Steady laminar flow over a rounded-tip $2\,:\,1$ elliptic cone of 0.86 m length at zero angle of attack and yaw has been computed at Mach number $7.45$ and unit Reynolds number $Re^{\prime }=1.015\times 10^{7}~\text{m}^{-1}$. The flow conditions are selected to match the planned flight of the Hypersonic Flight Research Experimentation HIFiRE-5 test geometry at an altitude of 21.8 km. Spatial linear BiGlobal modal instability analysis of this flow has been performed at selected streamwise locations on planes normal to the cone symmetry axis, resolving the entire flow domain in a coupled manner while exploiting flow symmetries. Four amplified classes of linear eigenmodes have been unravelled. The shear layer formed near the cone minor-axis centreline gives rise to amplified symmetric and antisymmetric centreline instability modes, classified as shear-layer instabilities. At the attachment line formed along the major axis of the cone, both symmetric and antisymmetric instabilities are also discovered and identified as boundary-layer second Mack modes. In both cases of centreline and attachment-line modes, symmetric instabilities are found to be more unstable than their antisymmetric counterparts. Furthermore, spatial BiGlobal analysis is used for the first time to resolve oblique second modes and cross-flow instabilities in the boundary layer between the major- and minor-axis meridians. Contrary to predictions for the incompressible regime for swept infinite wing flow, the cross-flow instabilities are not found to be linked to the attachment-line instabilities. In fact, cross-flow modes peak along most of the surface of the cone, but vanish towards the attachment line. On the other hand, the leading oblique second modes peak near the leading edge and their associated frequencies are in the range of the attachment-line instability frequencies. Consequently, the attachment-line instabilities are observed to be related to oblique second modes at the major-axis meridian. The linear amplification of centreline and attachment-line instability modes is found to be strong enough to lead to laminar–turbulent flow transition within the length of the test object. The predictions of global linear theory are compared with those of local instability analysis, also performed here under the assumption of locally parallel flow, where use of this assumption is permissible. Fair agreement is obtained for symmetric centreline and symmetric attachment-line modes, while for all other classes of linear disturbances use of the proposed global analysis methodology is warranted for accurate linear instability predictions.

This paper analyses numerically the quasi-two-dimensional flow of an incompressible electrically conducting viscous fluid past a localized zone of applied magnetic field, denominated a magnetic obstacle. The applied field is produced by the superposition of two parallel magnetized square surfaces, uniformly polarized in the normal direction, embedded in the insulating walls that contain the flow. The area of these surfaces is only a small fraction of the total flow domain. By considering inertial effects in the analysis under the low magnetic Reynolds number approximation, it is shown that the flow past a magnetic obstacle may develop vortical structures and eventually instabilities similar to those observed in flows interacting with bluff bodies. In the small zone where the oncoming uniform flow encounters the non-negligible magnetic field, the induced electric currents interact with the field, producing a non-uniform Lorentz force that brakes the fluid and creates vorticity. The effect of boundary layers is introduced through a friction term. Due to the localization of the applied magnetic field, this term models either the Hartmann braking within the zone of high magnetic field strength or a Rayleigh friction in zones where the magnetic field is negligible. Finite difference numerical computations have been conducted for Reynolds numbers $\hbox{\it Re}\,{=}\,100$ and 200, and Hartmann numbers in the range $1 \le \hbox{\it Ha} \le 100$ ($\hbox{\it Re}$ and $\hbox{\it Ha}$ are based on the side length of the magnetized square surfaces). Under these conditions, a wake is formed behind the obstacle. It may display two elongated streamwise vortices that remain steady as long as the Hartmann number does not exceed a critical value. Once this value is reached, the wake becomes unstable and a vortex shedding process similar to the one observed in the flow past bluff bodies is established. Similarities and differences with the flow around solid obstacles are discussed.

Direct measurements of the energy transfer spectrum in locally isotropic grid turbulence have been used to determine the extent of validity for grid turbulence of the dynamical equation for the three-dimensional energy spectrum in isotropic turbulence. The extent of applicability of the isotropic energy balance is consistent with the usual local isotropy criterion based on energy spectra alone.

The present results are in general agreement with some previous measurements by Uberoi, who determined the transfer spectrum assuming the strict validity of the isotropic dynamical equation. The measured energy transfer spectra are quantitatively similar to those calculated by Kraichnan using the direct-interaction approximation.

As in a previous theory (Longuet-Higgins 1963) parasitic capillary waves are considered as a perturbation due to the local action of surface tension forces on an otherwise pure progressive gravity wave. Here the theory is improved by: (i) making use of our more accurate knowledge of the profile of a steep Stokes wave; (ii) taking account of the influence of gravity on the capillary waves themselves, through the effective gravitational acceleration g* for short waves riding on longer waves.

Nonlinearity in the capillary waves themselves is not included, and certain other approximations are made. Nevertheless, the theory is shown to be in essential agreement with experiments by Cox (1958), Ebuchi, Kawamura & Toba (1987) and Perlin, Lin & Ting (1993).

A principal result is that for gravity waves of a given length L > 5 cm there is a critical steepness parameter (AK)c at which the surface velocity (in a frame of reference moving with the phase-speed) equals the minimum (local) speed of capillary-gravity waves. On subcritical gravity waves, with steepness AK < (AK)c, capillary waves may be generated at all points of the wave surface. On supercritical waves, with AK > (AK)c, capillary waves can only be generated in the wave troughs; they are trapped between two caustics near the crests. Generally, the amplitude of the parasitic capillaries is greatest on gravity waves of near critical (but not maximum) steepness.

Finite-difference calculations with random and single-mode perturbations are used to study the three-dimensional instability of vortex rings. The basis of current understanding of the subject consists of a heuristic inviscid model (Widnall, Bliss & Tsai 1974) and a rigorous theory which predicts growth rates for thin-core uniform vorticity rings (Widnall & Tsai 1977). At sufficiently high Reynolds numbers the results correspond qualitatively to those predicted by the heuristic model: multiple bands of wavenumbers are amplified, each band having a distinct radial structure. However, a viscous correction factor to the peak inviscid growth rate is found. It is well described by the first term, 1 – α1(β)/Res, for a large range of Res. Here Res is the Reynolds number defined by Saffman (1978), which involves the curvature-induced strain rate. It is found to be the appropriate choice since then α1(β) varies weakly with core thickness β. The three most nonlinearly amplified modes are a mean azimuthal velocity in the form of opposing streams, an n = 1 mode (n is the azimuthal wavenumber) which arises from the interaction of two second-mode bending waves and the harmonic of the primary second mode. When a single wave is excited, higher harmonics begin to grow successively later with nonlinear growth rates proportional to n. The modified mean flow has a doubly peaked azimuthal vorticity. Since the curvature-induced strain is not exactly stagnation-point flow there is a preference for elongation towards the rear of the ring: the outer structure of the instability wave forms a long wake consisting of n hairpin vortices whose waviness is phase shifted π/n relative to the waviness in the core. Whereas the most amplified linear mode has three radial layers of structure, higher radial modes having more layers of radial structure (hairpins piled upon hairpins) are excited when the initial perturbation is large, reminiscent of visualization experiments on the formation of a turbulent ring at the generator.

Theoretical and experimental studies have been performed on fully developed twodimensional turbulent channel flows in the low Reynolds number range that are subjected to system rotation. The turbulence is affected by the Coriolis force and the low Reynolds number simultaneously. Using dimensional analysis, the relevant parameters of this flow are found to be Reynolds number Re* = u*D/v (u* is the friction velocity, D the channel half-width) and Ωv/u2* (Ω is the angular velocity of the channel) for the inner region, and Re* and ΩD/u* for the core region. Employing these parameters, changes of skin friction coefficients and velocity profiles compared to nonrotating flow can be reasonably well understood. A Coriolis region where the Coriolis force effect predominates is shown to exist in addition to conventional regions such as viscous and buffer regions. A flow regime diagram that indicates ranges of these regions as a function of Re* and |Ω|v/u2* is given from which the overall flow structure in a rotating channel can be obtained.

Experiments have been made in the range of 56 ≤ Re* ≤ 310 and -0.0057 ≤ Ωv/u2* ≤ 0.0030 (these values correspond to Re = 2UmD/v from 1700 to 10000 and rotation number R0 = 2|Ω|D/Um up to 0.055; Um is bulk mean velocity). The characteristic features of velocity profiles and the variation of skin friction coefficients are discussed in relation to the theoretical considerations.

An investigation is made into the evolution, from a sinusoidal initial wave train, of long periodic waves of small but finite amplitude propagating in one direction over water in a uniform channel. The spatially periodic surface displacement is expanded in a Fourier series with time-dependent coefficients. Equations for the Fourier coefficients are derived from three sources, namely the Korteweg–de Vries equation, the regularized long-wave equation proposed by Benjamin, Bona & Mahony (1972) and the relevant nonlinear boundary-value problem for Laplace's equation. Solutions are found by analytical and by numerical methods, and the three models of the system are compared. The surface displacement is found to take the form of an almost linear superposition of wave trains of the same wavelength as the initial wave train.

The orientation dynamics of small anisotropic tracer particles in turbulent flows is studied using direct numerical simulation (DNS) and results are compared with Lagrangian stochastic models. Generalizing earlier analysis for axisymmetric ellipsoidal particles (Parsa et al., Phys. Rev. Lett., vol. 109, 2012, 134501), we measure the orientation statistics and rotation rates of general, triaxial–ellipsoidal tracer particles using Lagrangian tracking in DNS of isotropic turbulence. Triaxial ellipsoids that are very long in one direction, very thin in another and of intermediate size in the third direction exhibit reduced rotation rates that are similar to those of rods in the ellipsoid’s longest direction, while exhibiting increased rotation rates that are similar to those of axisymmetric discs in the thinnest direction. DNS results differ significantly from the case when the particle orientations are assumed to be statistically independent from the velocity gradient tensor. They are also different from predictions of a Gaussian process for the velocity gradient tensor, which does not provide realistic preferred vorticity–strain-rate tensor alignments. DNS results are also compared with a stochastic model for the velocity gradient tensor based on the recent fluid deformation approximation (RFDA). Unlike the Gaussian model, the stochastic model accurately predicts the reduction in rotation rate in the longest direction of triaxial ellipsoids since this direction aligns with the flow’s vorticity, with its rotation perpendicular to the vorticity being reduced. For disc-like particles, or in directions perpendicular to the longest direction in triaxial particles, the model predicts noticeably smaller rotation rates than those observed in DNS, a behaviour that can be understood based on the probability of vorticity orientation with the most contracting strain-rate eigendirection in the model.

Autorotation of an elliptic cylinder about an axis fixed perpendicular to a parallel flow is explained in this paper by means of numerical solutions of the Navier-Stokes equations. Potential-flow theory predicts, for constant angular velocity, half a period in which a torque supports rotation and half a period in which it opposes rotation, with vanishing torque in the average. This balance is disturbed by viscous-flow effects in such a way that, for a given angular velocity, vortex shedding either damps rotation or, under certain conditions, favours rotation. The proper interplay of those conditions, which include synchronization of vortex shedding and rate of rotation, results in auto-rotation. The numerical results for Re [les ] 400 are compared with experimental data for Re = 90000 from the literature. The agreement of the force coefficients and the large-scale flow patterns is surprisingly good.

The instability of oscillatory plane Poiseuille flow, in which the pressure gradient is time-periodically modulated, is investigated by a perturbation technique. The Floquet exponents (i.e. the complex growth rates of the disturbances to the oscillatory flow) are computed by series expansions, in powers of the oscillatory to steady flow velocity amplitude ratio, about the values of the growth rates of the disturbances of the steady flow. It is shown that the oscillatory flow is more stable than the steady flow for values of Reynolds number and disturbance wave number in the vicinity of the steady flow critical point and for values of frequencies of imposed oscillation greater than about one tenth of the frequency of the steady flow neutral disturbance. At very high and low values of imposed oscillation frequency, the unsteady flow is slightly less stable than the steady flow. These results hold for the values of the velocity amplitude ratio at least up to 0·25.

The instability of a plane layer of non-conducting fluid which is in hydrostatic equilibrium between two semi-infinite conducting fluids with surface charges is discussed for both inviscid and viscous fluid models. It is shown that for both the inviscid and viscous fluid cases, the criteria for instability are the same. Consideration is given to the relevance of the results in explaining the mechanism by which the presence of an electric field promotes more readily the coalescence of water droplets on a water surface by viewing the onset of disruption of the air film as the instability of the air film under the action of the electrostatic field produced by the surface charges on the water surfaces.

This paper considers the transient motion of a viscous fluid in a container rotating with constant angular velocity. The principal objective is to study the manner in which an arbitrary initial state of motion becomes a rigid rotation. In order to concentrate on the effects of viscosity, only the spherical container is studied here in great detail. A general theory will be presented in a subsequent publication.

Several sources of non-uniform behaviour make the analysis difficult and complex. In particular, there are three important time scales, viscous boundary layers, boundary-layer resonances at critical latitudes and intricate side-wall effects. The basic aproach consists of an expansion procedure by means of which the general inviscid solution is corrected for viscous effects and is made uniformly valid in time through the critical spin-up phase. Uniform validity is effected through the elimination of secular terms with unacceptable growth rates arising from the asymptotic perturbation series.

The interior (inviscid) motion leads to a non-self-adjoint partial differential equation eigenvalue problem with many intriguing properties. The general expansion theorem, orthogonality relationships, and viscous decay factors are deduced and used to solve the arbitrary intial-value problem. It is shown that the depth averaged circulation about circular contours, x2 + y2 = r2, is extracted from the fluid in the spin-up time scale T = L(Ωv)½. This is accomplished by a secondary non-oscillatory convective motion produced by suction into the Ekman layer. The excess circulation not eliminated in this way excites inviscid inertial oscillations which are also caused to decay by the boundary layers in the same time scale. Some very small residual effects decay in the ordinary viscous diffusion time, but all the essential processes are concluded in the much shorter interval. All modal oscillations in a sphere are determined and several specific calculations of frequency and decay rate are made and compared with experimental data. Perhaps the most important of these concerns the mode corresponding to rigid internal motion about another axis which can be produced by impulsively changing the rotation axis of the container. Agreement between theory and experiment is very good in all cases compared thus far.

The effects of almost neutrally buoyant plastic particles and heavy glass particles on grid-generated turbulence were studied experimentally in a water flow facility. From measured velocities of both the solid and liquid phases, drag and slip velocities of the particles and energy spectra and dissipation rates of the liquid phase were estimated. A monotonic increase in the dissipation rate of the turbulence energy with particle concentration was observed. The increase in energy dissipation rate for suspensions of glass particles was about twice that of suspensions of plastic particles. The increase in dissipation was larger than that predicted by a simple model based on the slip velocities between the phases. It is speculated that the particles enhance the transfer of energy to smaller eddies extending the dissipation spectrum to smaller scales. Since only part of the high wavenumber end of the energy spectrum could be resolved, this speculation cannot be conclusively demonstrated by the present experimental data. The measured velocity spectra do show that the particles increase the isotropy of the flow field and modify the high wavenumber end of the turbulence energy spectrum.

This paper concerns a narrow basin of uniform cross-section open to the sea at one end and closed at the other. An incident long wave of prescribed general form is supposed to enter from the sea and to represent the combination of tide and surge as generated in the sea. The solution of the linear terms of the equations of continuity and motion gives the reflection of this wave at the head of the estuary. This paper gives the next approximation when the non-linear terms are retained, the second-order motion being made determinate by the condition that, at the mouth, it reduces to a progression towards the sea.

The chief results relate to the surface elevation at the head of the estuary. When the first order elevation there increases steadily to a maximum, the effect of the ‘shallow water terms’ is to make high water higher and earlier, while the effect of the ‘frictional term’ is to make high water lower and later. For a short estuary, the interaction of the tide on a surge, due to a given sequence of meteorological conditions over the sea, is to make it higher when its maximum occurs at the time of tidal high water than when its maximum occurs at the time of tidal low water. This is directly opposite to the corresponding result when the estuary is of infinite length.

We performed large-eddy simulations of flow over a series of three-dimensional dunes at laboratory scale (Reynolds number based on the average channel depth and streamwise velocity was 18 900) using the Lagrangian dynamic eddy-viscosity subgrid-scale model. The bedform three-dimensionality was imposed by shifting a standard two-dimensional dune shape in the streamwise direction according to a sine wave. The statistics of the flow are discussed in 10 cases with in-phase and staggered crestlines, different deformation amplitudes and wavelengths. The results are validated qualitatively against experiments. The three-dimensional separation of flow at the crestline alters the distribution of wall pressure, which in turn may cause secondary flow across the stream, which directs low-momentum fluid, near the bed, toward the lobe (the most downstream point on the crestline) and high-momentum fluid, near the top surface, toward the saddle (the most upstream point on the crestline). The mean flow is characterized by a pair of counter-rotating streamwise vortices, with core radius of the order of the flow depth. However, for wavelengths smaller than the flow depth, the secondary flow exists only near the bed and the mean flow away from the bed resembles the two-dimensional case. Staggering the crestlines alters the secondary motion; the fastest flow occurs between the lobe and the saddle planes, and two pairs of streamwise vortices appear (a strong one, centred about the lobe, and a weaker one, coming from the previous dune, centred around the saddle). The distribution of the wall stress and the focal points of separation and attachment on the bed are discussed. The sensitivity of the average reattachment length, depends on the induced secondary flow, the streamwise and spanwise components of the channel resistance (the skin friction and the form drag), and the contribution of the form drag to the total resistance are also studied. Three-dimensionality of the bed increases the drag in the channel; the form drag contributes more than in the two-dimensional case to the resistance, except for the staggered-crest case. Turbulent-kinetic energy is increased in the separated shear layer by the introduction of three-dimensionality, but its value normalized by the plane-averaged wall stress is lower than in the corresponding two-dimensional dunes. The upward flow on the stoss side and higher deceleration of flow on the lee side over the lobe plane lift and broaden the separated shear layer, respectively, affecting the turbulent kinetic energy.

The bubbly two-phase flow and electric current density distribution along a single, vertical, gas-evolving electrode are modelled and the results of a boundary layer analysis are presented. Existing empirical models for particle transport in sheared and sedimenting suspensions are adopted for the bubble mixture to close the two-phase model. Ionic species concentrations are shown to be essentially homogeneous as the mixing effect of the bubble suspension usually is much larger than dispersion by molecular diffusion even at laminar flow conditions. The non-uniformity of the bubble distribution along the electrode results in a non-uniform current density distribution, which agrees well with existing experimental findings in the literature.

The motion of a spherical drop with a bulk-insoluble surfactant immersed in a background flow in the limits of low surface Péclet number and low Reynolds number is investigated. We develop a reciprocal theorem that applies to any prescribed background flow and provide a specific example of an unbounded Poiseuille flow. Analytical formulas for the migration velocity of the drop are obtained perturbatively in powers of the surface Péclet number. We show that the redistribution of surfactant due to the background flow acts to retard the motion of the drop, with the magnitude of this slip velocity being independent of the drop’s position in the Poiseuille flow. Moreover, a surfactant-induced cross-streamline migration of the drop occurs towards the centre of the Poiseuille flow, with its magnitude depending linearly on the distance of the drop from the centre of the Poiseuille flow.

The behaviour of a nominally two-dimensional ventilated partial cavity was examined over a wide range of size scales and flow speeds to determine the influence of Froude, Reynolds, and Weber number on the cavity shape, dynamics, and gas entrainment rate. Two geometrically similar experiments were conducted with a 14:1 length scale ratio. The results were compared to a two-dimensional semi-analytical model of the cavity flow, and Froude scaling was found to be sufficient to match basic cavity shapes. However, the air flux required to maintain a stable cavity did not scale with Froude number alone, as the dynamics of the cavity closure changed with increasing Reynolds number. The required air flux differed over one order of magnitude between the lowest and highest Reynolds number flows. But, for sufficiently high Reynolds numbers, the rate of scaled entrainment appeared to approach Reynolds number independence. Modest changes in surface tension of the small-scale experiment suggested that the Weber number was important only at the lowest speeds and smaller length scale. Otherwise, the Weber numbers of the flows were sufficiently high to make the effects of interfacial tension negligible. We also observed that modest unsteadiness in the inflow to the large-scale cavity led to a significant increase in the required air flux needed to maintain a stable cavity, with the required excess gas flux nominally proportional to the flow’s perturbation amplitude. Finally, discussion is provided on how these results relate to model testing of partial cavity drag reduction (PCDR) systems for surface ships.

In this short note we use the results of numerical solutions of the unsteady Navier–Stokes equations to study the occurrence of separation in unsteady flow. The structure of a flow cycle depends strongly on the Strouhal number, whose value can be classified as small, intermediate or large. In each of these regions different effects are calculated, and we give guidelines for the boundaries of these regions. In particular, we show that the small-Strouhal-number region, in which quasi-steady flow is observed, is very small and decreases rapidly as the peak Reynolds number increases.