The scattering of an incident acoustic wave by a non-uniform mean flow resulting from a heat source is investigated. The heat source produces gradients in the mean flow and the speed of sound that scatter the incident duct acoustic mode into vortical, entropic, and higher-order acoustic modes. Linear solutions utilizing the compact source limit and nonlinear solutions to the Euler equations are computed to understand how variations in the amplitude and axial extent of the heat source as well as the incident acoustic wave propagation angle and amplitude modify the scattered solution. For plane wave excitation, significant entropy waves are produced as the net heat addition increases at the expense of the transmitted acoustic energy. When the net heat addition is held constant, increasing the axial extent of the heat source results in a reduction of the entropy waves produced downstream and a corresponding increase in the downstream scattered acoustic energy. For circumferential acoustic mode excitations the incident acoustic wave angle, characterized by the cutoff ratio, significantly modifies the scattered acoustic energy. As the propagating mode cutoff ratio approaches unity, a rise in the scattered vortical disturbance and a decrease in the entropic disturbance amplitude is observed. As the cutoff ratio increases, the scattered solution approaches the plane wave results. Moreover, incident acoustic waves with different frequencies and circumferential mode orders but similar cutoff ratios yield similar scattered wave coefficients. Finally, for large amplitude incident acoustic waves the scattered solution is modified by nonlinear effects. The pressure field exhibits nonlinear steepening of the wavefront and the nonlinear interactions produce higher harmonic frequency content which distorts the sinusoidal variation of the outgoing scattered acoustic waves.

We define as an optimal mixer a mixing device able to deliver a uniformly optimal mixing performance over a wide range of operating and initial conditions. We consider the conceptual problem of designing an optimal mixer starting from a well-known reference mixer, the sine flow. We characterize the mixing performance of the reference mixer, and show that it performs poorly and erratically over a wide range of operating conditions and is quite sensitive to the geometry of the initial concentration field. We define as a target performance the best mixing performance the reference mixer is able to achieve. In steps we modify the design of the reference mixer. First, we optimize the time sequence of the switching protocols and show that the mixing performance of the time-optimized mixer, although substantially improved with respect to the reference mixer, is still far from achieving the target performance and being insensitive to the geometry of the initial concentration field. The analysis of the performance of the time-optimized mixer brings to light the deficiency of the actuating system used, which delivers always the same amount of shear at the same locations. We modify the actuating system by allowing the stirring velocity fields to shift along their coordinate axes. A new mixer, the space-optimized mixer, is created by equipping the reference mixer with the new actuating system and optimizing the shift of the stirring velocity field at each iteration. The space-optimized mixer is able to deliver the target performance over the upper two-thirds of the operating range. In the lower one-third, the performance of the space-optimized mixer deteriorates because of the use of a periodic protocol. A optimal mixer is finally obtained using the actuating system of the space-optimized mixer and coupling the time and shift optimizations. The resulting optimal mixer is able to deliver a uniform target performance, insensitive to the geometry of the initial conditions, over the entire operating range.

This work experimentally studies the behaviour of underwater explosion bubbles near different boundaries. The results are compared with theoretical and experimental data on cavitation bubbles. Although explosion and cavitation bubbles behave similarly on a macroscopic level, there are still some differences, most of which are from the explosive nature of the explosion bubble. The relationship between bubble migration and the Kelvin impulse, surface inertia m* and surface stiffness k* is investigated. We found that none of them comprehensively predicts the migration of both cavitation and explosion bubbles when boundary elasticity is considered. This elasticity should be considered as a relative value with respect to bubble size. On the other hand, the phase between local vibration of boundaries and the pulsation of bubbles could be a useful predictive index of bubble migration. When using research results developed for cavitation bubbles in relation to explosion bubbles, the material presented here may be useful for pointing out their similarities and differences.

The dynamics of neutrally buoyant particles transported by a turbulent flow is investigated for spherical particles with radii of the order of the Kolmogorov dissipative scale or larger. The pseudo-penalization spectral method that has been proposed by Pasquetti et al. (Appl. Numer. Math., vol. 58, 2008, pp. 946–954) is adapted to integrate numerically the simultaneous dynamics of the particle and of the fluid. Such a method gives a unique handle on the limit of validity of point-particle approximations, which are generally used in applicative situations. Analytical predictions based on such models are compared to result of very well-resolved direct numerical simulations. Evidence is obtained that Faxén corrections reproduce dominant finite-size effects on velocity and acceleration fluctuations for particle diameters up to four times the Kolmogorov scale. The dynamics of particles with larger diameters is consistent with predictions obtained from dimensional analysis.

The deformation of a cavitation bubble in shear and extensional flows is studied numerically. The Navier–Stokes equations are solved to observe the three-dimensional behaviour of the bubble as it grows and collapses. During the collapse phase of the bubble, two re-entrant jets are observed on two sides of the bubble. The re-entrant jets are not the result of interaction with a solid wall or free surface; rather, they are formed due to interaction of the bubble with the background flow. Effects of the viscosity, surface tension and shear rate on the formation and strength of re-entrant jets are investigated. Re-entrant jets with enough strength break up the bubble into smaller bubbles. Post-processing and analysis of the results are done to cast the disturbance by the bubble on the liquid velocity field in terms of spherical harmonics. It is found that quadrupole moments are created in addition to the monopole source.

The instability of a liquid layer coating the surface of a thin cylindrical wire is studied experimentally and numerically with negligible gravity effects. The initial uniform film is obtained as the residual of a sliding drop, and the thickness measurements are performed with an anamorphic optical system that compresses the vertical scale (allowing to observe several wavelengths) and widens the horizontal one (to follow in detail the evolution of local minima and maxima). Experimental timelines showing the growth and position of the maxima and minima are compared with linear theory and fully nonlinear simulations. A primary mode grows in the early stages of the instability, and its wavelength λ1 is not always in agreement with that corresponding to the maximum growth rate predicted by the linear theory λm. In later stages, a secondary mode appears, whose wavelength is half that of the primary mode. The behaviour of the secondary mode allows us to classify the experimental results into two cases, depending on whether it is linearly stable (case I) or unstable (case II). In case I, the amplitude of the secondary mode remains small compared with that of the primary one, while in case II both amplitudes may become very similar at the end. Thus, the distance between the final drops may be quite different from that seen between initial protuberances. The analysis of the experiments allows us to define a simple criterion based on the comparison between λ1 and λm. Contrary to the predictions of widely used previous quasi-static theories, experiments show that the relation between maximum and minimum of the primary mode is better approximated by a kinematic model based on the assumption that primary maxima increase as fast as the minima decrease. Numerical simulations confirm this hypothesis.

Complex wave patterns caused by unsteady heat release due to cloud formation in confined compressible flows are discussed. Two detailed numerical studies of condensation-induced waves are carried out. First, the response of a flow of nitrogen in a slender Laval nozzle to a sudden addition of water vapour at the nozzle entrance is considered. Condensation occurs just downstream of the nozzle throat, which initially leads to upstream- and downstream-moving shocks and an expansion fan downstream of the condensation front. Then, the flow becomes oscillatory and the expansion fan disappears, while upstream and much weaker downstream shocks are repeatedly generated. For a lower initial humidity, only a downstream starting shock is formed and a steady flow is established. Second, homogeneous condensation in an unsteady expansion fan in humid nitrogen is considered. In the initial phase, two condensation-induced shocks are found, moving upstream and downstream. The upstream-moving shock changes the shape of the expansion fan and has a strong influence on the condensation process itself. It is even quenching the nucleation process locally, which leads to a renewed condensation process more downstream. This process is repeated with asymptotically decreasing strength. The repeated interaction of the condensation-induced shocks with the main expansion wave leads to a distortion of the expansion wave towards its shape that can be expected on the basis of phase equilibrium, i.e. a self-similar wave structure consisting of dry part, a plateau of constant state and a wet part. The strengths of the condensation-induced waves, as well for the Laval nozzle flow as for the expansion fan, appear to be in qualitative agreement with the results from the analytical Rayleigh–Bartlmä model.

Turbulent swept flow over a cylindrical wire placed on a wall of a channel is investigated using direct numerical simulations. This geometry is a model of the flow through the wire-wrapped fuel pins, the heat exchanger, typical of many nuclear reactor designs. Mean flow along and across the wire axis is imposed, leading to the formation of separated flow regions. The Reynolds number based on the bulk velocity along the wire axis direction and the channel half height is 5400 and four cases are simulated with different flowrates across the wire. This configuration is topologically similar to backward-facing steps or slots with swept flow, except that the dominant flow is along the obstacle axis in the present study and the crossflow is smaller than the axial flow, i.e. the sweep angle is large. Mean velocities, turbulence statistics, wall shear stress and instantaneous flow structures are investigated. Particular attention is devoted to the statistics of the shear stress on the walls of the channel and the wire in the recirculation zone. The flow around the mean reattachment region, at the termination of the recirculating bubble, does not exhibit the typical decay of the mean shear stress observed in classical backward-facing step flows owing to the presence of a strong axial flow. The evolution of the mean wall shear stress angle after reattachment indicates that the flow recovers towards equilibrium at a rather slow rate, which decreases with sweep angle. Finally, the database is analysed to estimate resolution requirements, in particular around the recirculation zones, for large-eddy simulations. This has implications in more complete geometrical models of a wire-wrapped assembly, involving hundreds of fuel pins, where only turbulence modelling can be afforded computationally.

We describe, by means of asymptotic methods and direct numerical simulation, the structure of singularities developing at the interface between two perfect, inviscid and irrotational fluids of different densities ρ1 and ρ2 and under the action of gravity. When the lighter fluid is on top of the heavier fluid, one encounters the water-wave problem for fluids of different densities. In the limit when the density of the lighter fluid is zero, one encounters the classical water-wave problem. Analogously, when the heavier fluid is on top of the lighter fluid, one encounters the Rayleigh–Taylor problem for fluids of different densities, with this being the case when one of the densities is zero for the classical Rayleigh–Taylor problem. We will show that both water-wave and Rayleigh–Taylor problems develop singularities of the Moore-type (singularities in the curvature) when both fluid densities are non-zero. For the classical water-wave problem, we propose and provide evidence of the development of a singularity in the form of a logarithmic spiral, and for the classical Rayleigh–Taylor problem no singularities were found. The regularizing effects of surface tension are also discussed, and estimates of the size and wavelength of the capillary waves, bubbles or blobs that are produced are provided.

Helicity is a quadratic invariant of the Euler equation in three dimensions. As the energy, when present helicity cascades to smaller scales where it dissipates. However, the role played by helicity in the energy cascade is still unclear. In non-helical flows, the velocity and the vorticity tend to align locally creating patches with opposite signs of helicity. Also in helical flows helicity changes sign rapidly in space. Not being a positive definite quantity, global studies considering its spectral scaling in the inertial range are inconclusive, except for cases where one sign of helicity is dominant. We use the cancellation exponent to characterize the scaling laws followed by helicity fluctuations in numerical simulations of helical and non-helical turbulent flows, with different forcing functions and spanning a range of Reynolds numbers from ≈670 to ≈6200. The exponent can be related to the fractal dimension as well as to the first-order helicity scaling exponent. The results are consistent with the geometry of helical structures being filamentary. Further analysis indicates that statistical properties of helicity fluctuations in the simulations do not depend on the global helicity of the flow.

In this paper, we study the effect of the Reynolds number (Re) on the dynamics and vortex formation modes of spheres rising or falling freely through a fluid (where Re = 100–15000). Since the oscillation of freely falling spheres was first reported by Newton (University of California Press, 3rd edn, 1726, translated in 1999), the fundamental question of whether a sphere will vibrate, as it rises or falls, has been the subject of a number of investigations, and it is clear that the mass ratio m* (defined as the relative density of the sphere compared to the fluid) is an important parameter to define when vibration occurs. Although all rising spheres (m* < 1) were previously found to oscillate, either chaotically or in a periodic zigzag motion or even to follow helical trajectories, there is no consensus regarding precise values of the mass ratio (m*crit) separating vibrating and rectilinear regimes. There is also a large scatter in measurements of sphere drag in both the vibrating and rectilinear regimes.

In our experiments, we employ spheres with 133 combinations of m* and Re, to provide a comprehensive study of the sphere dynamics and vortex wakes occurring over a wide range of Reynolds numbers. We find that falling spheres (m* > 1) always move without vibration. However, in contrast with previous studies, we discover that a whole regime of buoyant spheres rise through a fluid without vibration. It is only when one passes below a critical value of the mass ratio, that the sphere suddenly begins to vibrate periodically and vigorously in a zigzag trajectory within a vertical plane. The critical mass is nearly constant over two ranges of Reynolds number (m*crit ≈ 0.4 for Re = 260–1550 and m*crit ≈ 0.6 for Re > 1550). We do not observe helical or spiral trajectories, or indeed chaotic types of trajectory, unless the experiments are conducted in disturbed background fluid. The wakes for spheres moving rectilinearly are comparable with wakes of non-vibrating spheres. We find that these wakes comprise single-sided and double-sided periodic sequences of vortex rings, which we define as the ‘R’ and ‘2R’ modes. However, in the zigzag regime, we discover a new ‘4R’ mode, in which four vortex rings are created per cycle of oscillation. We find a number of changes to occur at a Reynolds number of 1550, and we suggest the possibility of a resonance between the shear layer instability and the vortex shedding (loop) instability. From this study, ensuring minimal background disturbances, we have been able to present a new regime map of dynamics and vortex wake modes as a function of the mass ratio and Reynolds number {m*, Re}, as well as a reasonable collapse of the drag measurements, as a function of Re, onto principally two curves, one for the vibrating regime and one for the rectilinear trajectories.

We examine the integral properties of freely decaying homogeneous magnetohydrodynamic (MHD) turbulence subject to an imposed magnetic field B0 at low-magnetic Reynolds number. We confirm that, like conventional isotropic turbulence, the fully developed state possesses a Loitsyansky-like integral invariant, in this case I// = − ∫ r⊥2 〈 u⊥·u′⊥〉 dr, where 〈u(x) ·u(x + rc)〉 = 〈u·u′〉 is the usual two-point velocity correlation and the subscript ⊥ indicates components perpendicular to the imposed field. The conservation of I// for fully developed turbulence places a fundamental restriction on the way in which the integral scales can develop, i.e. it implies u⊥2 ℓ⊥4 ℓ// ≈ constant where u⊥, ℓ⊥ and ℓ// are integral scales. This constraint can be used to estimate the evolution of u⊥(t; B0), ℓ⊥(t; B0) and ℓ//(t; B0), and these theoretical decay laws are shown to be in good agreement with numerical simulations.

The sensitivity of the three-dimensional secondary instability of a circular-cylinder wake to a structural perturbation of the associated linear equations is investigated. In particular, for a given flow condition, the region of maximum coupling between the velocity components is localized by using the most unstable Floquet mode and its adjoint mode. The variation of this region in time is also found by considering a structural perturbation which is impulsively applied in time at a given phase of the vortex-shedding process. The analysis is carried out for both mode A and mode B types of transition in the wake of a circular cylinder using a finite-difference code. The resulting regions identified as the core of the instability are in full agreement with the results reported in the literature and with the a posteriori checks documented here.

Multi-point velocity measurements have been performed in turbulent pipe flow at ReD = 1.5 × 105 and combined with cross-spectral and proper orthogonal decomposition analysis to elucidate information on the structure of the large- and very-large-scale motions in the outer layer of wall-bounded flows. The results indicate that in the outer layer the large-scale motions (LSM) may be composed of detached eddies with a wide range of azimuthal scales, whereas in the logarithmic layer they are attached. The very-large-scale motions (VLSM) have large radial scales, are concentrated around a single azimuthal mode and make a smaller angle with the wall compared to the LSM. The results support a hypothesis that only the detached LSM in the outer layer align to form the VLSM.

The effect of an insoluble surfactant on the centrifugal and shear instability of a pair of radially stratified immiscible liquids in the annular gap between concentric two-fluid Taylor–Couette flow is investigated by a normal-mode linear analysis and complementary energy analysis. The interface is assumed to be concentric with the cylinders. The gravitational effects are ignored. Influences of density and viscosity stratification, surface tension, surfactant concentration distribution and Taylor–Couette shearing are considered comprehensively. The instability characteristics due to competition and interaction between various physical instability mechanisms are of principal concern. Neutral curves with upper and lower branches in the Reynolds number (Re1)/axial wavenumber (k) plane are obtained. A window of parameters is identified in which the flow is linearly stable. The Marangoni traction force caused by the gradient of surfactant concentration stabilizes the axisymmetric perturbations but initiates an instability corresponding to non-axisymmetric modes in the presence of basic Couette shearing flow. Co-rotation of the outer cylinder has a stabilizing effect in expanding the stable region, which dwindles in the counter-rotation situation.

We derive equations to describe the flow of multiple superposed layers of inviscid, incompressible fluids with constant densities over prescribed topography in a rotating frame. Motivated by geophysical applications, these equations incorporate the complete Coriolis force. We do not make the widely used ‘traditional approximation’ that omits the contribution to the Coriolis force from the locally horizontal part of the rotation vector. Our derivation is performed by averaging the governing Euler equations over each layer, and from two different forms of Hamilton's variational principle that differ in their treatment of the coupling between layers. The coupling may be included implicitly through the map from Lagrangian particle labels to particle coordinates, or explicitly by adding terms representing the work done on each layer by the pressure exerted by the layers above. The latter approach requires additional terms in the Lagrangian, but extends more easily to many layers. We show that our equations obey the expected conservation laws for energy, momentum and potential vorticity. The conserved momentum and potential vorticity are modified by non-traditional effects. The vertical component of the rotation vector that appears in the potential vorticity for each layer under the traditional approximation is replaced by the component perpendicular to the layer's midsurface. The momentum includes an additional contribution that reflects changes in angular momentum caused by changes in a fluid element's distance from the rotation axis as it is displaced vertically. Again, this effect is absent in the traditional approximation.

The results from a numerical investigation of inertial viscoelastic flow past a circular cylinder are presented which illustrate the significant effect that dilute concentrations of polymer additives have on complex flows. In particular, effects of polymer extensibility are studied as well as the role of viscoelasticity during three-dimensional cylinder wake transition. Simulations at two distinct Reynolds numbers (Re = 100 and Re = 300) revealed dramatic differences based on the choice of the polymer extensibility (L2 in the FENE-P model), as well as a stabilizing tendency of viscoelasticity. For the Re = 100 case, attention was focused on the effects of increasing polymer extensibility, which included a lengthening of the recirculation region immediately behind the cylinder and a sharp increase in average drag when compared to both the low extensibility and Newtonian cases. For Re = 300, a suppression of the three-dimensional Newtonian mode B instability was observed. This effect is more pronounced for higher polymer extensibilities where all three-dimensional structure is eliminated, and mechanisms for this stabilization are described in the context of roll-up instability inhibition in a viscoelastic shear layer.

Diffusive convection in an infinite two-dimensional fluid with linear vertical gradients of temperature and salinity is studied numerically and analytically. When the density gradient ratio exceeds a critical value above which diffusive convection grows according to the linear stability analysis, spontaneous layer formation is found to occur. At the first stage nearly steady oscillating motions, the horizontal scale of which is of the order of the buoyancy boundary layer scale δ, arise. After several tens of the oscillation cycle, a transition to the second stage occurs in which overturning convective motions develop and well-mixed regions are formed. The convective motions resemble Rayleigh–Bénard convection at a high Rayleigh number. The well-mixed regions are gradually organized into horizontal layers, a typical thickness of which is of the order of δ. A detailed analysis of the nonlinear process during the layer formation reveals that four modes are responsible for the layer formation: The first mode is the linear fastest-growing mode with wavenumber vector (k0, 0). The second mode with (k0, m0) is weakly growing. The third mode with (0, m0) is dissipating, and the fourth mode is its higher harmonic having (0, 2m0). It is shown that a truncated spectral model with the four modes well reproduces the results of the full numerical simulation.

Evolution of layers in an unbounded diffusively stratified two-component fluid and its dynamics are studied by means of a direct numerical simulation (DNS) and an analytical model. The numerical simulation shows that the layers grow by repeating mergings with the neighbouring layers. By analysing the results of the numerical simulation, the mechanism of the merging is examined in detail. Two modes of merging are found to exist: one is the layer vanishing mode and the other is the interface vanishing mode. The vanishings of layers and interfaces are caused by turbulent entrainment at the interfaces. Based on the analysis of the numerical model, a one-dimensional asymmetric entrainment model is proposed. In the model, each layer is assumed to interact with its neighbouring layers through simplified convective entrainment laws. The model is applied to two simple configurations of layers and is proved to reproduce the layer evolutions found in the DNS successfully.

The amplification of enstrophy is explored using cinematographic stereoscopic particle image velocimetry data. The enstrophy production rate is investigated by observation of the statistical tendency of the vorticity vector (ω) to align with the eigenvectors of the rate of strain tensor (ei). Previous studies have shown that ω preferentially aligns with the intermediate strain-rate eigenvector (e2) and is arbitrarily aligned with the extensive strain-rate eigenvector (e1). This study shows, however, that the nature of enstrophy amplification, whether it is positive (enstrophy production) or negative (enstrophy destruction), is dictated by the alignment between ω and e1. Parallel alignment leads to enstrophy production (ωiSijωj>0), whereas perpendicular alignment leads to enstrophy destruction (ωiSijωj<0). In this way, the arbitrary alignment between ω and e1 is the summation of the effects of enstrophy producing and enstrophy destroying regions. The structure of enstrophy production is also examined with regards to the intermediate strain-rate eigenvalue, s2. Enstrophy producing regions are found to be topologically ‘sheet-forming’, due to an extensive (positive) value of s2 in these regions, whereas enstrophy destroying regions are found to be ‘spotty’. Strong enstrophy producing regions are observed to occur in areas of strong local swirling as well as in highly dissipative regions. Instantaneous visualizations, produced from the volume of data created by Taylor's hypothesis, reveal that these ‘sheet-like’ strong enstrophy producing regions encompass the high enstrophy, strongly swirling ‘worms’. These ‘worms’ induce high local strain fields leading to the formation of dissipation ‘sheets’, thereby revealing enstrophy production to be a complex interaction between rotation and strain – the skew-symmetric and symmetric components of the velocity gradient tensor, respectively.