We derive a novel set of analytical solutions of the Navier–Stokes equations describing stationary plane motion of two immiscible fluids emitted from a two-dimensional point source. The solutions are two-fluid generalizations of Jeffery–Hamel flows. The presence of an interface yields an unexpectedly rich and complex structure of solutions. Each set of physical parameter values admits a hierarchy of three different types of solution. The solutions bifurcate when parameters of the flow change sufficiently, with each type of solution having a different bifurcation diagram.

Numerical experiments are conducted to study high-Rayleigh-number convective turbulence ($Ra$ ranging from $2\times 10^6$ up to $2\times 10^{11}$) in a $\Gamma=1/2$ aspect-ratio cylindrical cell heated from below and cooled from above and filled with gaseous helium ($Pr=0.7$). The numerical approach allows three-dimensional velocity, vorticity and temperature fields to be analysed. Furthermore, several numerical probes are placed within the fluid volume, permitting point-wise velocity and temperature time series to be extracted. Taking advantage of the data accessibility provided by the direct numerical simulation the flow dynamics has been explored and separated into its mean large-scale and fluctuating components, both in the bulk and in the boundary layer regions. The existence of large-scale structures creating a mean flow sweeping the horizontal walls has been confirmed. However, the presence of a single recirculation cell filling the whole volume was observed only for $Ra < 10^9 - 10^{10}$ and with reduced intensity compared to axisymmetric toroidal vortices attached to the horizontal plates. At larger $Ra$ the single cell is no longer observed, and the bulk recirculation breaks up into two counter-rotating asymmetric unity-aspect-ratio rolls. This transition has an appreciable impact on the boundary layer structure and on the global heat transfer properties. The large-scale structure signature is reflected in the statistics of the bulk turbulence as well, which, taking advantage of the large number of numerical probes available, is examined both in terms of frequency spectra and of temperature structure functions. The present results are also discussed within the framework of recent theoretical developments showing that the effect of the aspect ratio on the global heat transfer properties at large $Ra$ still remains an open question.

The flow produced in an enclosed cylinder of height-to-radius ratio of two by the counter-rotation of the top and bottom disks is numerically investigated. When the Reynolds number based on cylinder radius and disk rotation is increased, the axisymmetric basic state loses stability and different complex flows appear successively: steady states with an azimuthal wavenumber of 1; travelling waves; near-heteroclinic cycles; and steady states with an azimuthal wavenumber of 2. This scenario is understood in a dynamical system context as being due to a nearly codimension-two bifurcation in the presence of $O(2)$ symmetry. A bifurcation diagram is determined, together with the most dangerous eigenvalues as functions of the Reynolds number. Two distinct types of near-heteroclinic cycles are observed, with either two or four bursts per cycle. The physical mechanism for the primary instability could be the Kelvin–Helmholtz instability of the equatorial azimuthal shear layer of the basic state.

Green's functions for a source embedded in an isothermal transversely sheared mixing layer are compared with direct numerical simulation (DNS) at various frequencies. Based on the third-order convective wave equation (Lilley 1974), three types of wave responses are analysed. For direct waves, a vortex sheet is used in the low-frequency limit, while in the high-frequency limit the procedure derived by Goldstein (1982) is re-visited. For refracted arrival waves propagating in the zone of silence, the vortex sheet model derived by Friedland & Pierce (1969) is re-visited in the low-frequency limit, while in the high-frequency limit the finite thickness model derived by Suzuki & Lele (2002) is applied. Instability waves excited by a very low-frequency source are formulated in the linear regime using the normal mode decomposition: eigen-functions are normalized using the adjoint convective wave equation, and the receptivity of instability waves is predicted. These theoretical predictions are compared with numerical simulations in two dimensions: DNS are performed based on the full Navier–Stokes equations (the free-stream Mach number is $M_1=0.8$, and the ratios of the acoustic wavelength to the vorticity thickness $\lambda/\delta_V$ are 4.0, 1.0 and 0.25). The DNS results agree fairly well with the high-frequency limit in all three cases for direct waves, although the lowest-frequency case ($\lambda/\delta_V = 4.0$) indicates some features predicted in the low-frequency limit. For refracted arrival waves, the DNS data follow the low- and high-frequency limits to a reasonable degree of accuracy in all cases. Moreover, by setting $\lambda/\delta_V = 16.0$, instability waves are simulated, and a comparison with the theoretical prediction shows that the instability wave response is predicted well when a mixing-layer Reynolds number is high ($Re = 10^5$). They also reveal that the receptivity is fairly sensitive to the Reynolds number and the source position within the mixing layer.

Green's functions for a source embedded in an isothermal transversely sheared boundary layer are compared with direct numerical simulation (DNS) at various frequencies and free-stream Mach numbers. The procedures developed for a mixing layer in Part 1 (Suzuki & Lele 2003) are applied to derive the low- and high-frequency Green's functions for direct waves, i.e. the third-order convective wave equation is solved using asymptotic matching. In addition, channelled waves propagating downstream along the wall are analysed using the normal mode decomposition. By introducing an adjoint operator of the convective wave equation with a mixed-type boundary condition on the wall, the corresponding Hilbert space is defined and eigenfunctions of channelled waves are normalized. Furthermore, diffracted waves in the shadow zone are formulated in the high-frequency limit. These theoretical predictions are compared with numerical simulations in two dimensions: DNS are performed based on the full Navier–Stokes equations (the ratios between the acoustic wavelength and the boundary layer thickness are $\lambda/\delta_{BL}=4.0, 1.0, \hbox{ and } 0.25$ at a free-stream Mach number of $M_{\infty}=0.8$; and $\lambda/\delta_{BL}=1.0$ at $M_{\infty}=0.3 \hbox{ and } 1.2$). The DNS results generally agree with the theories: the pressure amplitudes of direct waves and diffracted waves follow the high-frequency limit with a reasonable degree of accuracy in the intermediate- and high-frequency cases ($\lambda/\delta_{BL}=1.0 \hbox{ and } 0.25$). The DNS results for channelled waves also agree with the theoretical predictions fairly well. In addition, the acoustic impedance on the wall under a strongly sheared viscous boundary layer is derived asymptotically based on the modal analysis.

We have studied the stability of ink-jet printed lines of liquid with zero receding contact angle on a homogeneous substrate. Such lines can become unstable by forming a series of liquid bulges, at various wavelengths, connected by a ridge of liquid. The instability was studied with a simple dynamic model. It was shown that the line becomes unstable when the contact angle of the liquid with the substrate is larger than the advancing contact angle. This condition, however, is not a sufficient condition. When the transported flow rate is sufficiently small compared to the applied flow rate a printed line can be shown to be stable, i.e. the width of the printed line is constant. This was found to depend on the advancing contact angle of the liquid over the substrate.

This paper reports the results of lattice Boltzmann simulations of the rotation behaviour of neutrally buoyant spheroidal particles in a three-dimensional Couette flow. We find several distinctive states depending on the Reynolds number range and particle shape. As the Reynolds number increases, rotation may change from one state to another. For a prolate spheroid, two rotation transitions are found. In the low Reynolds number range $0<R<R_1\approx 205$, the prolate spheroid rotates around its minor axis, which is parallel to the vorticity vector of the flow. The rate of rotation is a periodic function of time. In the intermediate Reynolds number range $R_1 < R < R_2\approx 345$, the prolate spheroid precesses about the vorticity direction with a nutational motion. The angular velocities are periodic functions of time. The mean nutation angle between the major axis and the vorticity increases monotonically as the Reynolds number increases. In the high Reynolds number range $R_2 < R < 467$, the prolate spheroid rotates with a constant rate around its major axis, which is parallel to the vorticity. For an oblate spheroid, only one rotation transition is observed. In the lower Reynolds number range $0 < R < R_1' \approx 220$, the oblate spheroid finally spins with a constant rate around its minor axis (the symmetric axis of the revolution), which is parallel to the vorticity vector. In the higher Reynolds number range $220 \approx R_1' < R < 467$, the oblate spheroid still spins with a constant rate around its minor axis but there is a finite inclination angle between the minor axis and the vorticity vector. This angle increases as the Reynolds number increases.

The unsteady flow evolution in a porous chamber with surface mass injection simulating propellant burning in a nozzleless solid rocket motor has been investigated by means of a large-eddy simulation (LES) technique. Of particular importance is the turbulence-transition mechanism in injection-driven compressible flows with high injection rates in a chamber closed at one end and connected to a divergent nozzle at the exit. The spatially filtered and Favre-averaged conservation equations of mass, momentum and energy are solved for resolved scales. The effect of unresolved subgrid scales is treated by using a dynamic Smagorinsky model extended to compressible flows. Three successive regimes of flow development are observed: laminar, transitional, and fully developed turbulent flow. Surface transpiration facilitates the formation of roller-like vortical structures close to the injection surface. The flow is essentially two-dimensional up to the mid-section of the chamber, with the dominant frequencies of vortex shedding governed by two-dimensional hydrodynamic instability waves. These two-dimensional structures are convected downstream and break into complex three-dimensional eddies. Transition to turbulence occurs further away from the wall than in standard channel flows without mass injection. The peak in turbulence intensity moves closer to the wall in the downstream direction until the surface injection prohibits further penetration of turbulence. The temporal and spatial evolution of the vorticity field obtained herein is significantly different from that of channel flow without transpiration.

The surgical technique of thread injection of medical implants is modelled by the axial pressure-gradient-driven flow between concentric cylinders with a moving core. The nonlinear stability of the basic flow is analysed theoretically at asymptotically large Reynolds number and it is found that non-axisymmetric finite-amplitude neutral modes can be supported over a wide range of thread radii and injection velocities. The axial force on the thread is calculated and it is found to be significantly less than that predicted by undisturbed-flow theory, in agreement with thread–annular experiments.

A simple mechanism of parametric excitation of internal gravity waves in a uniformly stably stratified flow under the inviscid Boussinesq approximation is presented. It consists in an oscillating planar irrotational strain field with frequency $\omega$ disturbed by three-dimensional plane waves. When the amplitude of the strain is weak, the problem is reduced to a Mathieu equation and a condition for parametric resonance is easily deduced. For a large-amplitude strain field equations are solved numerically with Floquet theory. In both cases, it is shown that parametric instabilities are excited when stratification is large enough, that is when $N > \frac{1}{2}\omega$, where $N$ is the Brunt–Väisälä frequency of the flow. On the other hand, when $N \leq \frac{1}{2}\omega$, the flow is shown to be stable for any periodic background excitation thanks to a theorem by Joukowski. Therefore, stratification promotes instability. In the strongly stratified case $N\,{\gg}\,\omega$, resonant waves satisfy the Billant–Chomaz self-similarity law and the resulting instabilities develop inside correlated quasi-horizontal layers. After discussion of the viscous effects, the theory of the paper is applied to the stability of an elliptical vortex in a rotating stratified medium.

The objective of this paper is to discuss and analyse the accuracy of various velocity formulations for water waves in the framework of Boussinesq theory. To simplify the discussion, we consider the linearized wave problem confined between the still-water datum and a horizontal sea bottom. First, the problem is further simplified by ignoring boundary conditions at the surface. This reduces the problem to finding truncated series solutions to the Laplace equation with a kinematic condition at the sea bed. The convergence and accuracy of the resulting expressions is analysed in comparison with the target cosh- and sinh-functions from linear wave theory. First, we consider series expansions in terms of the horizontal velocity variable at an arbitrary $z$-level, which can be varied from the sea bottom to the still-water datum. Second, we consider the classical possibility of expanding in terms of the depth-averaged velocity. Third, we analyse the use of a horizontal pseudo-velocity determined by interpolation between velocities at two arbitrary $z$-levels. Fourth, we investigate three different formulations based on two expansion variables, being the horizontal and vertical velocity variables at an arbitrary $z$-level. This is shown to have a remarkable influence on the convergence and to improve accuracy considerably. Fifth, we derive and analyse a new formulation which doubles the power of the vertical coordinate without increasing the order of the horizontal derivatives. Finally, we involve the kinematic and dynamic boundary conditions at the free surface and discuss the linear dispersion relation and a spectral solution for steady nonlinear waves.

The motion of a quantum vortex is considered in the context of the localized induction approximation (LIA). In this context, the instantaneous vortex velocity is proportional to the local curvature and is parallel to the vector which is a linear combination of the local binormal and the principal normal to the vortex line. This implies that the quantum vortex shrinks, which is in contrast to the classical vortex in an ideal fluid. The present work deals with a four-parameter class of static solutions of the equations governing the curvature and the torsion. Such solutions describe vortex lines, the motion of which is equivalent to an isometric transformation. In a particular case when the transformation is a pure translation, the analytic solutions for the curvature and the torsion are found. In the general case, when the transformation is a superposition of a non-trivial translation and rotation, the asymptotics of solutions is explicitly related to the parameters characterizing the transformation, and then to the initial conditions at the zero point of the vortex. In this case, the equations are solved numerically and the shape of a number of different vortices is reconstructed by numerical integration of Frenet–Seret equations.

A laboratory investigation of exchange flows near the two-layer hydraulic limit is used to examine the generation of shear instability at the interface dividing the two layers. The present experiments differ from many previous investigations into shear instability, in that the instabilities are an active part of a quasi-steady flow regime rather than the product of a controlled initial state. Regimes characterized by either Kelvin–Helmholtz or Holmboe's instability are found to be separated by a well-defined transition. Observations of the transition from Kelvin–Helmholtz to Holmboe's instability are compared to predictions from scaling arguments that draw on elements of both two-layer hydraulic theory and linear stability theory. The characteristics of unstable modes near the transition, and the structure of both classes of instability are examined in detail.

We present an improved upper bound on the energy dissipation rate in plane Couette flow. This is achieved through the numerical solution of the ‘background field’ variational problem formulated by Constantin and Doering with a one-dimensional unidirectional background field. The upper bound presented here both exhausts the bounding potential of the one-dimensional background field problem and also solves the provably equivalent problem formulated by Busse. The solution is calculated up to asymptotically large Reynolds number where we can estimate that the energy dissipation rate $\e \le 0.008553$ as $Re \rightarrow \infty$ (in units of $V^3/d$ where $V$ is the velocity difference across the plates separated by a distance $d$ and $Re = V d /\nu$, with $\nu$ kinematic viscosity). This represents a 21% improvement over the previous best value due to Nicodemus et al. A comparison is drawn between this numerical solution and the so-called multi-alpha asymptotic solutions discovered by Busse.

We investigate the stability of a binary alloy directionally solidifying at a constant rate and rotating with spin and/or precession about an inclined axis. Results show that, prior to the onset of instability, a flow is induced by the inclination and modified by the rotation, having a velocity profile like a spiral Ekman flow. The induced flow moves steadily relative to the system when the system rotates with precession only, while it changes direction periodically when the system rotates with spin (even if precession is included). Based on this flow, the effects of inclined rotation on the stability of the system are examined by linear analyses. We find that there are five mechanisms affecting the stability due to inclined rotation: the reduction of both buoyancy and the rotation vector along the height of the system are stabilizing, the gravity component along the melt/solid interface is destabilizing, and the inclination-induced flow and precession combine to play a stabilizing or a destabilizing role, depending on their relative orientation and amplitude ratio. In general, the morphological mode is slightly stabilized whereas the convective and mixed modes are significantly stabilized. For inclined precession, the instability mode moving aligned with the gravity component along the melt/solid interface is most unstable. For inclined spin, all the stability-affecting mechanisms act equally in all directions so that the stability thresholds for the instability modes moving in different directions are equal. For directional solidification applications, the present results suggest that to prevent compositional non-uniformities in the solid, inclined spin is more effective than inclined precession.