JFM Papers
Radiative decay of the nonlinear oscillations of an adiabatic spherical bubble at small Mach number
- Warren R. Smith, Qianxi Wang
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- Published online by Cambridge University Press:
- 19 December 2017, pp. 1-18
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A theoretical study is carried out for bubble oscillation in a compressible liquid with significant acoustic radiation based on the Keller–Miksis equation using a multi-scaled perturbation method. The leading-order analytical solution of the bubble radius history is obtained to the Keller–Miksis equation in a closed form including both compressible and surface tension effects. Some important formulae are derived including: the average energy loss rate of the bubble system for each cycle of oscillation, an explicit formula for the dependence of the oscillation frequency on the energy, and an implicit formula for the amplitude envelope of the bubble radius as a function of the energy. Our theory shows that the frequency of oscillation does not change on the inertial time scale at leading order, the energy loss rate on the long compressible time scale being proportional to the Mach number. These asymptotic predictions have excellent agreement with experimental results and the numerical solutions of the Keller–Miksis equation over very long times. A parametric analysis is undertaken using the above formula for the energy of the bubble system, frequency of oscillation and minimum/maximum bubble radii in terms of the dimensionless initial pressure of the bubble gases (or, equivalently, the dimensionless equilibrium radius), Weber number and polytropic index of the bubble gas.
Three-dimensional Rayleigh–Taylor instability under a unidirectional curved substrate
- Gioele Balestra, Nicolas Kofman, P.-T. Brun, Benoit Scheid, François Gallaire
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- 19 December 2017, pp. 19-47
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We investigate the Rayleigh–Taylor instability of a thin liquid film coated on the inside of a cylinder whose axis is orthogonal to gravity. We are interested in the effects of geometry on the instability, and contrast our results with the classical case of a thin film coated under a flat substrate. In our problem, gravity is the destabilizing force at the origin of the instability, but also yields the progressive drainage and stretching of the coating along the cylinder’s wall. We find that this flow stabilizes the film, which is asymptotically stable to infinitesimal perturbations. However, the short-time algebraic growth that these perturbations can achieve promotes the formation of different patterns, whose nature depends on the Bond number that prescribes the relative magnitude of gravity and capillary forces. Our experiments indicate that a transverse instability arises and persists over time for moderate Bond numbers. The liquid accumulates in equally spaced rivulets whose dominant wavelength corresponds to the most amplified mode of the classical Rayleigh–Taylor instability. The formation of rivulets allows for a faster drainage of the liquid from top to bottom when compared to a uniform drainage. For higher Bond numbers, a two-dimensional stretched lattice of droplets is found to form on the top part of the cylinder. Rivulets and the lattice of droplets are inherently three-dimensional phenomena and therefore require a careful three-dimensional analysis. We found that the transition between the two types of pattern may be rationalized by a linear optimal transient growth analysis and nonlinear numerical simulations.
The regular reflection→Mach reflection transition in unsteady flow over convex surfaces
- M. Geva, O. Ram, O. Sadot
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- Published online by Cambridge University Press:
- 19 December 2017, pp. 48-79
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The non-stationary transition from regular reflection (RR) to Mach reflection (MR) over convex segments has been the focus of many recent studies. Until recently, the problem was thought to be very complicated because it was believed that many parameters such as the radius of curvature, initial angle and geometrical shape of the reflecting surface influenced this process. In this study, experiments and inviscid numerical computations were performed in air ($\unicode[STIX]{x1D6FE}=1.4$) at an incident shock-wave Mach number of 1.3. The incident shock waves were reflected over cylindrical and elliptical convex surfaces. The computations were validated by high-resolution experiments, which enabled the detection of features in the flow having characteristic lengths as small as 0.06 mm. Therefore, the RR →MR transition and Mach stem growth were successfully validated in the early stages of the Mach stem formation and closer to the surface than ever before. The evolution of the RR, the transition to MR and the Mach stem growth were found to depend only on the radius of the reflecting surface. The reflected shock wave adjusts itself to the changing angles of the reflecting surface. This feature, which was demonstrated at Mach numbers 1.3 and 1.5, distinguishes the unsteady case from the self-similar pseudo-steady case and requires the formulation of the conservation equations. A modification of the standard two-shock theory (2ST) is presented to predict the flow properties behind a shock wave that propagates over convex surfaces. Until recently, the determination of the time-dependent flow properties was possible solely by numerical computations. Moreover, this derivation explains the controversial issue on the delay in the transition from the RR to the MR that was observed by many researchers. It turns out that the entire RR evolution and the particular moment of transition to MR, are based on the essential ‘no-penetration’ condition of the flow. Therefore, we proposed a simple geometrical criterion for the RR →MR transition.
Predicting viscous-range velocity gradient dynamics in large-eddy simulations of turbulence
- Perry L. Johnson, Charles Meneveau
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- 20 December 2017, pp. 80-114
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The detailed dynamics of small-scale turbulence are not directly accessible in large-eddy simulations (LES), posing a modelling challenge, because many micro-physical processes such as deformation of aggregates, drops, bubbles and polymers dynamics depend strongly on the velocity gradient tensor, which is dominated by the turbulence structure in the viscous range. In this paper, we introduce a method for coupling existing stochastic models for the Lagrangian evolution of the velocity gradient tensor with coarse-grained fluid simulations to recover small-scale physics without resorting to direct numerical simulations (DNS). The proposed approach is implemented in LES of turbulent channel flow and detailed comparisons with DNS are carried out. An application to modelling the fate of deformable, small (sub-Kolmogorov) droplets at negligible Stokes number and low volume fraction with one-way coupling is carried out and results are again compared to DNS results. Results illustrate the ability of the proposed model to predict the influence of small-scale turbulence on droplet micro-physics in the context of LES.
Gel-controlled droplet spreading
- M. Jalaal, C. Seyfert, B. Stoeber, N. J. Balmforth
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- 19 December 2017, pp. 115-128
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Spreading and stationary droplets of a thermally responsive fluid on a heated surface are studied. The fluid undergoes a reversible gel formation at elevated temperature. The spatio-temporal pattern of gel formation within the droplet is examined using an experimental method based on spectral domain optical coherence tomography and time varying speckle patterns. Two stages of gel formation can be distinguished: first, a thin crust appears starting at the contact line. Second, a gel layer appears above the heated plate and then expands upward. We attribute the first stage of gel formation to solvent evaporation and heating through the air and the second to thermal conduction through the fluid from the base. Gel formation at the contact line is likely responsible for the arrest of spreading droplets, but was not detectable with our experimental protocol at the time of contact line arrest, suggesting that this arose over a microscopic length scale. Overall, substrate heating provides an effective way to control the final shape of droplets of thermo-responsive fluids.
Vortex-ring-induced stratified mixing: mixing model
- Jason Olsthoorn, Stuart B. Dalziel
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- 20 December 2017, pp. 129-146
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The study of vortex-ring-induced mixing has been significant for understanding stratified turbulent mixing in the absence of a mean flow. Renewed interest in this topic has prompted the development of a one-dimensional model for the evolution of a stratified system in the context of isolated mixing events. This model is compared to numerical simulations and physical experiments of vortex rings interacting with a stratification. Qualitative agreement between the evolution of the density profiles is observed, along with close quantitative agreement of the mixing efficiency. This model highlights the key dynamical features of such isolated mixing events.
Formation, evolution and scaling of plasma synthetic jets
- Haohua Zong, Marios Kotsonis
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- 20 December 2017, pp. 147-181
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Plasma synthetic jet actuators (PSJAs), capable of producing high-velocity pulsed jets at high frequency, are well suited for high-Reynolds-number subsonic and supersonic flow control. The effects of energy deposition and actuation frequency on the formation and evolution characteristics of plasma synthetic jets (PSJs) are investigated in detail by high-speed phase-locked particle imaging velocimetry (PIV). Increasing jet intensity with energy deposition is mainly contributed by the increasing peak jet velocity ($U_{p}$), while decreasing jet intensity with actuation frequency is attributed to both the reduced cavity density (primary factor) and the shortened jet duration (secondary factor). The total energy efficiency of the considered PSJA ($O(0.01\,\%)$) reduces monotonically with increasing frequency, while the time-averaged thrust produced by the PSJA is positively proportional to both the deposition energy and the frequency. A simplified theoretical model is derived and reveals a scaling power law between the peak jet velocity and the non-dimensional deposition energy (exponent $1/3$). The propagation velocity of the vortex ring attached at the jet front shows a non-monotonic behaviour of initial sharp increase and subsequent mild decay. The peak values for both the propagation velocity and the circulation of the front vortex ring are reached approximately two exit diameters away from the exit. Finally, analysis of the time-averaged flow fields of the issuing PSJ indicates that the axial decay rate of the centreline velocity is proportional to the actuation frequency whereas it is invariant with the energy deposition. The jet spreading rate of the PSJ is found to be higher than steady jets but lower than piezoelectric synthetic jets. Similarly, the entrainment coefficients of the PSJ are found to be twice as high as the values for comparable steady jets.
Numerical investigation of the saturation process in an incompressible cavity flow
- N. Vinha, F. Meseguer-Garrido, J. de Vicente, E. Valero
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- 20 December 2017, pp. 182-209
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A numerical study of the saturation process inside a rectangular open cavity is presented. Previous experiments and linear stability analysis of the problem completely described the flow in its onset, as well as in a saturated regime, characterized by three-dimensional centrifugal modes. The morphology of the modes found in the experiments matched the ones predicted by linear analysis, but with a shift in frequencies for the oscillating modes. A three-dimensional incompressible direct numerical simulation (DNS) is employed for a detailed investigation of the saturation process inside a cavity with dimensions similar to the one used in the experiments, to further explain the behaviour of these modes. In this work, periodic boundary conditions are first imposed to better understand the effect of the saturation process far from the walls. Then, the effects of spanwise solid wall boundary conditions are investigated with a DNS reproducing the full dynamics of the experiments. The main flow structures are identified using the dynamic mode decomposition technique and compared with previous experimental and linear stability analysis results. The main reason for the aforementioned shift in frequency is explained in this paper, as it is a function of the velocity of the main recirculating vortex.
A semi-infinite hydraulic fracture with leak-off driven by a power-law fluid
- E. V. Dontsov, O. Kresse
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- 20 December 2017, pp. 210-229
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This study investigates the problem of a semi-infinite hydraulic fracture that propagates steadily in a permeable formation. The fracturing fluid rheology is assumed to follow a power-law behaviour, while the leak-off is modelled by Carter’s model. A non-singular formulation is employed to effectively analyse the problem and to construct a numerical solution. The problem under consideration features three limiting analytic solutions that are associated with dominance of either toughness, leak-off or viscosity. Transitions between all the limiting cases are analysed and the boundaries of applicability of all these limiting solutions are quantified. These bounds allow us to determine the regions in the parametric space, in which these limiting solutions can be used. The problem of a semi-infinite fracture, which is considered in this study, provides the solution for the tip region of a hydraulic fracture and can be used in hydraulic fracturing simulators to facilitate solving the moving fracture boundary problem. To cater for such applications, for which rapid evaluation of the solution is necessary, the last part of this paper constructs an approximate closed form solution for the problem and evaluates its accuracy against the numerical solution inside the parametric space.
The diffusive sheet method for scalar mixing
- D. Martínez-Ruiz, P. Meunier, B. Favier, L. Duchemin, E. Villermaux
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- 20 December 2017, pp. 230-257
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The diffusive strip method (DSM) is a near-exact numerical method for mixing computations initially developed in two dimensions (Meunier & Villermaux, J. Fluid Mech., vol. 662, 2010, pp. 134–172). The method, which consists of following stretched material lines to compute the resulting scalar field a posteriori, is extended here to three-dimensional flows. We describe the procedure and its three-dimensional peculiarity, which relies on the Lagrangian advection of a triangulated surface from which the stretching rate is extracted to infer the scalar field. The method is first validated at moderate Péclet number against a classical pseudospectral method solving the advection–diffusion equation for a Batchelor vortex, and then applied to a simple Taylor–Couette experimental configuration with non-rotating boundary conditions at the top-end disk, bottom-end disk and outer cylinder. This motion, producing an elaborate although controlled steady three-dimensional flow, relies on Ekman pumping arising from the rotation of the inner cylinder. A recurrent two-cell structure is separated by the horizontal mid-plane and formed by stream tubes shaped as nested tori under laminar flow conditions. A scalar blob in the flow experiences a Lagrangian oscillating dynamics undergoing stretchings and compressions, driving the mixing process. The DSM enables the calculation of the blob elongation and scalar concentration distributions through a single variable computation along the advected blob surface, capturing the rich evolution observed in the experiments. Interestingly, the mixing process in this axisymmetric and steady three-dimensional flow leads to a linear growth of surfaces in time similar to the one obtained in a two-dimensional shear. The potentialities, limits and extension of the method to more general flows are finally discussed.
Vortex-induced vibration of a rotating sphere
- A. Sareen, J. Zhao, D. Lo Jacono, J. Sheridan, K. Hourigan, M. C. Thompson
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- 20 December 2017, pp. 258-292
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Vortex-induced vibration (VIV) of a sphere represents one of the most generic fundamental fluid–structure interaction problems. Since vortex-induced vibration can lead to structural failure, numerous studies have focused on understanding the underlying principles of VIV and its suppression. This paper reports on an experimental investigation of the effect of imposed axial rotation on the dynamics of vortex-induced vibration of a sphere that is free to oscillate in the cross-flow direction, by employing simultaneous displacement and force measurements. The VIV response was investigated over a wide range of reduced velocities (i.e. velocity normalised by the natural frequency of the system): $3\leqslant U^{\ast }\leqslant 18$, corresponding to a Reynolds number range of $5000<Re<30\,000$, while the rotation ratio, defined as the ratio between the sphere surface and inflow speeds, $\unicode[STIX]{x1D6FC}=|\unicode[STIX]{x1D714}|D/(2U)$, was varied in increments over the range of $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 7.5$. It is found that the vibration amplitude exhibits a typical inverted bell-shaped variation with reduced velocity, similar to the classic VIV response for a non-rotating sphere but without the higher reduced velocity response tail. The vibration amplitude decreases monotonically and gradually as the imposed transverse rotation rate is increased up to $\unicode[STIX]{x1D6FC}=6$, beyond which the body vibration is significantly reduced. The synchronisation regime, defined as the reduced velocity range where large vibrations close to the natural frequency are observed, also becomes narrower as $\unicode[STIX]{x1D6FC}$ is increased, with the peak saturation amplitude observed at progressively lower reduced velocities. In addition, for small rotation rates, the peak amplitude decreases almost linearly with $\unicode[STIX]{x1D6FC}$. The imposed rotation not only reduces vibration amplitudes, but also makes the body vibrations less periodic. The frequency spectra revealed the occurrence of a broadband spectrum with an increase in the imposed rotation rate. Recurrence analysis of the structural vibration response demonstrated a transition from periodic to chaotic in a modified recurrence map complementing the appearance of broadband spectra at the onset of bifurcation. Despite considerable changes in flow structure, the vortex phase ($\unicode[STIX]{x1D719}_{vortex}$), defined as the phase between the vortex force and the body displacement, follows the same pattern as for the non-rotating case, with the $\unicode[STIX]{x1D719}_{vortex}$ increasing gradually from low values in Mode I of the sphere vibration to almost $180^{\circ }$ as the system undergoes a continuous transition to Mode II of the sphere vibration at higher reduced velocity. The total phase ($\unicode[STIX]{x1D719}_{total}$), defined as the phase between the transverse lift force and the body displacement, only increases from low values after the peak amplitude response in Mode II has been reached. It reaches its maximum value (${\sim}165^{\circ }$) close to the transition from the Mode II upper plateau to the lower plateau, reminiscent of the behaviour seen for the upper to lower branch transition for cylinder VIV. Hydrogen-bubble visualisations and particle image velocimetry (PIV) performed in the equatorial plane provided further insights into the flow dynamics near the sphere surface. The mean wake is found to be deflected towards the advancing side of the sphere, associated with an increase in the Magnus force. For higher rotation ratios, the near-wake rear recirculation zone is absent and the flow is highly vectored from the retreating side to the advancing side, giving rise to large-scale shedding. For a very high rotation ratio of $\unicode[STIX]{x1D6FC}=6$, for which vibrations are found to be suppressed, a one-sided large-scale shedding pattern is observed, similar to the shear-layer instability one-sided shedding observed previously for a rigidly mounted rotating sphere.
What is the final size of turbulent mixing zones driven by the Faraday instability?
- B.-J. Gréa, A. Ebo Adou
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- 21 December 2017, pp. 293-319
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Miscible fluids of different densities subjected to strong time-periodic accelerations normal to their interface can mix due to Faraday instability effects. Turbulent fluctuations generated by this mechanism lead to the emergence and the growth of a mixing layer. Its enlargement is gradually slowed down as the resonance conditions driving the instability cease to be fulfilled. The final state corresponds to a saturated mixing zone in which the turbulence intensity progressively decays. A new formalism based on second-order correlation spectra for the turbulent quantities is introduced for this problem. This method allows for the prediction of the final mixing zone size and extends results from classical stability analysis limited to weakly nonlinear regimes. We perform at various forcing frequencies and amplitudes a large set of homogeneous and inhomogeneous numerical simulations, extensively exploring the influence of initial conditions. The mixing zone widths, measured at the end of the simulations, are satisfactorily compared to the predictions, and bring a strong support to the proposed theory. The flow dynamics is also studied and reveals the presence of sub-harmonic as well as harmonic modes depending on the initial parameters in the Mathieu phase diagram. Important changes in the flow anisotropy, corresponding to the large scale structures of turbulence, occur. This phenomenon appears directly related to the orientation of the most amplified gravity waves excited in the system, evolving due to the enlargement of the mixing zone.
Transport of anisotropic particles under waves
- Michelle H. DiBenedetto, Nicholas T. Ouellette, Jeffrey R. Koseff
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- 21 December 2017, pp. 320-340
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Using a numerical model, we analyse the effects of shape on both the orientation and transport of anisotropic particles in wavy flows. The particles are idealized as prolate and oblate spheroids, and we consider the regime of small Stokes and particle Reynolds numbers. We find that the particles preferentially align into the shear plane with a mean orientation that is solely a function of their aspect ratio. This alignment, however, differs from the Jeffery orbits that occur in the residual shear flow (that is, the Stokes drift velocity field) in the absence of waves. Since the drag on an anisotropic particle depends on its alignment with the flow, this preferred orientation determines the effective drag on the particles, which in turn impacts their net downstream transport. We also find that the rate of alignment of the particles is not constant and depends strongly on their initial orientation; thus, variations in initial particle orientation result in dispersion of anisotropic-particle plumes. We show that this dispersion is a function of the particle’s eccentricity and the ratio of the settling and wave time scales. Due to this preferential alignment, we find that a plume of anisotropic particles in waves is on average transported farther but dispersed less than it would be if the particles were randomly oriented. Our results demonstrate that accurate prediction of the transport of anisotropic particles in wavy environments, such as microplastic particles in the ocean, requires the consideration of these preferential alignment effects.
Frontogenesis and frontal arrest of a dense filament in the oceanic surface boundary layer
- Peter P. Sullivan, James C. McWilliams
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- 21 December 2017, pp. 341-380
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The evolution of upper ocean currents involves a set of complex, poorly understood interactions between submesoscale turbulence (e.g. density fronts and filaments and coherent vortices) and smaller-scale boundary-layer turbulence. Here we simulate the lifecycle of a cold (dense) filament undergoing frontogenesis in the presence of turbulence generated by surface stress and/or buoyancy loss. This phenomenon is examined in large-eddy simulations with resolved turbulent motions in large horizontal domains using ${\sim}10^{10}$ grid points. Steady winds are oriented in directions perpendicular or parallel to the filament axis. Due to turbulent vertical momentum mixing, cold filaments generate a potent two-celled secondary circulation in the cross-filament plane that is frontogenetic, sharpens the cross-filament buoyancy and horizontal velocity gradients and blocks Ekman buoyancy flux across the cold filament core towards the warm filament edge. Within less than a day, the frontogenesis is arrested at a small width, ${\approx}100~\text{m}$, primarily by an enhancement of the turbulence through a small submesoscale, horizontal shear instability of the sharpened filament, followed by a subsequent slow decay of the filament by further turbulent mixing. The boundary-layer turbulence is inhomogeneous and non-stationary in relation to the evolving submesoscale currents and density stratification. The occurrence of frontogenesis and arrest are qualitatively similar with varying stress direction or with convective cooling, but the detailed evolution and flow structure differ among the cases. Thus submesoscale filament frontogenesis caused by boundary-layer turbulence, frontal arrest by frontal instability and frontal decay by forward energy cascade, and turbulent mixing are generic processes in the upper ocean.
A hybrid lattice Boltzmann and finite difference method for droplet dynamics with insoluble surfactants
- Haihu Liu, Yan Ba, Lei Wu, Zhen Li, Guang Xi, Yonghao Zhang
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- 21 December 2017, pp. 381-412
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Droplet dynamics in microfluidic applications is significantly influenced by surfactants. It remains a research challenge to model and simulate droplet behaviour including deformation, breakup and coalescence, especially in the confined microfluidic environment. Here, we propose a hybrid method to simulate interfacial flows with insoluble surfactants. The immiscible two-phase flow is solved by an improved lattice Boltzmann colour-gradient model which incorporates a Marangoni stress resulting from non-uniform interfacial tension, while the convection–diffusion equation which describes the evolution of surfactant concentration in the entire fluid domain is solved by a finite difference method. The lattice Boltzmann and finite difference simulations are coupled through an equation of state, which describes how surfactant concentration influences interfacial tension. Our method is first validated for the surfactant-laden droplet deformation in a three-dimensional (3D) extensional flow and a 2D shear flow, and then applied to investigate the effect of surfactants on droplet dynamics in a 3D shear flow. Numerical results show that, at low capillary numbers, surfactants increase droplet deformation, due to reduced interfacial tension by the average surfactant concentration, and non-uniform effects from non-uniform capillary pressure and Marangoni stresses. The role of surfactants on the critical capillary number ($Ca_{cr}$) of droplet breakup is investigated for various confinements (defined as the ratio of droplet diameter to wall separation) and Reynolds numbers. For clean droplets, $Ca_{cr}$ first decreases and then increases with confinement, and the minimum value of $Ca_{cr}$ is reached at a confinement of 0.5; for surfactant-laden droplets, $Ca_{cr}$ exhibits the same variation in trend for confinements lower than 0.7, but, for higher confinements, $Ca_{cr}$ is almost a constant. The presence of surfactants decreases $Ca_{cr}$ for each confinement, and the decrease is also attributed to the reduction in average interfacial tension and non-uniform effects, which are found to prevent droplet breakup at low confinements but promote breakup at high confinements. In either clean or surfactant-laden cases, $Ca_{cr}$ first remains almost unchanged and then decreases with increasing Reynolds number, and a higher confinement or Reynolds number favours ternary breakup. Finally, we study the collision of two equal-sized droplets in a shear flow in both surfactant-free and surfactant-contaminated systems with the same effective capillary numbers. It is identified that the non-uniform effects in the near-contact interfacial region immobilize the interfaces when two droplets are approaching each other and thus inhibit their coalescence.
Gas slip flow in a fracture: local Reynolds equation and upscaled macroscopic model
- Tony Zaouter, Didier Lasseux, Marc Prat
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- 21 December 2017, pp. 413-442
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The slightly compressible flow of a gas in the slip regime within a rough fracture featuring a heterogeneous aperture field is analysed in depth in this work. Starting from the governing Navier–Stokes, continuity and gas state law equations together with a first-order slip boundary condition at the impermeable walls of the fracture, the two-dimensional slip-corrected Reynolds model is first derived, which is shown to be second-order-accurate in the local slope of the roughness asperities while being first-order-accurate in the Knudsen number. Focusing the interest on the flow-rate to pressure-gradient relationship over a representative element of the fracture, an upscaling procedure is applied to the local Reynolds equation using the method of volume averaging, providing a macroscopic model for which the momentum conservation equation has a Reynolds-like form. The effective macroscopic transmissivity tensor, which is characteristic of the representative element, is shown to be given by a closure problem that is non-intrinsic to the geometrical structure of the fracture only due to the slip effect. An expansion to the first order in the Knudsen number is carried out on the closure, yielding a decomposition of the effective transmissivity tensor into its purely viscous part and its slip correction, both being given by the solution of intrinsic closure subproblems. Numerical validations of the solution to the closure problem are performed with analytical predictions for simple fracture geometries. Comparison between the macroscopic transmissivity tensor, obtained from the solution of the closure problem, and its first-order approximation is illustrated on a randomly rough correlated Gaussian fracture.
Thermal transfer in Rayleigh–Bénard cell with smooth or rough boundaries
- E. Rusaouën, O. Liot, B. Castaing, J. Salort, F. Chillà
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- 28 December 2017, pp. 443-460
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Several Rayleigh–Bénard experiments in water are performed with smooth or rough boundaries. We present new thermal transfer measurements obtained with large roughness elements arranged in a square lattice. The data are compared to previous data obtained with smaller elements in the same cell (Tisserand et al., Phys. Fluids, vol. 23, 2011). Experiments in the same apparatus without roughness are presented, as reference results, to allow for comparison. In the rough case, several regimes of heat transfer are identified: one similar to the smooth case, an enhanced heat transfer regime characterized by a modification of the Nusselt versus Rayleigh number relation and a third part where the relation can be similar to a smooth one with a corrected prefactor.
Coupling performance of tandem flexible inverted flags in a uniform flow
- Haibo Huang, Heng Wei, Xi-Yun Lu
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- 28 December 2017, pp. 461-476
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The interaction of tandem inverted flexible flags in a uniform flow is investigated. For the inverted flags, their ends are fixed with their heads freely flapping. A direct numerical simulation is performed for which the Reynolds number is of order 200. Large flapping amplitude as well as large drag force is preferred because more energy may be harvested if more bending energy is generated. For the simple case of two tandem inverted flags, the drag force and flapping amplitude of the rear flag are found to be smaller than those of an isolated inverted flag due to the destructive merging mode of vortices. However, it is still unknown whether more bending energy can be generated when coupled inverted flags are arranged properly. To explore the possibility, inverted flags are proposed to be arranged as two rows, which indicate two lines of inverted flags perpendicular to the direction of the incoming flow, and flags in the front and rear rows are in-line or staggered. First the results for infinite flags with periodic boundary condition are presented. In both the in-line and the staggered arrangements, due to the interactions between the front–rear flags, the flapping amplitude or the maximum bending deformation and bending energy of a flag in the rear row can be enhanced, which may be significantly higher than those of an isolated case. Meanwhile, the bending energy of a flag in the front row is close to that of an isolated case. Second, results for finite inverted flag groups show that antiphase synchronization is preferred. When the group number is large enough, the bending energies of the front and rear flags in the inner groups are close to those in the infinite case. This finding may be helpful for the designing of an efficient energy harvesting device using inverted flags.
Multiple states in turbulent plane Couette flow with spanwise rotation
- Zhenhua Xia, Yipeng Shi, Qingdong Cai, Minping Wan, Shiyi Chen
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- 28 December 2017, pp. 477-490
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Turbulence is ubiquitous in nature and engineering applications. Although Kolmogorov’s (C. R. Acad. Sci. URSS, vol. 30, 1941a, pp. 301–305; Dokl. Akad. Nauk URSS, vol. 30, 1941b, pp. 538–540) theory suggested a unique turbulent state for high Reynolds numbers, multiple states were reported for several flow problems, such as Rayleigh–Bénard convection and Taylor–Couette flows. In this paper, we report that multiple states also exist for turbulent plane Couette flow with spanwise rotation through direct numerical simulations at rotation number $Ro=0.2$ and Reynolds number $Re_{w}=1300$ based on the angular velocity in the spanwise direction and half of the wall velocity difference. With two different initial flow fields, our results show that the flow statistics, including the mean streamwise velocity and Reynolds stresses, show different profiles. These different flow statistics are closely related to the flow structures in the domain, where one state corresponds to two pairs of roll cells, and the other shows three pairs. The present result enriches the studies on multiple states in turbulence.
Solitary waves on falling liquid films in the inertia-dominated regime
- Fabian Denner, Alexandros Charogiannis, Marc Pradas, Christos N. Markides, Berend G. M. van Wachem, Serafim Kalliadasis
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- 04 January 2018, pp. 491-519
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We offer new insights and results on the hydrodynamics of solitary waves on inertia-dominated falling liquid films using a combination of experimental measurements, direct numerical simulations (DNS) and low-dimensional (LD) modelling. The DNS are shown to be in very good agreement with experimental measurements in terms of the main wave characteristics and velocity profiles over the entire range of investigated Reynolds numbers. And, surprisingly, the LD model is found to predict accurately the film height even for inertia-dominated films with high Reynolds numbers. Based on a detailed analysis of the flow field within the liquid film, the hydrodynamic mechanism responsible for a constant, or even reducing, maximum film height when the Reynolds number increases above a critical value is identified, and reasons why no flow reversal is observed underneath the wave trough above a critical Reynolds number are proposed. The saturation of the maximum film height is shown to be linked to a reduced effective inertia acting on the solitary waves as a result of flow recirculation in the main wave hump and in the moving frame of reference. Nevertheless, the velocity profile at the crest of the solitary waves remains parabolic and self-similar even after the onset of flow recirculation. The upper limit of the Reynolds number with respect to flow reversal is primarily the result of steeper solitary waves at high Reynolds numbers, which leads to larger streamwise pressure gradients that counter flow reversal. Our results should be of interest in the optimisation of the heat and mass transport characteristics of falling liquid films and can also serve as a benchmark for future model development.