completely change the route to non-vertical paths as well as the boundaries between the various path regimes. The transition from the straight vertical path to the planar fluttering regime is found to exhibit complex dynamics: a bistable behaviour of the system is detected within some parameter range and several intermediate regimes are observed in which, although the wake is unstable, the path barely deviates from vertical. By varying independently the body-to-fluid inertia ratio and the relative magnitude of inertial and viscous effects over a significant range, we set up a comprehensive map of the corresponding styles of path followed by an infinitely thin disk. We observe the four types of planar regimes already reported in experiments but also identify two additional fully three-dimensional regimes in which the body experiences a slow horizontal precession superimposed onto zigzagging or tumbling motions.

This paper reports a fundamental study of the fluid dynamics inside a triangular (attic-shaped) enclosure with cold upper wall and warm horizontal bottom wall. The study was undertaken in three distinct parts. In the first part, the flow and temperature fields in the cavity are determined theoretically on the basis of an asymptotic analysis valid for shallow spaces (H/L → 0, where H and L are the attic height and length). It is shown that in the H/L → 0 limit the circulation consists of a single elongated cell driven by the cold upper wall. The net heat transfer in this limit is dominated by pure conduction. In the second part of the study, the transient behaviour of the attic fluid is examined, based on a scaling analysis. The transient phenomenon begins with the sudden cooling of the upper sloped wall. It is shown that both walls develop thermal and viscous layers whose thicknesses increase towards steady-state values. The criterion for the existence of distinct thermal layers in the steady state is (H/L)½RaH¼ > 1, where RaH is the Rayleigh number based on attic height. The corresponding criterion for distinct viscous wall jets is (H/L)½RaH¼ Pr−½ > 1, where Pr is the Prandtl number. The third phase of this study focused on a complete sequence of transient numerical simulations covering the ranges H/L = 0.2, 0.4, 1; RaH/Pr = 10, 103, 105; Pr = 0.72, 6. The numerical experiments verify the flow features described theoretically in the first two parts of the study. The effect of thermal convection on the net heat transfer between the bottom and top walls is illustrated numerically. Finally, the transient numerical experiments show that in the present parametric domain the single-cell circulation pattern is stable with respect to the Bénard instability expected in fluid layers heated from below.

A primary motivation for this work arises from the contradictory results obtained in some recent measurements of the zero-crossing frequency of turbulent fluctuations in shear flows. A systematic study of the various factors involved in zero-crossing measurements shows that the dynamic range of the signal, the discriminator characteristics, filter frequency and noise contamination have a strong bearing on the results obtained. These effects are analysed, and explicit corrections for noise contamination have been worked out. New measurements of the zero-crossing frequency N0 have been made for the longitudinal velocity fluctuation in boundary layers and a wake, for wall shear stress in a channel, and for temperature derivatives in a heated boundary layer. All these measurements show that a zero-crossing microscale, defined as Λ = (2πN0)−1, is always nearly equal to the well-known Taylor microscale λ (in time). These measurements, as well as a brief analysis, show that even strong departures from Gaussianity do not necessarily yield values appreciably different from unity for the ratio Λ/λ. Further, the variation of N0/N0 max across the boundary layer is found to correlate with the familiar wall and outer coordinates; the outer scaling for N0 max is totally inappropriate, and the inner scaling shows only a weak Reynolds-number dependence. It is also found that the distribution of the interval between successive zero-crossings can be approximated by a combination of a lognormal and an exponential, or (if the shortest intervals are ignored) even of two exponentials, one of which characterizes crossings whose duration is of the order of the wall-variable timescale ν/U2*, while the other characterizes crossings whose duration is of the order of the large-eddy timescale δ/U∞. The significance of these results is discussed, and it is particularly argued that the pulse frequency of Rao, Narasimha & Badri Narayanan (1971) is appreciably less than the zero-crossing rate.

The wake states from a circular cylinder undergoing controlled sinusoidal oscillation transverse to the free stream are examined. As the frequency of oscillation passes through the natural Kármán frequency there is a transition between two distinctly different wake states: the low- and high-frequency states. The transition corresponds to a change in the structure of the near wake and is also characterized by a jump in the phase and amplitude of both the total and vortex lift. Over the range of flow and oscillation parameters studied the wake states exhibit a number of universal features. The phases of the vortex lift and drag forces have characteristic values for the low- and high-frequency states, which appear to be directly related to the phase of vortex shedding. A split force concept is employed, whereby instantaneous force traces and images allow discrimination between the actual loading and the physics, and their conventional time-averaged representations. The wake states for the forced oscillations show some remarkable similarities to the response branches of elastically mounted cylinders. The equivalence between forced and self-excited oscillations is addressed in detail using concepts of energy transfer.

Multiple states of spatially localized steady convection are found in numerical simulations of water–ethanol mixtures in two dimensions. Realistic boundary conditions at the top and bottom are used, with periodic boundary conditions in the horizontal. The states form by a mechanism similar to the pinning region around a Maxwell point in variational systems, but are located in a parameter regime in which the conduction state is overstable. Despite this the localized states can be stable. The properties of the localized states are described in detail, and the mechanism of their destruction with increasing or decreasing Rayleigh number is elucidated. When the Rayleigh number becomes too large the fronts bounding the state at either end unpin and move apart, allowing steady convection to invade the domain. In contrast, when the Rayleigh number is too small the fronts move inwards, and eliminate the localized state which decays into dispersive chaos. Out of this state spatially localized states re-emerge at irregular times before decaying again. Thus an interval of Rayleigh numbers exists that is characterized by relaxation oscillations between localized convection and dispersive chaos.

This paper is a continuation of previous work (Ogura 1962a, b) on the dynamical consequence of the hypothesis that fourth-order mean values of the fluctuating velocity components are related to second-order mean values as they would be for a normal joint-probability distribution. The equations derived by Tatsumi (1957) for isotropic turbulence on the basis of this hypothesis are integrated numerically for specific intitial conditions. The initial values of the Reynolds number, $R_ \lambda = (u^{\overline{2}})^{\frac {1}{2}} \lambda|v$, assigned in this investigation are 28·8, 14·4, 7·2 and 1·8, where $(u^{\overline{2}})^{\frac {1}{2}}$ is the root-mean-square turbulent velocity, λ the dissipation length and v the kinematic viscosity coefficient.

The result of such computations is that the energy spectrum does develop negative values for Rλ = 28·8 and 14·4. This first occurs at a time approximately 2·8 for Rλ = 28·8 and 4·2 for Rλ = 14·4 The time-scale here is $(E_0 k^3_0)^-{\frac{1}{2}}$, where k0 is a wave-number scale typical of the energy-containing velocity component and E10, a typical value of the energy spectrum, is given by $4 \pi ^-{\frac{1}{2}}k_0^{-1}\overline{u^2}$

There is no evidence of the energy distribution tending to become negative for Rλ = 7·2 and 1·8. It is observed that inertial effects are relatively weak at Rλ = 7·2 and the decay process is largely controlled by viscous effects. For Rλ = 1·8 a purely viscous calculation is found to be adequate to account for the numerically integrated results.

Detailed measurements of the velocity field in the symmetry plane of two jets and two jet flames in a crossflow are obtained using particle image velocimetry. The jets issue into a wind tunnel at density-weighted jet-to-crossflow velocity ratios r = 10 and r = 21, with corresponding Reynolds numbers 6000 and 12 800. Ensemble statistics of the velocity field are presented, and some interesting features of the entrainment process in transverse jets are discussed. Deviations from the simple behaviour predicted by the similarity analysis presented in Part 1 are highlighted. Simultaneous planar laser-induced fluorescence imaging of the OH radical is performed in selected regions of the flames. Results suggests that flame/flow interaction is strong near the lifted flamebase, but increasingly weaker further downstream.

The collapse of a cavitation bubble near a rigid boundary induces a high-speed transient jet accelerating liquid onto the boundary. The shear flow produced by this event has many applications, examples of which are surface cleaning, cell membrane poration and enhanced cooling. Yet the magnitude and spatio-temporal distribution of the wall shear stress are not well understood, neither experimentally nor by simulations. Here we solve the flow in the boundary layer using an axisymmetric compressible volume-of-fluid solver from the OpenFOAM framework and discuss the resulting wall shear stress generated for a non-dimensional distance, $\unicode[STIX]{x1D6FE}=1.0$ ( $\unicode[STIX]{x1D6FE}=h/R_{max}$ , where $h$ is the distance of the initial bubble centre to the boundary, and $R_{max}$ is the maximum spherical equivalent radius of the bubble). The calculation of the wall shear stress is found to be reliable once the flow region with constant shear rate in the boundary layer is determined. Very high wall shear stresses of 100 kPa are found during the early spreading of the jet, followed by complex flows composed of annular stagnation rings and secondary vortices. Although the simulated bubble dynamics agrees very well with experiments, we obtain only qualitative agreement with experiments due to inherent experimental challenges.

Although the vortex is ubiquitous in nature, its definition is somewhat ambiguous in the field of fluid dynamics. In this absence of a rigorous mathematical definition, considerable confusion appears to exist in visualizing and understanding the coherent vortical structures in turbulence. Cited in the previous studies, a vortex cannot be fully described by vorticity, and vorticity should be further decomposed into a rotational and a non-rotational part to represent the rotation and the shear, respectively. In this paper, we introduce several new concepts, including local fluid rotation at a point and the direction of the local fluid rotation axis. The direction and the strength of local fluid rotation are examined by investigating the kinematics of the fluid element in two- and three-dimensional flows. A new vector quantity, which is called the vortex vector in this paper, is defined to describe the local fluid rotation and it is the rotational part of the vorticity. This can be understood as that the direction of the vortex vector is equivalent to the direction of the local fluid rotation axis, and the magnitude of vortex vector is the strength of the location fluid rotation. With these new revelations, a vortex is defined as a connected region where the vortex vector is not zero. In addition, through direct numerical simulation (DNS) and large eddy simulation (LES) examples, it is demonstrated that the newly defined vortex vector can fully describe the complex vertical structures of turbulence.

Schooling, an archetype of collective behaviour, emerges from the interactions of fish responding to sensory information mediated by their aqueous environment. A fundamental and largely unexplored question in fish schooling concerns the role of hydrodynamics. Here, we investigate this question by modelling swimmers as vortex dipoles whose interactions are governed by the Biot–Savart law. When we enhance these dipoles with behavioural rules from classical agent-based models, we find that they do not lead robustly to schooling because of flow-mediated interactions. We therefore propose to use swimmers equipped with adaptive decision-making that adjust their gaits through a reinforcement learning algorithm in response to nonlinearly varying hydrodynamic loads. We demonstrate that these swimmers can maintain their relative position within a formation by adapting their strength and school in a variety of prescribed geometrical arrangements. Furthermore, we identify schooling patterns that minimize the individual and collective swimming effort, through an evolutionary optimization. The present work suggests that the adaptive response of individual swimmers to flow-mediated interactions is critical in fish schooling.

We demonstrate that data from direct numerical simulation of turbulent boundary layers at Mach 3 exhibit the same large-scale coherent structures that are found in supersonic and subsonic experiments, namely elongated, low-speed features in the logarithmic region and hairpin vortex packets. Contour plots of the streamwise mass flux show very long low-momentum structures in the logarithmic layer. These low-momentum features carry about one-third of the turbulent kinetic energy. Using Taylor's hypothesis, we find that these structures prevail and meander for very long streamwise distances. Structure lengths on the order of 100 boundary layer thicknesses are observed. Length scales obtained from correlations of the streamwise mass flux severely underpredict the extent of these structures, most likely because of their significant meandering in the spanwise direction. A hairpin-packet-finding algorithm is employed to determine the average packet properties, and we find that the Mach 3 packets are similar to those observed at subsonic conditions. A connection between the wall shear stress and hairpin packets is observed. Visualization of the instantaneous turbulence structure shows that groups of hairpin packets are frequently located above the long low-momentum structures. This finding is consistent with the very large-scale motion model of Kim & Adrian (1999).

The properties of the time- and span-averaged mean wake recirculation region are investigated in separated flows over several different two-dimensional bluff bodies. Ten different cases are considered and they divide into two groups: cylindrical geometries of circular, elliptic and square cross-sections and the normal plate. A wide Reynolds number range from 250 to 140000 is considered, but in all the cases the attached portion of the boundary layer remains laminar until separation. The lower Reynolds number data are from direct numerical simulations, while the data at the higher Reynolds number are obtained from large-eddy simulation and the experimental work of Cantwell & Coles (1983), Krothapalli (1996, personal communication), Leder (1991) and Lyn et al. (1995). Unlike supersonic and subsonic separations with a splitter plate in the wake, in all the cases considered here there is strong interaction between the shear layers resulting in Kármán vortex shedding. The impact of this fundamental difference on the distribution of Reynolds stress components and pressure in relation to the mean wake recirculation region (wake bubble) is considered. It is observed that in all cases the contribution from Reynolds normal stress to the force balance of the wake bubble is significant. In fact, in the cylinder geometries this contribution can outweigh the net force from the shear stress, so that the net pressure force tends to push the bubble away from the body. In contrast, in the case of normal plate, owing to the longer wake, the net contribution from shear stress outweighs that from the normal stress. At higher Reynolds numbers, separation of the Reynolds stress components into incoherent contributions provides more insight. The behaviour of the coherent contribution, arising from the dominant vortex shedding, is similar to that at lower Reynolds numbers. The incoherent contribution to Reynolds stress, arising from small-scale activity, is compared with that of a canonical free shear layer. Based on these observations a simple extension of the wake model (Sychev 1982; Roshko 1993a, b) is proposed.

The Sun and the giant planets rotate and possess deep shells of convection. Some basic aspects of prototypical global convection have been studied with a laboratory model operated in the microgravity environment of Spacelab 3 that flew on the space shuttle Challenger in May 1985. This experiment studied thermally driven circulations within a rotating hemispherical shell of fluid across which are imposed radial and latitudinal temperature gradients. The radial force of gravity is modelled by imposing a strong electric field across the shell, with dielectric polarization forces producing radial accelerations proportional to temperature. When the influence of rotation is large, the experiments yield north-south oriented columnar convection in equatorial and subequatorial regions. As the differential heating is increased, these roll-like cells interact with mid-latitude waves, ultimately being destroyed by turbulent, horizontally isotropic convection that moves down from the pole. When a significant equator-to-pole temperature difference is imposed on the boundaries, spiral waves develop on top of a strong meridional circulation. Intricate, non-axisymmetric, convective patterns that propagate in longitude and evolve in time are described. Schlieren visualizations of these laboratory flows are compared with three-dimensional nonlinear simulations.

The propagation of long, weakly nonlinear interfacial waves in a two-layer fluid of slowly varying depth is studied. The governing equations are formulated to include cubic nonlinearity, which dominates quadratic nonlinearity in some parametric neighbourhood of equal layer depths. Numerical solutions are obtained for an initial profile corresponding to either a single solitary wave or a rank-ordered pair of such waves incident in a monotonic transition between two regions of constant depth. The numerical solutions, supplemented by inverse-scattering theory, are used to investigate the change of polarity of the incident waves as they pass through a ‘turning point’ of approximately equal layer depths. Our results exhibit significant differences from those reported by Knickerbocker & Newell (1980), which were based on a model equation. In particular, we find that more than one wave of reversed polarity may emerge.

The creeping flow of a liquid film along an inclined periodic wall of arbitrary geometry is considered. The problem is formulated using the boundary-integral method for Stokes flow. This method is extended to two-dimensional flows involving free surfaces, and is implemented in an iterative numerical procedure. Detailed calculations for flow along a sinusoidal wall are perfomed. The free-surface profile is studied as a function of flow rate, inclination angle, wave amplitude, and surface tension, and is compared with previous asymptotic solutions. The results include streamline patterns, velocity profiles and wall-shear-stress distributions, and establish criteria for flow reversal. For specified wall geometry, the asymptotic behaviour for very small flow rates is shown to be a strong function of surface tension. It is demonstrated that these results are valid in a qualitative sense for general wall geometries. The analogy between gravity-driven flow and the flow of a liquid layer on a rotating disk (spin coating) is also discussed.

Wind-tunnel turbulence behind a parallel-rod grid with jets evenly distributed along each rod is nearly isotropic. Homogeneity improvement over prior related experiments was attained by the use of controllable nozzles. Compared with the ‘passive’ case, the downwind-jet ‘active’ grid has a smaller static pressure drop across it and gives a smaller turbulence level at a prescribed distance from it, while the upwind-jet grid gives a larger static pressure drop and larger turbulence level. ‘Counterflow injection’ generates larger turbulence energy and larger scales, both events being evidently associated with instability of the jet system. This behaviour is much like that commonly observed behind passive grids of higher solidities.

If the turbulent kinetic energy is approximated as an inverse power law in distance, the (positive) exponent decreases with increasing (downwind or upwind) jet strength, corresponding to slower absolute decay rates. No peculiar decay behaviour occurs when the jet grid is ‘self-propelled’ (zero net average force), or when the static pressure drop across it is zero.

The injection does not change the general behaviour of the energy spectra, although the absolute spectra change inasmuch as the turbulence kinetic energy changes.

Bounds on the bulk rate of energy dissipation in body-force-driven steady-state turbulence are derived directly from the incompressible Navier–Stokes equations. We consider flows in three spatial dimensions in the absence of boundaries and derive rigorous a priori estimates for the time-averaged energy dissipation rate per unit mass, ε, without making any further assumptions on the flows or turbulent fluctuations. We prove

ε [les ] c1vU2/l2 + c2U3/l,

where v is the kinematic viscosity, U is the root-mean-square (space and time averaged) velocity, and l is the longest length scale in the applied forcing function. The prefactors c1 and c2 depend only on the functional shape of the body force and not on its magnitude or any other length scales in the force, the domain or the flow. We also derive a new lower bound on ε in terms of the magnitude of the driving force F. For large Grashof number Gr = Fl3/v2, we find

c3vFl/λ2 [les ] ε

where λ = √vU2/ε is the Taylor microscale in the flow and the coefficient c3 depends only on the shape of the body force. This estimate is seen to be sharp for particular forcing functions producing steady flows with λ/l ∼ O(1) as Gr → 1. We interpret both the upper and lower bounds on ε in terms of the conventional scaling theory of turbulence – where they are seen to be saturated – and discuss them in the context of experiments and direct numerical simulations.

Particulate flows have mainly been studied under the simplifying assumption of a one-way coupling regime where the disperse phase does not modify the carrier fluid. A more complete view of multiphase flows can be gained calling into play two-way coupling effects, i.e. by accounting for the inter-phase momentum exchange, which is certainly relevant at increasing mass loading. In this paper we present a new methodology rigorously designed to capture the inter-phase momentum exchange for particles smaller than the smallest hydrodynamical scale, e.g. the Kolmogorov scale in a turbulent flow. The momentum coupling mechanism exploits the unsteady Stokes flow around a small rigid sphere, where the transient disturbance produced by each particle is evaluated in a closed form. The particles are described as lumped point masses, which would lead to the appearance of singularities. A rigorous regularization procedure is conceived to extract the physically relevant interactions between the particles and the fluid which avoids any ‘ad hoc’ assumption. The approach is suited for high-efficiency implementation on massively parallel machines since the transient disturbance produced by the particles is strongly localized in space. We will show that hundreds of thousands of particles can be handled at an affordable computational cost, as demonstrated by a preliminary application to a particle-laden turbulent shear flow.

An ageostrophic version of Phillips’ model is studied. All instabilities found are systematically interpreted in terms of resonance of wave components. The instability occurs if there is a pair of wave components which propagate in the opposite direction to the basic flow and these wave components have almost the same Doppler-shifted frequency. A new instability, identified as a resonance between the Kelvin wave and the Rossby waves, is found at Froude number F ≈ 0.7. The Rossby waves are almost completely in geostrophic balance while the ageostrophic Kelvin wave is the same as in a one-layer system. Doppler shifting matches frequencies which would otherwise be very different. This instability is presumably the mechanism of the frontal instability observed by Griffiths & Linden (1982) in a laboratory experiment. Ageostrophic, baroclinic instability with non-zero phase speed is also observed in the numerical calculation. This instability is caused by resonance between different geostrophic modes.