The inviscid compressible instability of columnar-vortex flows to three-dimensional perturbations with large wavenumbers is considered. The sufficient conditions for instability obtained are compared with recently published results for incompressible fluids.

Measurements of a steady free-stream, nominally two-dimensional, separating turbulent boundary layer have been reported in earlier parts of this work. Here measurements are reported that show the effects of frequency on sinusoidal unsteadiness of the free-stream velocity on this separating turbulent boundary layer at reduced frequencies of 0.61 and 0.90. The ratio of oscillation amplitude to mean velocity is about 1/3 for each flow.

Upstream of flow detachment, hot-wire anemometer measurements were obtained. A surface hot-wire anemometer was used to measure the phase-averaged skin friction. Measurements in the detached-flow zone of phase-averaged velocities and turbulence quantities were obtained with a directionally sensitive laser anemometer. The fraction of time that the flow moves downstream was measured by the LDV and by a thermal flow-direction probe.

Upstream of any flow reversal or backflow, each flow behaves in a quasisteady manner, i.e. the phase-averaged flow is described by the steady free-stream flow structure. The semilogarithmic law-of-the-wall velocity profiles applies at each phase of the cycle. The Perry & Schofield (1973) velocity-profile correlations fit the mean and ensemble-averaged velocity profiles near detachment.

After the beginning of detachment, large amplitude and phase variations develop through each flow. Unsteady effects produce hysteresis in relationships between flow parameters. As the free-stream velocity during a cycle begins to increase, the detached shear layer decreases in thickness, and the fraction of time $\hat{\gamma}_{{\rm p}u} $ that the flow moves downstream increases as backflow fluid is washed downstream. As the free-stream velocity nears the maximum value in a cycle, the increasingly adverse pressure gradient causes progressively greater near-wall backflow at downstream locations while $\hat{\gamma}_{{\rm p}u}$ remains high at the upstream part of the detached flow. After the free-stream velocity begins to decelerate, the detached shear layer grows in thickness, and the location where flow reversal begins moves upstream. This cycle is repeated as the free-stream velocity again increases.

In both unsteady flows, the ensemble-averaged detached-flow velocity profiles agree with steady free-stream profiles for the same $\hat{\gamma}_{{\rm p}u\min} $ value near the wall when $\partial\hat{\gamma}_{{\rm p}u\min}/\partial\hat{t} < 0 $. However, the reduced-frequency k = 0.90 flow has much larger hysteresis in ensemble-averaged velocity profile shapes when $\partial\hat{\gamma}_{{\rm p}u\min}/\partial{t} \geqslant 0 $. Larger and negative values of the profile shape factor $\hat{H}$ occur for this flow during phases when the non-dimensional backflow is greater and $\hat{\gamma}_{{\rm p}u\min}\rightarrow 0.01$.

The hydrodynamic oscillations of a cyclone separator – in particular the precessing vortex core (PVC) phenomena – are investigated by measuring their radiated sound spectra. Strong coherence was observed between internal flow oscillations measured via hot wire anemometry and the external acoustic field measured via microphone. This means that the oscillations can be characterised by using acoustics as a proxy. The oscillations cause narrow-band noise, referred to as cyclone hum. System characterisation by dimensional analysis used velocity and length scales of the vortex core region as scaling parameters. The relevant non-dimensional parameters are a Strouhal number for the cyclone hum centre frequency, a Reynolds number, a geometry based swirl number and numerous geometric scales defining the shape of the device. Cyclones with multiple sizes of inlets and outlets were tested at different flow rates using external microphones to detect the cyclone hum. The results produce an excellent collapse of the data, yielding a simple relationship for Strouhal number as a function of swirl number and the outlet diameter ratio. The non-invasive method of examining oscillations that is presented in this paper could be applied to other swirling systems.

When particles settle through a stable temperature or salinity gradient they can drive an instability known as sedimentary fingering convection. This phenomenon is thought to occur beneath sediment-rich river plumes in lakes and oceans, in the context of marine snow where decaying organic materials serve as the suspended particles or in the atmosphere in the presence of aerosols or volcanic ash. Laboratory experiments of Houk & Green (Deep-Sea Res., vol. 20, 1973, pp. 757–761) and Green (Sedimentology, vol. 34(2), 1987, pp. 319–331) have shown sedimentary fingering convection to be similar to the more commonly known thermohaline fingering convection in many ways. Here, we study the phenomenon using three-dimensional direct numerical simulations. We find evidence for layer formation in sedimentary fingering convection in regions of parameter space where it does not occur for non-sedimentary systems. This is due to two complementary effects. Sedimentation affects the turbulent fluxes and broadens the region of parameter space unstable to the $\unicode[STIX]{x1D6FE}$-instability (Radko, J. Fluid Mech., vol. 497, 2003, pp. 365–380) to include systems at larger density ratios. It also gives rise to a new layering instability that exists in $\unicode[STIX]{x1D6FE}$-stable regimes. The former is likely quite ubiquitous in geophysical systems for sufficiently large settling velocities, while the latter probably grows too slowly to be relevant, at least in the context of sediments in water.

An analysis of the motion and deformation of red blood cells between two parallel flat plates is presented. The motion is driven by an imposed pressure gradient in the surrounding fluid. Mammalian red cells are highly flexible, but deform at constant volume because the contents of the cell are incompressible, and at nearly constant surface area because the membrane strongly resists dilatation. Consequently, a minimum spacing between the plates exists, below which passage of intact cells is not possible. We consider spacings slightly larger than this minimum. The shape of the cell in this case is a disk with a rounded edge. The flow of the surrounding fluid is described using lubrication theory. Under the approximation that the distance between the plates is small compared with the cell diameter, cell shapes, pressure distributions, membrane stresses and cell velocities are deduced as functions of geometrical parameters. It is found that the narrow gaps between the cell and the plate are not uniform in width, and that as a result, membrane shear stresses are generated which increase in proportion to flow velocity. This contrasts with axisymmetric configurations, in which membrane shear stress remains bounded as cell velocity increases. The variation of cell velocity with spacing of the plates is similar to that previously demonstrated for rigid disk-shaped particles of corresponding dimensions.

In this work, we employ a kinetic-theory-based approach to predict the hydrodynamic forces on electromechanical resonators operating in gaseous media. Using the Boltzmann–BGK equation, we investigate the influence of the resonator geometry on the fluid resistance in the entire range of non-dimensional frequency variation 0 ≤ τω ≤ ∞; here the fluid relaxation time τ = μ/p is determined by the gas viscosity μ and pressure p at thermodynamic equilibrium, and ω is the (angular) oscillation frequency. Our results here capture two important aspects of recent experimental measurements that covered a broad range of experimental parameters. First, the experimentally observed transition from viscous to viscoelastic flow in simple gases at τω ≈ 1 emerges naturally in the numerical data. Second, the calculated effects of resonator geometry are in agreement with experimental observations.

The flow of a fluid in a narrow spherical annulus is considered. When the outer sphere remains fixed and the angular velocity of the inner one is increased beyond a critical value an instability resembling Taylor vortices appears. This instability is investigated by expanding the solution in powers of the small parameter ε, the ratio of the gap thickness to the radius, and assuming that two length scales, O(1) and O(ε), are important in the latitudinal direction.

The perturbation then takes the form of cells which are of roughly square cross-section, at least near the equator, but whose amplitude decays rapidly with latitude; it is also subject to a slow spatial modulation. The critical value of the Taylor number at which the instability first appears is shown to be that for infinite concentric cylinders plus a correction O(ε) due to secondary motions and a correction not greater than O(ε) due to the domain being bounded.

The free-surface flow past a semi-infinite horizontal plate in a finite-depth fluid is considered. It is assumed that the fluid is incompressible and inviscid and that the flow approaches a uniform shear flow downstream. Exact relations are derived using conservation of mass and momentum for the case where the downstream free surface is flat. The complete nonlinear problem is solved numerically using a boundary-integral method and these waveless solutions are shown to exist only when the height of the plate above the bottom is greater than the height of the uniform shear flow. Interesting results are found for various values of the constant vorticity. Solutions with downstream surface waves are also considered, and nonlinear results of this type are compared with linear results found previously. These solutions can be used to model the flow near the stern of a (two-dimensional) ship.

An approximate solution is devised for the one-dimensional motion following the impact of a shock wave on a wall which is free to move. The approximate solution neglects changes in entropy occurring through the reflected and transmitted shocks, thus reducing the problem to one of a simple wave type. The asymptotic behaviour of the system is considered and it is shown by exact physical argument that the transmitted shock eventually attains the same strength as the incident shock and that the reflected shock ultimately decays to a sound wave.

An experimental investigation of the interaction was made, using thin walls of cellulose acetate, in a shock tube at an incident shock Mach number of 1·50. Agreement between the theoretical and experimental results, especially for the path followed by the wall, was found to be good.

The stability with respect to initial condition perturbations of solitary travelling-wave solutions of the Euler equations for continuously, stably stratified, near two-layer fluids is examined numerically and analytically for a set of parameters of relevance for laboratory experiments. Numerical travelling-wave solutions of the Dubreil–Jacotin–Long equation are first obtained with a variant of Turkington, Eyland and Wang’s iterative code by testing convergence on the equation’s residual. In this way, stationary solutions with very thin pycnoclines (and small Richardson numbers) approaching the near two-layer configurations used in experiments can be obtained, allowing for a stability study free of non-stationary effects, introduced by lack of numerical resolution, which develop when these solutions are used as initial conditions in a time-dependent evolution code. The thin pycnoclines in this study permit analytical results to be derived from strongly nonlinear models and their predictions compared with carefully controlled numerical simulations. This brings forth shortcomings of simple criteria for shear instability manifestations based on parallel shear approximations due to subtle higher-order effects. In particular, evidence is provided that the fore–aft asymmetric growth observed in all simulations requires non-parallel shear analysis. Collectively, the results of this study reveal that while the wave-induced shear can locally reach unstable configurations and give rise to local convective instability, the global wave/self-generated shear system is in fact stable, even for extreme cases of thin pycnoclines and near-maximum-amplitude waves.

We investigate three-dimensional flagellar swimming in a fluid with a sparse network of stationary obstacles or fibres. The Brinkman equation is used to model the average fluid flow where a flow-dependent term, including a resistance parameter that is inversely proportional to the permeability, captures the effects of the fibres on the fluid. To solve for the local linear and angular velocities that are coupled to the flagellar motion, we extend the method of regularized Brinkmanlets to incorporate a Kirchhoff rod, discretized as point forces and torques along a centreline. Representing a flagellum as a Kirchhoff rod, we investigate emergent waveforms for different preferred strain and twist functions. Since the Kirchhoff rod formulation allows for out-of-plane motion, in addition to studying a preferred planar sine wave configuration, we also study a preferred helical configuration. Our numerical method is validated by comparing results to asymptotic swimming speeds derived for an infinite-length cylinder propagating planar or helical waves. Similar to the asymptotic analysis for both planar and helical bending, we observe that with small amplitude bending, swimming speed is always enhanced relative to the case with no fibres in the fluid (Stokes) as the resistance parameter is increased. For regimes not accounted for with asymptotic analysis, i.e. large amplitude planar and helical bending, our model results show a non-monotonic change in swimming speed with respect to the resistance parameter; a maximum swimming speed is observed when the resistance parameter is near one. The non-monotonic behaviour is due to the emergent waveforms; as the resistance parameter increases, the swimmer becomes incapable of achieving the amplitude of its preferred configuration. We also show how simulation results of slower swimming speeds for larger resistance parameters are actually consistent with the asymptotic swimming speeds if work in the system is fixed.

Cylindrical gravity currents, produced by a full-depth lock release, in a rotating system are investigated by means of three-dimensional high-resolution simulations of the incompressible variable-density Navier–Stokes equations with the Coriolis term and using the Boussinesq approximation for a small density difference. Here, the depth of the fluid is chosen to be the same as the radius of the cylindrical lock and the ambient fluid is non-stratified. Our attention is focused on the situation when the ratio of Coriolis to inertia forces is not large, namely $0.1\leqslant {\mathcal{C}}\leqslant 0.3$, and the non-rotating case, namely ${\mathcal{C}}=0$, is also briefly considered. The simulations reproduce the major features observed in the laboratory and provide more detailed flow information. After the heavy fluid contained in a cylindrical lock is released in a rotating system, the influence of the Coriolis effects is not significant during the initial one-tenth of a revolution of the system. During the initial one-tenth of a revolution of the system, Kelvin–Helmholtz vortices form and the rotating cylindrical gravity currents maintain nearly perfect axisymmetry. Afterwards, three-dimensionality of the flow quickly develops and the outer rim of the spreading heavy fluid breaks away from the body of the current, which gives rise to the maximum dissipation rate in the system during the entire adjustment process. The detached outer rim of heavy fluid then continues to propagate outward until a maximum radius of propagation is attained. The body of the current exhibits a complex contraction–relaxation motion and new outwardly propagating pulses form regularly in a period slightly less than half-revolution of the system. Depending on the ratio of Coriolis to inertia forces, such a contraction–relaxation motion may be initiated after or before the attainment of a maximum radius of propagation. In the contraction–relaxation motion of the heavy fluid, energy is transformed between potential energy and kinetic energy, while it is mainly the kinetic energy that is consumed by the dissipation. As a new pulse initially propagates outward, the potential energy in the system increases at the expense of decreasing kinetic energy, until a local maximum of potential energy is reached. During the latter part of the new pulse propagation, the kinetic energy in the system increases at the expense of decreasing potential energy, until a local minimum of potential energy is reached and another new pulse takes form. With the use of three-dimensional high-resolution simulations, the lobe-and-cleft structure at the advancing front can be clearly observed. The number of lobes is maintained only for a limited period of time before merger between existing lobes occurs when a maximum radius of propagation is approached. The high-resolution simulations complement the existing shallow-water formulation, which accurately predicts many important features and provides insights for rotating cylindrical gravity currents with good physical assumptions and simple mathematical models.

Extensive experimental studies are presented of the effects of prolonged streamline divergence on developing turbulent boundary layers. The experiment was arranged as source flow over a flat plate with a maximum divergence parameter of about 0.075. Mild, but alternating in sign, upstream-pressure-gradient effects on diverging boundary layers are also discussed.

It appears that two overlapping stages of development are involved. The initial stage covers a distance of about 20 initial boundary-layer thicknesses (δ0) from the start of divergence, where the coupled effects of pressure gradient and divergence are present. In this region there is a fairly large reduction in divergence parameter, Rθ (Reynolds number based on momentum thickness) remains constant (≈ 1400) and the boundary-layer properties change rapidly. In the second region, which lasts nearly to the end of the diverging section, the pressure-gradient effects are negligible, the rate of decrease in divergence parameter is very small and Rθ increases gradually. Up to the last measurement station (≈ 100δ0) the flow is still considered to be at a low Reynolds number (Rθ ≈ 2000). For almost the entire length of this region, the profiles of non-dimensional eddy viscosity appear to be self-similar, but have larger values than for the unperturbed flow. Also in this region, beyond 35δ0, the wake parameter, which has reduced significantly, becomes nearly constant and independent of Rθ. On the other hand the entrainment rate attains a constant value at around 50δ0. It appears that the boundary layer reaches a state of equilibrium. It is suggested that this is the result of an enhanced turbulent diffusion to the outer layer. Spectral measurements show that divergence affects mainly the low-wavenumber, large-scale motions. However, there is no change in large-eddy configurations, since the dimensionless structure parameters show only negligible deviations from the unperturbed values.

Viscoelastic properties of complex fluids are usually extracted by applying an oscillatory shear rate ( where is a constant which is small for a linear response to make sense) to the fluid. This leads to a complex effective viscosity where its real part carries information on viscous effects while its imaginary part informs us on elastic properties. We show here theoretically, by taking a dilute vesicle suspension as an example, that application of a pure shearing oscillation misses several interesting microscopic features of the suspension. It is shown that if, in addition to the oscillatory part, a basic constant shear rate is applied to the suspension (so that the total shear rate is , with a constant), then the complex viscosity reveals much more insightful properties of the suspension. First, it is found that the complex viscosity exhibits a resonance for tank-treading vesicles as a function of the frequency of oscillation. This resonance is linked to the fact that vesicles, while being in the stable tank-treading regime (with their main axis having a steady orientation with respect to the flow direction), possess damped oscillatory modes. Second, in the region of parameter space where the vesicle exhibits either vacillating-breathing (permanent oscillations of the main axis about the flow direction and breathing of the shape) or tumbling modes, the complex viscosity shows an infinite number of resonances as a function of the frequency. It is shown that these behaviours markedly differ from that obtained when only the classical oscillation is applied. The results are obtained numerically by solution of the analytical constitutive equation of a dilute vesicle suspension and confirmed analytically by a linear-response phenomenological theory. It is argued that the same type of behaviour is expected for any suspension of soft entities (capsules, red blood cells, etc.) that exhibit periodic motion under constant shear flow. We shall also discuss the reason why this type of behaviour could not have been captured by existing constitutive laws of complex fluids.

While non-Boussinesq hexagonal convection patterns are known to be stable close to threshold (i.e. for Rayleigh numbers $R \,{\approx}\, R_c$), it has often been assumed that they are always unstable to rolls for slightly higher Rayleigh numbers. Using the incompressible Navier–Stokes equations for parameters corresponding to water as the working fluid, we perform full numerical stability analyses of hexagons in the strongly nonlinear regime ($\epsilon\,{\equiv}\, (R-R_c)/R_c\,{=}\,{O}(1)$). We find ‘re-entrant’ behaviour of the hexagons, i.e. as $\epsilon$ is increased they can lose and regain stability. This can occur for values of $\epsilon$ as low as $\epsilon\,{=}\,0.2$. We identify two factors contributing to the re-entrance: (i) far above threshold there exists a hexagon attractor even in Boussinesq convection as has been shown recently and (ii) the non-Boussinesq effects increase with $\epsilon$. Using direct simulations for circular containers we show that the re-entrant hexagons can prevail even for sidewall conditions that favour convection in the form of competing stable rolls. For sufficiently strong non-Boussinesq effects hexagons even become stable over the whole $\epsilon$-range considered, $0 \,{\le}\, \epsilon \,{\le}\, 1.5$.

In this study, the theory of micropolar fluids is employed to study the stability problem of flow between two concentric rotating cylinders. The field equations subject to no-slip conditions (non-zero velocity and microrotation velocity components) at the wall surfaces are solved. The analytical solutions of the velocity and microrotation velocity fields as well as the shear stress difference, couple stress and strain rate for basic flow are obtained. The equations with respect to non-axisymmetric disturbances are derived and solved by a direct numerical procedure. It is found that non-zero wall-surface microrotation velocity makes the flow faster and more unstable. Moreover, it tends to reduce the limits of critical non-axisymmetric disturbances. The effect on the stability characteristics can be magnified by increasing the microstructure or couple-stress parameter or the microinertia parameter for the cases of corotating cylinders and a stationary outer cylinder or by decreasing the radius ratio or the microinertia parameter for the case of counterrotating cylinders.

We present a model describing the evolution of the small-scale Navier–Stokes turbulence due to its stochastic distortion by much larger turbulent scales. This study is motivated by numerical findings (Laval et al. Phys. Fluids vol. 13, 2001, p. 1995) that such interactions of separated scales play an important role in turbulence intermittency. We introduce a description of turbulence in terms of the moments of $k$-space quantities using a method previously developed for the kinematic dynamo problem (Nazarenko et al. Phys. Rev. E vol. 68, 2003, 0266311). Working with the $k$-space moments allows us to introduce new useful measures of intermittency such as the mean polarization and the spectral flatness. Our study of the small-scale two-dimensional turbulence shows that the Fourier moments take their Gaussian values in the energy cascade range whereas the enstrophy cascade is intermittent. In three dimensions, we show that the statistics of turbulence wavepackets deviates from Gaussianity toward dominance of the plane polarizations. Such turbulence is formed by ellipsoids in the $k$-space centred at its origin and having one large, one neutral and one small axis with the velocity field pointing parallel to the smallest axis.

We present a detailed analysis of temperature statistics in an oceanographic observational dataset. The data are collected using a moored array of thermistors, $100~\text{m}$ tall and starting $5~\text{m}$ above the bottom, deployed during four months above the slopes of a Seamount in the north-eastern Atlantic Ocean. Turbulence at this location is strongly affected by the semidiurnal tidal wave. Mean stratification is stable in the entire dataset. We compute structure functions, of order up to 10, of the distributions of temperature increments. Strong intermittency is observed, in particular, during the downslope phase of the tide, and farther from the solid bottom. In the lower half of the mooring during the upslope phase, the temperature statistics are consistent with those of a passive scalar. In the upper half of the mooring, the temperature statistics deviate from those of a passive scalar, and evidence of turbulent convective activity is found. The downslope phase is generally thought to be more shear-dominated, but our results suggest on the other hand that convective activity is present. High-order moments also show that the turbulence scaling behaviour breaks at a well-defined scale (of the order of the buoyancy length scale), which is however dependent on the flow state (tidal phase, height above the bottom). At larger scales, wave motions are dominant. We suggest that our results could provide an important reference for laboratory and numerical studies of mixing in geophysical flows.