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The dynamics of flowing, concentrated suspensions of non-colloidal particles continues to surprise, despite decades of work and the widespread importance of suspension transport properties to industrial processes and natural phenomena. Blanc, Lemaire & Peters (J. Fluid Mech., 2014, vol. 746, R4) report a striking example. They probed the time-dependent dynamics of concentrated suspensions of rigid and neutrally buoyant spheres by simultaneously measuring the oscillatory rheology and the sedimentation rate of a falling ball. The sedimentation velocity of the ball through the suspension depends strongly on the frequency of oscillation, though the rheology was found to be independent of frequency. The results demonstrate the complexities of suspension flows and highlight opportunities for improving models by exploring suspension dynamics and rheology over a wide range of conditions, beyond steady and unidirectional ones.
A two-dimensional D-shaped cylinder and heaving foil were mounted in tandem and used to simulate the main body and tail, respectively, of a natural swimmer. Thrust/drag measurements of the force on the foil and particle image velocimetry measurements of the flow downstream of the swimming system were conducted at a Reynolds number of about $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}10^4$ in a water channel with a constant free stream current speed. Two main sets of measurements were conducted: one set with the swimming system locked at a fixed streamwise location in the water channel as the heave frequency of the foil was varied; the other set with the system freely swimming to a desired set-point position from different upstream and downstream locations. When the freely swimming system reached and maintained its set-point position, so that its swimming speed matched that of the current, the oscillation frequency of the heaving foil corresponded to a Strouhal number of 0.36. Phase portraits of the measured thrust/drag forces reveal limit cycle oscillations for all swimming cases studied, which suggests that self-regulation drives the selection of this Strouhal number. No coupling was observed between the vortices shed by the D-shaped cylinder and the self-selected frequency of the heaving foil during free swimming. An examination of the ratio of the phase-locking indices for the input heaving motion of the foil and the coupled fluidic thrust/drag response reveals that it approaches a value of 0.5 over time when the freely swimming system is released from rest and allowed to achieve steady-state cruising. The jet produced by the freely swimming foil was inclined at an angle of approximately $4^\circ $ with respect to the direction of the mean flow.
Self-propelled jumping upon drop coalescence has been observed on a variety of textured superhydrophobic surfaces, where the jumping motion follows the capillary–inertial velocity scaling as long as the drop radius is above a threshold. In this paper, we report an experimental study of the self-propelled jumping on a Leidenfrost surface, where the heated substrate gives rise to a vapour layer on which liquid drops float. For the coalescence of identical water drops, we have tested initial drop radii ranging from 20 to $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}500\ \mu \mathrm{m}$ , where the lower bound is related to the spontaneous takeoff of individual drops and the upper bound to gravitational effects. Regardless of the approaching velocity prior to coalescence, the measured jumping velocity is around 0.2 when scaled by the capillary–inertial velocity. This constant non-dimensional velocity holds for the experimentally accessible range of drop radii, and we have found no cutoff radius for the scaling, in contrast to prior experiments on textured superhydrophobic surfaces. The Leidenfrost experiments quantitatively agree with our numerical simulations of drop coalescence on a flat surface with a contact angle of 180°, suggesting that the cutoff is likely to be due to drop–surface interactions unique to the textured superhydrophobic surfaces.
Coalescing drops spontaneously jump out of plane on a variety of biological and synthetic superhydrophobic surfaces, with potential applications ranging from self-cleaning materials to self-sustained condensers. To investigate the mechanism of self-propelled jumping, we report three-dimensional phase-field simulations of two identical spherical drops coalescing on a flat surface with a contact angle of 180°. The numerical simulations capture the spontaneous jumping process, which follows the capillary–inertial scaling. The out-of-plane directionality is shown to result from the counter-action of the substrate to the impingement of the liquid bridge between the coalescing drops. A viscous cutoff to the capillary–inertial velocity scaling is identified when the Ohnesorge number of the initial drops is around 0.1, but the corresponding viscous cutoff radius is too small to be tested experimentally. Compared to experiments on both superhydrophobic and Leidenfrost surfaces, our simulations accurately predict the nearly constant jumping velocity of around 0.2 when scaled by the capillary–inertial velocity. By comparing the simulated drop coalescence processes with and without the substrate, we attribute this low non-dimensional velocity to the substrate intercepting only a small fraction of the expanding liquid bridge.
The long-wave behaviour of perfectly conducting liquid films flowing down a vertical fibre in a radial electric field was investigated by an asymptotic model. The validity of the asymptotic model was verified by the fully linearized problem, which showed that results were in good agreement in the long-wave region. The linear stability analysis indicated that, when the ratio (the radius of the outer cylindrical electrode over the radius of the liquid film) $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\beta <e$ , the electric field enhanced the long-wave instability; when $\beta >e$ , the electric field impeded the long-wave instability; when $\beta =e$ , the electric field did not affect the long-wave instability. The nonlinear evolution study of the asymptotic model compared well with the linear theory when $\beta <e$ . However, when $\beta =e$ , the nonlinear evolution study showed that the electric field enhanced the instability which may cause the interface to become singular. When $\beta >e$ , the nonlinear evolution studies showed that the influence of the electric field on the nonlinear behaviour of the interface was complex. The electric field either enhanced or impeded the interfacial instability. In addition, an interesting phenomenon was observed by the nonlinear evolution study that the electric field may cause an oscillation in the amplitude of permanent waves when $\beta \ge e$ . Further study on steady travelling waves was conducted to reveal the influence of electric field on the wave speed. Results showed that the electric field either increased or decreased the wave speed as well as the wave amplitude and flow rate. In some situations, the wave speed may increase/decrease while its amplitude decreased/increased as the strength of the external electric field increased.
The global linear stability of steady axisymmetric flow through a model fusiform aneurysm is studied numerically. The aneurysm is modelled as a Gaussian-shaped inflation on a vessel of circular cross-section. The fluid is assumed to be Newtonian, and the flow far upstream and downstream of the inflation is a Hagen–Poiseuille flow. The model aneurysm is characterized by a maximum height $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ and width $W$ , non-dimensionalized by the upstream vessel diameter, and the steady flow is characterized by the Reynolds number of the upstream flow. The base flow through the model aneurysms is determined for non-dimensional heights and widths in the physiologically relevant ranges $0.1 \leq H \leq 1.0$ and $0.25 \leq W \leq 2.0$ , and for Reynolds numbers up to 7000, corresponding to peak values recorded during pulsatile flows under physiological conditions. It is found that the base flow consists of a core of relatively fast-moving fluid, surrounded by a slowly recirculating fluid that fills the inflation; for larger values of the ratio $H/W$ , a secondary recirculation region is observed. The wall shear stress (WSS) in the inflation is vanishingly small compared to the WSS in the straight vessels. The global linear stability of the base flows is analysed by determining the eigenfrequencies of a modal representation of small-amplitude perturbations and by looking at the energy transfer between the base flow and the perturbations. Relatively shallow aneurysms (of relatively large width) become unstable by the lift-up mechanism and have a perturbation flow which is characterized by stationary, growing modes. More localized aneurysms (with relatively small width) become unstable at larger Reynolds numbers, presumably by an elliptic instability mechanism; in this case the perturbation flow is characterized by oscillatory modes.
Based on the method of moments, we derive a general theoretical expression for the time-dependent dispersion of an initial point concentration in steady and unsteady laminar flows through long straight channels of any constant cross-section. We retrieve and generalize previous case-specific theoretical results, and furthermore predict new phenomena. In particular, for the transient phase before the well-described steady Taylor–Aris limit is reached, we find anomalous diffusion with a dependence of the temporal scaling exponent on the initial release point, generalizing this finding in specific cases. During this transient we furthermore identify maxima in the values of the dispersion coefficient which exceed the Taylor–Aris value by amounts that depend on channel geometry, initial point release position, velocity profile and Péclet number. We show that these effects are caused by a difference in relaxation time of the first and second moments of the solute distribution and may be explained by advection-dominated dispersion powered by transverse diffusion in flows with local velocity gradients.
We study the shape and dynamics of cavities created by the explosion of firecrackers at the surface of a large pool of water. Without confinement, the explosion generates a hemispherical air cavity which grows, reaches a maximum size and collapses in a generic w-shape to form a final central jet. When a rigid open tube confines the firecracker, the explosion produces a cylindrical cavity that expands without ever escaping the free end of the tube. We discuss a potential flow model, which captures most of these features.
We consider the log-law layer of both smooth- and rough-wall boundary layers at large Reynolds number. A scaling theory is proposed for low-order structure functions (say $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n \leq 6$ ) in the range of scales $\eta \ll r \ll \delta $ , where $\eta $ is the Kolmogorov length and $\delta $ is the boundary layer thickness. This theory rests on the hypothesis that the turbulence in this intermediate range of scales depends only on the scale $r$ , the local dissipation rate and the shear velocity. Crucially, the structure of the turbulence is assumed to be independent of the distance from the wall, $y$ , except to the extent that $y$ sets the value of the local dissipation rate. A detailed comparison is made between the predictions of the theory and data taken from both smooth- and rough-wall boundary layers. The data support the hypothesis that it is the dissipation rate, and not $y$ , that controls the structure of the turbulence for this range of eddy sizes. Our findings provide the first unified scaling theory for both smooth- and rough-wall turbulence.
Gravitationally and viscously unstable miscible displacements in vertical Hele-Shaw cells are investigated via three-dimensional Navier–Stokes simulations. The velocity of the two-dimensional base-flow displacement fronts generally increases with the unfavourable viscosity contrast and the destabilizing density difference. Displacement fronts moving faster than the maximum velocity of the Poiseuille flow far downstream exhibit a single stagnation point in a moving reference frame, consistent with earlier observations for corresponding capillary tube flows. Gravitationally stable fronts, on the other hand, can move more slowly than the Poiseuille flow, resulting in more complex streamline patterns and the formation of a spike at the tip of the front, in line with earlier findings. A two-dimensional pinch-off governed by dispersion is observed some distance behind the displacement front. Three-dimensional simulations of viscously and gravitationally unstable vertical displacements show a strong vorticity quadrupole along the length of the finger, similar to recent observations for neutrally buoyant flows. This quadrupole results in an inner splitting instability of vertically propagating fingers. Even though the quadrupole’s strength increases for larger destabilizing density differences, the inner splitting is delayed due to the presence of a secondary, outer quadrupole which counteracts the inner one. For large unstable density differences, the formation of a secondary, downward-propagating front is observed, which is also characterized by inner and outer vorticity quadrupoles. This front develops an anchor-like shape as a result of the flow induced by these quadrupoles. Increased spanwise wavelengths of the initial perturbation are seen to result in the formation of the well-known tip-splitting instability. For suitable initial conditions, the inner and tip-splitting instabilities can be seen to develop side by side, affecting different regions of the flow field.
It is well known that water slowly issued vertically downward exhibits a hysteresis phenomenon. A jetting-to-dripping transition appearing upon a stepwise decrease in jet issue speed was used to identify the origin of the Plateau–Rayleigh unstable wave elements which disintegrate the jetting liquid. In the present laboratory experiment using a stainless steel nozzle of inner radius 1 mm and length 30 mm, the transition occurred at a dimensionless jet issue speed of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\sqrt{\rho U^{2}a_{0} /\sigma } =0.8,$ where $\rho $ and $\sigma $ respectively denote the density and surface tension coefficient of the liquid issued at the speed $U$ from the nozzle of radius $a_0$ . The jet length gradually shortened with an oscillation of considerably large amplitude and period. High-speed camera images show that this oscillation is caused by tip contraction capillary wave (TCCW) elements which are elongated by the gravitationally accelerating jet flow and become Plateau–Rayleigh unstable wave elements. The jet length increases while the jet tip experiences end-pinching and radiates TCCW elements upstream. Only those TCCW elements destabilized at appropriate locations can grow sufficiently to shorten the jet. Since the unstable wave elements produced nearer the nozzle exit have much smaller amplitude at the jet tip, the end-pinching becomes effective. Thus, these processes are repeatable and constitute a self-destabilizing loop. The observed jetting-to-dripping transition has nothing to do with the random nozzle disturbances which were believed to be the origin of the Plateau–Rayleigh unstable wave in conventional instability theories. It is also different from the feature conjectured from current absolute/convective instability analysis. The underling physics of the self-destabilizing loop are explored in detail by numerical simulations based on a one-dimensional model.
A theoretical study of linear global instability of incompressible flow over a rectangular spanwise-periodic open cavity in an unconfined domain is presented. Comparisons with the limited number of results available in the literature are shown. Subsequently, the parameter space is scanned in a systematic manner, varying Reynolds number, incoming boundary-layer thickness and length-to-depth aspect ratio. This permits documenting the neutral curves and leading eigenmode characteristics of this flow. Correlations constructed using the results obtained collapse all available theoretical data on the three-dimensional instabilities.
In this theoretical and numerical paper, we derive the adjoint equations for a thermo-acoustic system consisting of an infinite-rate chemistry diffusion flame coupled with duct acoustics. We then calculate the thermo-acoustic system’s linear global modes (i.e. the frequency/growth rate of oscillations, together with their mode shapes), and the global modes’ receptivity to species injection, sensitivity to base-state perturbations and structural sensitivity to advective-velocity perturbations. Some of these could be found by finite difference calculations but the adjoint analysis is computationally much cheaper. We then compare these with the Rayleigh index. The receptivity analysis shows the regions of the flame where open-loop injection of fuel or oxidizer will have the greatest influence on the thermo-acoustic oscillation. We find that the flame is most receptive at its tip. The base-state sensitivity analysis shows the influence of each parameter on the frequency/growth rate. We find that perturbations to the stoichiometric mixture fraction, the fuel slot width and the heat-release parameter have most influence, while perturbations to the Péclet number have the least influence for most of the operating points considered. These sensitivities oscillate, e.g. positive perturbations to the fuel slot width either stabilizes or destabilizes the system, depending on the operating point. This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. The structural-sensitivity analysis shows the influence of perturbations to the advective flame velocity. The regions of highest sensitivity are around the stoichiometric line close to the inlet, showing where velocity models need to be most accurate. This analysis can be extended to more accurate models and is a promising new tool for the analysis and control of thermo-acoustic oscillations.
Non-axisymmetric drops can significantly alter impact dynamics via rebound suppression when compared to axisymmetric drops. In this study, we focus on ellipsoidal drop impact on a non-wetting surface and investigate the effects of the geometric aspect ratio ( $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{AR}$ ) and the Weber number ( ${\mathit{We}}$ ) on the dynamics and outcomes of impacts, both experimentally and numerically. Non-axisymmetric spreading features are characterized by scrutinizing the maximal extensions along the $x$ -axis ( $D_{mx}$ ) and $y$ -axis ( $D_{my}$ ) with respect to ${AR}$ and ${\mathit{We}}$ . The ratio of the maximal extensions depends strongly on ${AR}$ , following our scaling relation $D_{mx}/D_{my} \sim {AR}^{1/2}$ . Experimental and numerical studies show that increasing ${AR}$ induces a high degree of axis switching during retraction, thereby resulting in the prevention of drop rebound, where axis switching denotes alternate expansion and contraction along the principal axes. We determine the transition between rebound and deposition (rebound suppression) over the ${AR}$ and ${\mathit{We}}$ domains and discuss the transition based on a non-axial distribution of the kinetic energy. The understanding of ellipsoidal drop impacts will potentially provide applications to surface patterning, cleaning, and cooling.
We present an experimental and numerical study of the upstream internal wavefield in a channel generated by constant density intrusions propagating into a linearly stratified ambient fluid during the initial phase of translation. Using synthetic schlieren imaging and two-dimensional direct numerical simulations, we quantify this wave motion within the ambient stratified fluid ahead of the advancing front. We show that the height of the neutral buoyancy surface in the ambient fluid determines the vertical modal response with the predominant waves being mode 2 for intrusions near the mid-depth of the channel and mode 1 waves being produced by intrusions nearer the top or bottom of the domain. All higher vertical modes travel slower than the intrusion and so do not appear upstream ahead of the intrusion front. We find the energy flux into this upstream wavefield to be approximately constant, and to be between 10 and 30 % of the rate of available potential energy transfer into the flow.
A combined approach using system identification and feed-forward control design has been applied to experimental laminar channel flow in an effort to reduce the naturally occurring disturbance level. A simple blowing/suction strategy was capable of reducing the standard deviation of the measured sensor signal by 45 %, which markedly exceeds previously obtained results under comparable conditions. A comparable reduction could be verified over a significant streamwise extent, implying an improvement over previous, more localized disturbance control. The technique is effective, flexible, and robust, and the obtained results encourage further explorations of experimental control of convection-dominated flows.
Direct numerical simulation of the gravity-induced settling of finite-size particles in triply periodic domains has been performed under dilute conditions. For a single solid-to-fluid-density ratio of 1.5 we have considered two values of the Galileo number corresponding to steady vertical motion ( $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ga}=121$ ) and to steady oblique motion ( $\mathit{Ga}=178$ ) in the case of one isolated sphere. For the multiparticle system we observe strong particle clustering only in the latter case. The geometry and time scales related to clustering are determined from Voronoï tessellation and particle-conditioned averaging. As a consequence of clustering, the average particle settling velocity is increased by 12 % as compared with the value of an isolated sphere; such a collective effect is not observed in the non-clustering case. By defining a local (instantaneous) fluid velocity average in the vicinity of the finite-size particles it is shown that the observed enhancement of the settling velocity is due to the fact that the downward fluid motion (with respect to the global average) which is induced in the cluster regions is preferentially sampled by the particles. It is further observed that the variance of the particle velocity is strongly enhanced in the clustering case. With the aid of a decomposition of the particle velocity it is shown that this increase is due to enhanced fluid velocity fluctuations (due to clustering) in the vicinity of the particles. Finally, we discuss a possible explanation for the observation of a critical Galileo number marking the onset of clustering under dilute conditions.
Strong interactions of shock waves with boundary layers lead to flow separations and enhanced heat transfer rates. When the approaching boundary layer is hypersonic and transitional the problem is particularly challenging and more reliable data is required in order to assess changes in the flow and the surface heat transfer, and to develop simplified models. The present contribution compares results for transitional interactions on a flat plate at Mach 6 from three different experimental facilities using the same instrumented plate insert. The facilities consist of a Ludwieg tube (RWG), an open-jet wind tunnel (H2K) and a high-enthalpy free-piston-driven reflected shock tunnel (HEG). The experimental measurements include shadowgraph and infrared thermography as well as heat transfer and pressure sensors. Direct numerical simulations (DNS) are carried out to compare with selected experimental flow conditions. The combined approach allows an assessment of the effects of unit Reynolds number, disturbance amplitude, shock impingement location and wall cooling. Measures of intermittency are proposed based on wall heat flux, allowing the peak Stanton number in the reattachment regime to be mapped over a range of intermittency states of the approaching boundary layer, with higher overshoots found for transitional interactions compared with fully turbulent interactions. The transition process is found to develop from second (Mack) mode instabilities superimposed on streamwise streaks.
In a low Morton number ( $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ ) regime, the stability of a single drop rising in an immiscible viscous liquid is experimentally and computationally examined for varying viscosity ratio $\eta $ (the viscosity of the drop divided by that of the suspending fluid) and varying Eötvös number ( $\mathit{Eo}$ ). Three-dimensional computations, rather than three-dimensional axisymmetric computations, are necessary since non-axisymmetric unstable drop behaviour is studied. The computations are performed using the sharp-interface coupled level-set and volume-of-fluid (CLSVOF) method in order to capture the deforming drop boundary. In the lower $\eta $ regimes, $\eta = 0.02 $ or 0.1, and when $\mathit{Eo}$ exceeds a critical threshold, it is observed that a rising drop exhibits nonlinear lateral/tilting motion. In the higher $\eta $ regimes, $\eta = 0.1$ , 1.94, 10 or 100, and when $\mathit{Eo}$ exceeds another critical threshold, it is found that a rising drop becomes unstable and breaks up into multiple drops. The type of breakup, either ‘dumbbell’, ‘intermediate’ or ‘toroidal’, depends intimately on $\eta $ and $\mathit{Eo}$ .
We present findings from an experimental investigation into the impact of solid cone-shaped bodies onto liquid pools. Using a variety of cone angles and liquid physical properties, we show that the ejecta formed during the impact exhibits self-similarity for all impact speeds for very low surface tension liquids, whilst for high-surface tension liquids similarity is only achieved at high impact speeds. We find that the ejecta tip can detach from the cone and that this phenomenon can be attributed to the air entrainment phenomenon. We analyse of a range of cone angles, including some ogive cones, and impact speeds in terms of the spatiotemporal evolution of the ejecta tip. Using superhydrophobic cones, we also examine the entry of cones which entrain an air layer.