The dynamics of gravity currents are believed to be strongly influenced by dissipation due to turbulence and mixing between the current and the surrounding ambient fluid. This paper describes new theory and experiments on gravity currents produced by lock exchange which suggest that dissipation is unimportant when the Reynolds number is sufficiently high. Although there is mixing, the amount of energy dissipated is small, reducing the current speed by a few percent from the energy-conserving value. Benjamin (J. Fluid Mech. vol. 31, 1968, p. 209) suggests that dissipation is an essential ingredient in gravity current dynamics. We show that dissipation is not important at high Reynolds number, and provide an alternative theory that predicts the current speed and depth based on energy-conserving flow that is in good agreement with experiments. We predict that in a deep ambient the front Froude number is 1, rather than the previously accepted value of $\sqrt 2$. New experiments are reported for this case that support the new theoretical value.

A detailed parametric study of the noise from dual-stream jets has been carried out to assess the importance of the noise generated by the primary and secondary shear layers and their principal radiation directions under different operating conditions. Realistic geometries and engine conditions are chosen to enhance the relevance and the usefulness of the experimental results reported here. The effects of different operating conditions in the two streams on both the turbulent mixing noise and shock-associated noise are evaluated under static conditions as well as in the presence of an external co-flowing stream. The results indicate that the secondary-to-primary jet velocity ratio is an important parameter for mixing noise while its effect is negligible on shock-associated noise. The shock-associated noise, not surprisingly, is dependent on the geometric details of the nozzle. The characteristics of this noise component are very different depending on whether shocks are present in the primary or secondary stream. There is strong radiation of shock noise to the aft angles when the secondary stream is supersonic. This trend is troublesome since this component is transmitted into the aft cabin of certain aircraft with engines mounted close to the fuselage. The intensity of the shock-associated noise at the lower angles, in general, is proportional to the strength of the shocks and scales with ($M_{p}^{2}$ – $M_{d}^{2})^{2}$ or ($M_{s}^{2}$ – $M_{d}^{2})^{2 }$ ($M_{p}$ and $M_{s}$ denote the Mach numbers of the primary and secondary streams and $M_{d}$ is the design Mach number). Just as for a single jet, the effect of forward flight on mixing noise and shock-associated noise is very different. While there is a progressive reduction in levels of mixing noise with increasing free-stream velocity, there is amplification of the shock-associated noise. This is particularly so for the aft-radiated component of shock noise from the secondary stream. This effect could further exacerbate the interior noise in the aft cabin.

The behaviour of an inviscid gravity current released from a lock and propagating over a horizontal boundary at the base of a stratified ambient fluid is considered in the framework of a one-layer shallow-water formulation. Solutions for two-dimensional rectangular and axisymmetric geometries, with emphasis on the rotation of the latter, were obtained by a Lax–Wendroff scheme. Box-model approximations are also discussed. The axisymmetric and rotating case admits steady-state lens structures, for which approximate and numerical solutions are presented. In general, the stratification reduces the velocity of propagation and enhances the Coriolis effects in a rotating system (in particular, the maximal radius of propagation decreases). Comparisons of the shallow-water results with Navier–Stokes simulations and laboratory experiments indicate good agreement, at least for the initial period of propagation. The major deficiency of this shallow-water model is the lack of incorporation of internal waves. In particular, if the propagation is at subcritical speed, the applicability of the model is restricted to the time prior to the first effective interaction between the head of the gravity current and the lowest-order internal wave; an estimate of this position is presented.

An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent diffusivity, which is the molecular diffusivity that would be required to achieve the same level of fluctuations in the scalar concentration in the absence of stirring, for the same source distribution. The bound on the equivalent diffusivity depends only on the functional ‘shape’ of both the source and the advecting field. Direct numerical simulations performed for a simple advecting flow to test the bounds are reported.

When a gas bubble rises in an impure liquid, its surface often has an upper spherical cap with negligible shear stress, a lower spherical cap with negligible tangential velocity, and a very small transition region between the two caps.

This paper gives the diffusion boundary-layer theory for the distribution of surfactant around a stagnant-cap bubble, allowing for slowness of both adsorption and diffusion. The resulting singular Volterra integro-differential equations are solved numerically for creeping flow (small Reynolds number). The main result is the relation between the surface pressure of surfactant in the bulk solution, the cap angle and Péclet number of the bubble, and the adsorption depth and adsorption speed of the surfactant. The values of the latter two parameters affect the validity of the approximations much more than the numerical results.

A flow of electrically conducting fluid in the presence of a steady magnetic field has a tendency to become quasi-two-dimensional, i.e. uniform in the direction of the magnetic field, except in thin so-called Hartmann boundary layers. The condition for this tendency is that of a strong magnetic field, corresponding to large values of the dimensionless Hartmann number ($\hbox{\it Ha} \,{\gg}\, 1$). This is analogous to the case of low-Ekman-number rotating flows, with Ekman layers replacing Hartmann layers. This has been at the origin of the homogeneous model for flows in a rotating frame of reference, with its rich structure: geostrophic contours and shear layers. In magnetohydrodynamics, the characteristic surfaces introduced by Kulikovskii (Isv. Akad. Nauk SSSR Mekh. Zhidk Gaza, vol. 3, 1968, p. 3) play a role similar to that of the geostrophic contours. However, a general theory for quasi-two-dimensional magnetohydrodynamics is lacking. In this paper, a model is proposed which provides a general framework. Not only can this model account for otherwise disconnected past results, but it is also used to predict a new type of shear layer, of typical thickness $\hbox{\it Ha}^{-1/4}$. Two practical cases are then considered: the classical problem of a fringing transverse magnetic field across a circular pipe flow, treated by Holroyd & Walker (J. Fluid Mech. vol. 84, 1978, p. 471), and the problem of a rectangular cross-section duct flow in a slowly varying transverse magnetic field. For the first problem, the existence of thick shear layers of dimensionless thickness of order of magnitude $\hbox{\it Ha}^{-1/4}$ explains why the flow expected at large Hartmann number was not observed in experiments. The second problem exemplifies a situation where an analytical solution had been obtained in the past Walker & Ludford (J. Fluid Mech. vol. 56, 1972, p. 481) for the so-called ‘M-shaped’ velocity profile, which is here understood as an aspect of general quasi-two-dimensional magnetohydrodynamics.

An asymptotic method for analysing slender non-axisymmetric drops, bubbles and jets in a general straining flow is developed. The method relies on the slenderness of the geometry to reduce the three-dimensional equations to a sequence of weakly coupled, quasi-two-dimensional Stokes flow problems for the cross-sectional evolution. Exact solution techniques for the flow outside a bubble in two-dimensional Stokes flow are generalized to solve for the transverse flow field, allowing large non-axisymmetric deformations to be described. A generalization to the case where the interior of the bubble contains a slightly viscous fluid is also presented.

Our method is used to compute steady non-axisymmetric solution branches for inviscid bubbles and slightly viscous drops. We also present unsteady numerical solutions showing how the eccentricity of the cross-section adjusts to a non-axisymmetric external flow. Finally, we use our theory to investigate how the pinch-off of a jet of relatively inviscid fluid is affected by a two-dimensional straining cross-flow.

Electro-osmotic flows in the vicinity of surface charge density transitions are studied using a two-dimensional model problem. The analysed configuration consists of an electrolyte solution in contact with a dielectric planar wall, on which the density transition is approximated by a finite jump. The flow field in the semi-infinite fluid domain is driven by an external electric field which is applied parallel to the charged wall, in the direction of the jump. The problem is analysed using both the thin-Debye-layer (TDL) formulation, which does not resolve the fine details of the Debye-layer structure, and an approximate electrokinetic model.

Real turbulent flows are difficult to classify as either spatially homogeneous or isotropic. Nonetheless these idealizations allow the identification of certain universal features associated with the small-scale motions almost invariably observed in a variety of different conditions. The single most significant aspect is a flux of energy through the spectrum of inertial scales related to the phenomenology commonly referred to as the Richardson cascade. Inhomogeneity, inherently present in near-wall turbulence, generates additional energy fluxes of a different nature, corresponding to the spatial redistribution of turbulent kinetic energy. Traditionally the spatial flux is associated with a single-point observable, namely the turbulent kinetic energy density. The flux through the scales is instead classically related to two-point statistics, given in terms of an energy spectrum or, equivalently, in terms of the second-order moment of the velocity increments. In the present paper, starting from a suitably generalized form of the classical Kolmogorov equation, a scale-by-scale balance for the turbulent fluctuations is evaluated by examining in detail how the energy associated with a specific scale of motion – hereafter called the scale energy – is transferred through the spectrum of scales and, simultaneously, how the same scale of motion exchanges energy with a properly defined spatial flux. The analysis is applied to a data set taken from a direct numerical simulation (DNS) of a low-Reynolds-number turbulent channel flow. The detailed scale-by-scale balance is applied to the different regions of the flow in the various ranges of scales, to understand how – i.e. through which mechanisms, at which scales and in which regions of the flow domain – turbulent fluctuations are generated and sustained. A complete and formally precise description of the dynamics of turbulence in the different regions of the channel flow is presented, providing rigorous support for previously proposed conceptual models.

In this paper the scaling of the mean velocity profile and Reynolds stresses is considered for the case of turbulent near-wall flows subjected to strong pressure gradients. Strong pressure gradients are defined as those in which the streamwise pressure gradient non-dimensionalized with inner variables, $p_x^+$, is greater than 0.005. A range of values of this parameter (${-}0.02\,{<}\,p_x^+\,{<}\,0.06$) is examined in this paper. An appropriate functional form for the mean velocity profile is developed and used to parameterize available data. A physical model for the parametric variation with pressure gradient is then developed. This model is based on the concept of a universal critical Reynolds number for the sublayer which explains (both qualitatively and quantitatively) the variation of the important parameters in the inner flow. In particular this gives an explanation for the shift in the apparent log-law due to pressure-gradient effects and provides an appropriate scaling for the Reynolds stresses. It is shown that this model is not only physically plausible but is also consistent with the available data.

The effect of an insoluble surfactant on the stability of the gravity-driven flow of a liquid film down an inclined plane is investigated by a normal-mode analysis. Numerical solutions of the Orr–Sommerfeld equation reveal the occurrence of a stable Marangoni mode and a possibly unstable Yih mode, and demonstrate that the primary role of the surfactant is effectively to raise the critical Reynolds number at which instability is first encountered.

Direct numerical simulations of the incompressible Navier–Stokes equations are employed to study the turbulent wall-shear stress in a turbulent channel flow forced by lateral sinusoidal oscillations of the walls. The objective is to produce a documented database of numerically computed friction reductions. To this aim, the particular numerical requirements for such simulations, owing for example to the time-varying direction of the skin-friction vector, are considered and appropriately accounted for.

A detailed analysis of the dependence of drag reduction on the oscillatory parameters allows us to address conflicting results hitherto reported in the literature. At the Reynolds number of the present simulations, we compute a maximum drag reduction of 44.7%, and we assess the possibility for the power saved to be higher than the power spent for the movement of the walls (when mechanical losses are neglected). A maximum net energy saving of 7.3% is computed.

Furthermore, the scaling of the amount of drag reduction is addressed. A parameter, which depends on both the maximum wall velocity and the period of the oscillation, is found to be linearly related to drag reduction, as long as the half-period of the oscillation is shorter than a typical lifetime of the turbulent near-wall structures. For longer periods of oscillation, the scaling parameter predicts that drag reduction will decrease to zero more slowly than the numerical data. The same parameter also describes well the optimum period of oscillation for fixed maximum wall displacement, which is smaller than the optimum period for fixed maximum wall velocity, and depends on the maximum displacement itself.

A new class of large-eddy simulation (LES) models (optimal LES) was previously introduced by the authors. These models are based on multi-point statistical information, which here is provided by direct numerical simulation (DNS). In this paper, the performance of these models in LES of forced isotropic turbulence is investigated. It is found that both linear and quadratic optimal models yield good simulation results, with an excellent match between the LES and filtered DNS for spectra, and low-order structure functions.

Optimal models were then used as a vehicle to investigate the effects of filter shape and the locality of model dependence on LES performance. Results indicate that a Fourier cutoff filter yields more accurate simulations than graded cutoff filters, leaving no motivation to use graded filters in spectral simulations. It was also found that optimal models formulated to depend on local information performed nearly as well as global models. This is important because in practical LES simulations in which spectral methods are not applicable, global model dependence would be prohibitively expensive.

This paper extends the Boltzmann hereditary delay integral approach previously used to predict the anelastic response of a floating flexible plate to a steadily moving load, to describe the evolution of the response when the load is impulsively started from rest. Asymptotic analysis demonstrates that the ultimate one-dimensional and two-dimensional deflections (due to a line load and a point load, respectively) are consistent with the results obtained from the earlier time-independent viscoelastic theory. Transients in the two-dimensional case decay much more rapidly than their counterparts in the one-dimensional case. Steady-state deflections are approached at all load speeds in the two-dimensional theory, including the critical load speed (coincident with the minimum phase speed of hybrid flexural–gravity waves) and the gravity wave speed.

The classical problem of linear stability of a regular $N$-gon of point vortices to infinitesimal space displacements from an equilibrium of the vortex configuration is generalized to the one for $N$ helical vortices (couple, triplet, etc., $N\,{>}\, 1$) for the first time. As a consequence of this consideration, the analytical form for the stability boundaries has been obtained. This solution allows an efficient analysis to be made of the existence of stable helical vortex arrays, which were repeatedly observed in practice.

Such a stability problem was earlier considered in theory, but only for the case of a plane polygonal array of $N$ point vortices. As for helical vortices, owing to their complexity, intensive study has been mainly on the self-induced motion of the vortex.

The algebraic representation for the velocity of motion of the $N$ helical vortex array was originally obtained as an additional intermediate result. The new formula allows accurate calculations to be made within the whole range of helical pitch variations and has a simpler form than the known asymptotic expressions.

Solution of these two classical problems of vortex dynamics has significance both for theoretic and applied mechanics, as well as for many other areas of natural science, where the rotor (vortex) concept is the basic one.

A method to decompose geophysical flows into a balanced flow (defined by its potential vorticity, PV) and an imbalanced component (inertia–gravity waves, IGWs) is introduced. The balanced flow, called the optimal potential vorticity (OPV) balance, is a solution of an IGW-permitting dynamics in which the amount of IGWs is minimal. The residual IGWs are those spontaneously generated by the vortical flow during the numerical integration in which the PV anomaly grows slowly over a sufficiently long ramp period toward a prescribed PV field. The OPV balanced flow is obtained, iteratively, in a cycle of backward and forward integrations where IGWs are removed and PV is restored in every loop. The method is applied to the flow of unsteady vortices in the three-dimensional baroclinic non-hydrostatic dynamics on the $f$-plane and to the single-layer shallow-water dynamics on the sphere. Both applications show that the iterative method converges strongly, after only a few iterations, to the balanced flow.

In this paper we consider the possibility of generating homogeneous flows with a nearly constant strain rate. This is achieved by stretching an almost cylindrical liquid bridge under microgravity. One key issue is the adjustability of the disk diameters, necessary for maintaining ideal boundary conditions. We first study the stretching of two different fluids by both numerical and experimental means. The numerical results are compared with the experimental data and very good agreement is found. The numerical method is then used to study the behaviour of liquid bridges for quite a large range of the flow parameters (capillary number $\Ca$ and Weber number $\We$) in order to detect those regimes with most suitable flow conditions.