Journal of Fluid Mechanics

Published by Cambridge University Press (CUP)
Online ISSN: 1469-7645
Publications
Article
We computationally investigate the unsteady pulsatile propagation of a finger of air through a liquid-filled cylindrical rigid tube using a combined boundary element method and lubrication theory approach. The flow-field is governed by the dimensionless parameters Ca(Q)(t) = Ca(M) + Ca(Omega) sin(Omegat) = muQ*(t*)/piR(2)gamma, Omega = muomegaR/gamma and A = 2Ca(Omega)/Omega. Here, Ca(Q)(t) consists of both mean (Ca(M)) and oscillatory (Ca(Omega)) components. It is shown that the behavior of this system is appropriately described by steady-state responses until the onset of reverse flow, wherein the system operates in the unsteady regime (Ca(Omega) > Ca(M)). When flows in this regime are considered, converging and diverging stagnation points move dynamically throughout the cycle and may temporarily separate from the interface at high Omega. We have also found that for Ca(Omega) < 10Ca(M) the bubble tip pressure drop DeltaP(tip) may be estimated accurately from the pressure measured downstream of the bubble tip when corrections for the pressure drop due to Poiseuille flow are applied. The normal stress gradient at the tube wall ( partial differentialtau(n)/ partial differentialz) is discussed in detail, as this is believed to be the primary factor in airway epithelial cell damage (Bilek et al 2003). In the unsteady regime we find that local film-thinning produces high partial differentialtau(n)/ partial differentialz at low Ca(Omega). Film thickening at moderate Ca(Omega) in the unsteady regime protects the tube wall from the large gradients near the bubble tip, therefore reducing partial differentialtau(n)/ partial differentialz. We find that the stress field is highly dynamic and exhibits intriguing spatial and temporal characteristics that may be of interest to our field of study, pulmonary airway reopening.
 
Article
We investigate numerically vortex-induced vibrations (VIV) of two identical two-dimensional elastically mounted cylinders in tandem in the proximity-wake interference regime at Reynolds number Re = 200 for systems having both one (transverse vibrations) and two (transverse and in-line) degrees of freedom (1-DOF and 2-DOF, respectively). For the 1-DOF system the computed results are in good qualitative agreement with available experiments at higher Reynolds numbers. Similar to these experiments our simulations reveal: (1) larger amplitudes of motion and a wider lock-in region for the tandem arrangement when compared with an isolated cylinder; (2) that at low reduced velocities the vibration amplitude of the front cylinder exceeds that of the rear cylinder; and (3) that above a threshold reduced velocity, large-amplitude VIV are excited for the rear cylinder with amplitudes significantly larger than those of the front cylinder. By analysing the simulated flow patterns we identify the VIV excitation mechanisms that lead to such complex responses and elucidate the near-wake vorticity dynamics and vortex-shedding modes excited in each case. We show that at low reduced velocities vortex shedding provides the initial excitation mechanism, which gives rise to a vertical separation between the two cylinders. When this vertical separation exceeds one cylinder diameter, however, a significant portion of the incoming flow is able to pass through the gap between the two cylinders and the gap-flow mechanism starts to dominate the VIV dynamics. The gap flow is able to periodically force either the top or the bottom shear layer of the front cylinder into the gap region, setting off a series of very complex vortex-to-vortex and vortex-to-cylinder interactions, which induces pressure gradients that result in a large oscillatory force in phase with the vortex shedding and lead to the experimentally observed larger vibration amplitudes. When the vortex shedding is the dominant mechanism the front cylinder vibration amplitude is larger than that of the rear cylinder. The reversing of this trend above a threshold reduced velocity is associated with the onset of the gap flow. The important role of the gap flow is further illustrated via a series of simulations for the 2-DOF system. We show that when the gap-flow mechanism is triggered, the 2-DOF system can develop and sustain large VIV amplitudes comparable to those observed in the corresponding (same reduced velocity) 1-DOF system. For sufficiently high reduced velocities, however, the two cylinders in the 2-DOF system approach each other, thus significantly reducing the size of the gap region. In such cases the gap flow is entirely eliminated, and the two cylinders vibrate together as a single body with vibration amplitudes up to 50% lower than the amplitudes of the corresponding 1-DOF in which the gap flow is active. Three-dimensional simulations are also carried out to examine the adequacy of two-dimensional simulations for describing the dynamic response of the tandem system at Re = 200. It is shown that even though the wake transitions to a weakly three-dimensional state when the gap flow is active, the three-dimensional modes are too weak to affect the dynamic response of the system, which is found to be identical to that obtained from the two-dimensional computations.
 
Article
The effects of vascular rarefaction (the loss of small arteries) on the circulation of blood are studied using a multiscale mathematical model that can predict blood flow and pressure in the systemic and pulmonary arteries. We augmented a model originally developed for the systemic arteries (Olufsen et al. 1998, 1999, 2000, 2004) to (a) predict flow and pressure in the pulmonary arteries, and (b) predict pressure propagation along the small arteries in the vascular beds. The systemic and pulmonary arteries are modelled as separate, bifurcating trees of compliant and tapering vessels. Each tree is divided into two parts representing the `large' and `small' arteries. Blood flow and pressure in the large arteries are predicted using a nonlinear cross-sectional area-averaged model for a Newtonian fluid in an elastic tube with inflow obtained from magnetic resonance measurements. Each terminal vessel within the network of the large arteries is coupled to a vascular bed of small `resistance' arteries, which are modelled as asymmetric structured trees with specified area and asymmetry ratios between the parent and daughter arteries. For the systemic circulation, each structured tree represents a specific vascular bed corresponding to major organs and limbs. For the pulmonary circulation, there are four vascular beds supplied by the interlobar arteries. This manuscript presents the first theoretical calculations of the propagation of the pressure and flow waves along systemic and pulmonary large and small arteries. Results for all networks were in agreement with published observations. Two studies were done with this model. First, we showed how rarefaction can be modelled by pruning the tree of arteries in the microvascular system. This was done by modulating parameters used for designing the structured trees. Results showed that rarefaction leads to increased mean and decreased pulse pressure in the large arteries. Second, we investigated the impact of decreasing vessel compliance in both large and small arteries. Results showed, that the effects of decreased compliance in the large arteries far outweigh the effects observed when decreasing the compliance of the small arteries. We further showed that a decrease of compliance in the large arteries results in pressure increases consistent with observations of isolated systolic hypertension, as occurs in ageing.
 
Article
Thrust performance and wake structure were investigated for a rigid rectangular panel pitching about its leading edge in a free stream. For Re(C) = O(10(4)), thrust coefficient was found to depend primarily on Strouhal number St and the aspect ratio of the panel AR. Propulsive efficiency was sensitive to aspect ratio only for AR less than 0.83; however, the magnitude of the peak efficiency of a given panel with variation in Strouhal number varied inversely with the amplitude to span ratio A/S, while the Strouhal number of optimum efficiency increased with increasing A/S. Peak efficiencies between 9 % and 21 % were measured. Wake structures corresponding to a subset of the thrust measurements were investigated using dye visualization and digital particle image velocimetry. In general, the wakes divided into two oblique jets; however, when operating at or near peak efficiency, the near wake in many cases represented a Kármán vortex street with the signs of the vortices reversed. The three-dimensional structure of the wakes was investigated in detail for AR = 0.54, A/S = 0.31 and Re(C) = 640. Three distinct wake structures were observed with variation in Strouhal number. For approximately 0.20 < St < 0.25, the main constituent of the wake was a horseshoe vortex shed by the tips and trailing edge of the panel. Streamwise variation in the circulation of the streamwise horseshoe legs was consistent with a spanwise shear layer bridging them. For St > 0.25, a reorganization of some of the spanwise vorticity yielded a bifurcating wake formed by trains of vortex rings connected to the tips of the horseshoes. For St > 0.5, an additional structure formed from a perturbation of the streamwise leg which caused a spanwise expansion. The wake model paradigm established here is robust with variation in Reynolds number and is consistent with structures observed for a wide variety of unsteady flows. Movies are available with the online version of the paper.
 
Article
Flow visualization is used to interrogate the wake structure produced by a rigid flat panel of aspect ratio (span/chord) 0.54 pitching in a free stream at a Strouhal number of 0.23. At such a low aspect ratio, the streamwise vorticity generated by the plate tends to dominate the formation of the wake. Nevertheless, the wake has the appearance of a three-dimensional von Kármán vortex street, as observed in a wide range of other experiments, and consists of horseshoe vortices of alternating sign shed twice per flapping cycle. The legs of each horseshoe interact with the two subsequent horseshoes in an opposite-sign, then like-sign interaction in which they become entrained. A detailed vortex skeleton model is proposed for the wake formation.
 
Article
Experiments are reported on the hydrodynamic performance of a flexible fin. The fin replicates some features of the pectoral fin of a batoid fish (such as a ray or skate) in that it is actuated in a travelling wave motion, with the amplitude of the motion increasing linearly along the span from root to tip. Thrust is found to increase with non-dimensional frequency, and an optimal oscillatory gait is identified. Power consumption measurements lead to the computation of propulsive efficiency, and an optimal efficiency condition is evaluated. Wake visualizations are presented, and a vortex model of the wake near zero net thrust is suggested. Strouhal number effects on the wake topology are also illustrated.
 
Article
'Bioconvection' is the name given to pattern-forming convective motions set up in suspensions of swimming micro-organisms. 'Gyrotaxis' describes the way the swimming is guided through a balance between the physical torques generated by viscous drag and by gravity operating on an asymmetric distribution of mass within the organism. When the organisms are heavier towards the rear, gyrotaxis turns them so that they swim towards regions of most rapid downflow. The presence of gyrotaxis means that bioconvective instability can develop from an initially uniform suspension, without an unstable density stratification. In this paper a continuum model for suspensions of gyrotactic micro-organisms is proposed and discussed; in particular, account is taken of the fact that the organisms of interest are non-spherical, so that their orientation is influenced by the strain rate in the ambient flow as well as the vorticity. This model is used to analyse the linear instability of a uniform suspension. It is shown that the suspension is unstable if the disturbance wavenumber is less than a critical value which, together with the wavenumber of the most rapidly growing disturbance, is calculated explicitly. The subsequent convection pattern is predicted to be three-dimensional (i.e. with variation in the vertical as well as the horizontal direction) if the cells are sufficiently elongated. Numerical results are given for suspensions of a particular algal species (Chlamydomonas nivalis); the predicted wavelength of the most rapidly growing disturbance is 5-6 times larger than the wavelength of steady-state patterns observed in experiments. The main reasons for the difference are probably that the analysis describes the onset of convection, not the final, nonlinear steady state, and that the experimental fluid layer has finite depth.
 
Article
Carefully timed tandem microbubbles have been shown to produce directional and targeted membrane poration of individual cells in microfluidic systems, which could be of use in ultrasound-mediated drug and gene delivery. This study aims at contributing to the understanding of the mechanisms at play in such an interaction. The dynamics of single and tandem microbubbles between two parallel plates is studied numerically and analytically. Comparisons are then made between the numerical results and the available experimental results. Numerically, assuming a potential flow, a three-dimensional boundary element method (BEM) is used to describe complex bubble deformations, jet formation, and bubble splitting. Analytically, compressibility and viscous boundary layer effects along the channel walls, neglected in the BEM model, are considered while shape of the bubble is not considered. Comparisons show that energy losses modify the bubble dynamics when the two approaches use identical initial conditions. The initial conditions in the boundary element method can be adjusted to recover the bubble period and maximum bubble volume when in an infinite medium. Using the same conditions enables the method to recover the full dynamics of single and tandem bubbles, including large deformations and fast re-entering jet formation. This method can be used as a design tool for future tandem-bubble sonoporation experiments.
 
Article
Fabrication of functionalized surfaces using polymer brushes is a relatively simple process and parallels the presence of glycocalyx filaments coating the luminal surface of our vasculature. In this paper, we perform atomistic-like simulations based on dissipative particle dynamics (DPD) to study both polymer brushes and glycocalyx filaments subject to shear flow, and we apply mean-field theory to extract useful scaling arguments on their response. For polymer brushes, a weak shear flow has no effect on the brush density profile or its height, while the slip length is independent of the shear rate and is of the order of the brush mesh size as a result of screening by hydrodynamic interactions. However, for strong shear flow, the polymer brush is penetrated deeper and is deformed, with a corresponding decrease of the brush height and an increase of the slip length. The transition from the weak to the strong shear regime can be described by a simple 'blob' argument, leading to the scaling γ̇0 ∝ σ(3/2), where γ̇0 is the critical transition shear rate and σ is the grafting density. Furthermore, in the strong shear regime, we observe a cyclic dynamic motion of individual polymers, causing a reversal in the direction of surface flow. To study the glycocalyx layer, we first assume a homogeneous flow that ignores the discrete effects of blood cells, and we simulate microchannel flows at different flow rates. Surprisingly, we find that, at low Reynolds number, the slip length decreases with the mean flow velocity, unlike the behaviour of polymer brushes, for which the slip length remains constant under similar conditions. (The slip length and brush height are measured with respect to polymer mesh size and polymer contour length, respectively.) We also performed additional DPD simulations of blood flow in a tube with walls having a glycocalyx layer and with the deformable red blood cells modelled accurately at the spectrin level. In this case, a plasma cell-free layer is formed, with thickness more than three times the glycocalyx layer. We then find our scaling arguments based on the homogeneous flow assumption to be valid for this physiologically correct case as well. Taken together, our findings point to the opposing roles of conformational entropy and bending rigidity - dominant effects for the brush and glycocalyx, respectively - which, in turn, lead to different flow characteristics, despite the apparent similarity of the two systems.
 
Article
A high-order accurate shock- and interface-capturing scheme is used to simulate the collapse of a gas bubble in water. In order to better understand the damage caused by collapsing bubbles, the dynamics of the shock-induced and Rayleigh collapse of a bubble near a planar rigid surface and in a free field are analysed. Collapse times, bubble displacements, interfacial velocities and surface pressures are quantified as a function of the pressure ratio driving the collapse and of the initial bubble stand-off distance from the wall; these quantities are compared to the available theory and experiments and show good agreement with the data for both the bubble dynamics and the propagation of the shock emitted upon the collapse. Non-spherical collapse involves the formation of a re-entrant jet directed towards the wall or in the direction of propagation of the incoming shock. In shock-induced collapse, very high jet velocities can be achieved, and the finite time for shock propagation through the bubble may be non-negligible compared to the collapse time for the pressure ratios of interest. Several types of shock waves are generated during the collapse, including precursor and water-hammer shocks that arise from the re-entrant jet formation and its impact upon the distal side of the bubble, respectively. The water-hammer shock can generate very high pressures on the wall, far exceeding those from the incident shock. The potential damage to the neighbouring surface is quantified by measuring the wall pressure. The range of stand-off distances and the surface area for which amplification of the incident shock due to bubble collapse occurs is determined.
 
Article
We present detailed results for the motion of a finite sized gas bubble in a blood vessel. The bubble (dispersed phase) size is taken to be such as to nearly occlude the vessel. The bulk medium is treated as a shear thinning Casson fluid and contains a soluble surfactant that adsorbs and desorbs from the interface. Three different vessel sizes, corresponding to a small artery, a large arteriole, and a small arteriole, in normal humans, are considered. The hematocrit (volume fraction of RBCs) has been taken to be 0.45. For arteriolar flow, where relevant, the Fahraeus-Lindqvist effect is taken into account. Bubble motion cause temporal and spatial gradients of shear stress at the cell surface lining the vessel wall as the bubble approaches the cell, moves over it and passes it by. Rapid reversals occur in the sign of the shear stress imparted to the cell surface during this motion. Shear stress gradients together with sign reversals are associated with a recirculation vortex at the rear of the moving bubble. The presence of the surfactant reduces the level of the shear stress gradients imparted to the cell surface as compared to an equivalent surfactant-free system. Our numerical results for bubble shapes and wall shear stresses may help explain phenomena observed in experimental studies related to gas embolism, a significant problem in cardiac surgery and decompression sickness.
 
Article
We investigate the influence of a soluble surfactant on the steady-state motion of a finger of air through a compliant channel. This study provides a basic model from which to understand the fluid–structure interactions and physicochemical hydrodynamics of pulmonary airway reopening. Airway closure occurs in lung diseases such as respiratory distress syndrome and acute respiratory distress syndrome as a result of fluid accumulation and surfactant insufficiency. This results in ‘compliant collapse’ with the airway walls buckled and held in apposition by a liquid occlusion that blocks the passage of air. Airway reopening is essential to the recovery of adequate ventilation, but has been associated with ventilator-induced lung injury because of the exposure of airway epithelial cells to large interfacial flow-induced pressure gradients. Surfactant replacement is helpful in modulating this deleterious mechanical stimulus, but is limited in its effectiveness owing to slow surfactant adsorption. We investigate the effect of surfactant on micro-scale models of reopening by computationally modelling the steady two-dimensional motion of a semi-infinite bubble propagating through a liquid-filled compliant channel doped with soluble surfactant. Many dimensionless parameters affect reopening, but we primarily investigate how the reopening pressure depends upon the capillary number (the ratio of viscous to surface tension forces), the adsorption depth parameter (a bulk concentration parameter) and the bulk Péclet number (the ratio of bulk convection to diffusion). These studies demonstrate a dependence of on , and suggest that a critical bulk concentration must be exceeded to operate as a low-surface-tension system. Normal and tangential stress gradients remain largely unaffected by physicochemical interactions – for this reason, further biological studies are suggested that will clarify the role of wall flexibility and surfactant on the protection of the lung from atelectrauma.
 
Article
The dynamic interaction of a shockwave (modelled as a pressure pulse) with an initially spherically oscillating bubble is investigated. Upon the shockwave impact, the bubble deforms non-spherically and the flow field surrounding the bubble is determined with potential flow theory using the boundary-element method (BEM). The primary advantage of this method is its computational efficiency. The simulation process is repeated until the two opposite sides of the bubble surface collide with each other (i.e. the formation of a jet along the shockwave propagation direction). The collapse time of the bubble, its shape and the velocity of the jet are calculated. Moreover, the impact pressure is estimated based on water-hammer pressure theory. The Kelvin impulse, kinetic energy and bubble displacement (all at the moment of jet impact) are also determined. Overall, the simulated results compare favourably with experimental observations of lithotripter shockwave interaction with single bubbles (using laser-induced bubbles at various oscillation stages). The simulations confirm the experimental observation that the most intense collapse, with the highest jet velocity and impact pressure, occurs for bubbles with intermediate size during the contraction phase when the collapse time of the bubble is approximately equal to the compressive pulse duration of the shock wave. Under this condition, the maximum amount of energy of the incident shockwave is transferred to the collapsing bubble. Further, the effect of the bubble contents (ideal gas with different initial pressures) and the initial conditions of the bubble (initially oscillating vs. non-oscillating) on the dynamics of the shockwave-bubble interaction are discussed.
 
Schematic diagram (not to scale) illustrating the mathematical problem of electromigration dispersion of a sample in CE. In addition to the sample ions shown, the capillary also contains a carrier electrolyte consisting of co-and counter-ions.
Time evolution of the normalized rate of increase of variance for three different values of the dimensionless electro-osmotic flow strength u * = ueo/v0. Horizontal dotted line is the effective diffusivity predicted by equation (35) in GC.  
Concentration profiles ¯ φ(x, t) at a fixed instant of time (v0t/w0 = 200) for several values of the electro-osmotic flow strength, u * = ueo/v0. The increased effective axial diffusivity due to Taylor dispersion " softens " the electromigration shock that tends to form at the leading edge. Here P = 50 and x = xc is the location of the centroid of the peak.  
Article
The differential migration of ions in an applied electric field is the basis for separation of chemical species by capillary electrophoresis. Axial diffusion of the concentration peak limits the separation efficiency. Electromigration dispersion is observed when the concentration of sample ions is comparable to that of the background ions. Under such conditions, the local electrical conductivity is significantly altered in the sample zone making the electric field, and therefore, the ion migration velocity concentration dependent. The resulting nonlinear wave exhibits shock like features, and, under certain simplifying assumptions, is described by Burgers' equation (S. Ghosal and Z. Chen Bull. Math. Biol. 201072, pg. 2047). In this paper, we consider the more general situation where the walls of the separation channel may have a non-zero zeta potential and are therefore able to sustain an electro-osmotic bulk flow. The main result is a one dimensional nonlinear advection diffusion equation for the area averaged concentration. This homogenized equation accounts for the Taylor-Aris dispersion resulting from the variation in the electro-osmotic slip velocity along the wall. It is shown that in a certain range of parameters, the electro-osmotic flow can actually reduce the total dispersion by delaying the formation of a concentration shock. However, if the electro-osmotic flow is sufficiently high, the total dispersion is increased because of the Taylor-Aris contribution.
 
Streamsurfaces of the potential function Y ( y, τ ) intersecting the vortex sheet boundary of the idealised jet. 
Free space radiation produced by throttled flow into the cavity of a Helmholtz resonator. The Fant equation is derived by consideration of the sound produced at an arbitrary point x within the body of the cavity. 
Numerical predictions of (4.2)-(4.4) for conditions (4.6) when p I = 0.5 kPa, typified by the non-special case where f o /f 1 = 0.6: (a) the glottal volume flux Q normalised by Q = 2p I A L /ρ o c o ; (b) the glottis area ratio A g /A L ; (c) the monopole, far-field acoustic pressure at distance r from the mouth. 
Dependence of the volume flux ratio Q/Q on glottal frequency f o near the cavity resonance frequency f 1 for the conditions of figure 4: (a) f o /f 1 = 1 (-), 0.95 (-), 0.997 (-), 1.003 (· · · · · · ); (b) f o /f 1 = 1 (-), 1.0003 (-). 
Throttled flow into the supraglottal tract modelled by a cylindrical tube of interior length L and cross-section A. The Fant equation is derived by consideration of the sound produced at an arbitrary point x within the body of the tract. 
Article
An analysis is made of the sound generated by the time-dependent throttling of a nominally steady stream of air through a small orifice into a flow-through resonant cavity. This is exemplified by the production of voiced speech, where air from the lungs enters the vocal tract through the glottis at a time variable volume flow rate Q(t) controlled by oscillations of the glottis cross-section. Voicing theory has hitherto determined Q from a heuristic, reduced complexity 'Fant' differential equation (G. Fant, Acoustic Theory of Speech Production, 1960). A new self-consistent, integro-differential form of this equation is derived in this paper using the theory of aerodynamic sound, with full account taken of the back-reaction of the resonant tract on the glottal flux Q. The theory involves an aeroacoustic Green's function (G) for flow-surface interactions in a time-dependent glottis, so making the problem non-self-adjoint. In complex problems of this type it is not usually possible to obtain G in an explicit analytic form. The principal objective of the paper is to show how the Fant equation can still be derived in such cases from a consideration of the equation of aerodynamic sound and from the adjoint of the equation governing G in the neighbourhood of the 'throttle'. The theory is illustrated by application to the canonical problem of throttled flow into a Helmholtz resonator.
 
Article
The inviscid instability of O(ε) two-dimensional free-surface gravity waves propagating along an O(1) parallel shear flow is considered. The modes of instability involve spanwise-periodic longitudinal vortices resembling oceanic Langmuir circulation. Here, not only are wave-induced mean effects important but also wave modulation, caused by velocity anomalies which develop in the streamwise direction. The former are described by a generalized Lagrangian-mean formulation and the latter by a modified Rayleigh equation. Since both effects are essential, the instability may be called 'generalized' Craik-Leibovich (CLg). Of specific interest is whether spanwise distortion of the wave field, both at the free surface and in the interior, acts to enhance or inhibit instability to longitudinal vortices. Also of interest is whether the instability gives rise to a preferred spacing for the vortices and whether that spacing concurs well or poorly with experiment. The layer depth is varied from much less than the e-folding depth of the O(ε) wave motion to infinity. Relative to an identical shear flow with rigid though wavy top boundary, it is found, inter alia, that wave modulation acts in concert with the free surface, at some wavenumbers, to increase the maximum growth rate of the instability. Indeed, two preferred spanwise spacings occur, one which gives rise to longitudinal vortices through a convective oscillatory bifurcation and a second, at higher wavenumber and growth rate, through a stationary bifurcation. The preferred spacings set by the stationary bifurcation concur well with those observed in laboratory experiments, with the implication that the instability acting in the experiments is very likely to be CLg.
 
Article
A new continuum model is formulated for dilute suspensions of swimming micro-organisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker-Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. In this paper we solve the Fokker-Planck equation exactly for ε = 0 (λ arbitrary) and also obtain the first-order solution for small ε. Using experimental data on Vc and D obtained with the swimming alga, Chlamydomonas nivalis, in the absence of bulk flow, the ε = 0 results can be used to estimate the value of λ for that species (λ ≈ 2.2; Dr ≈ 0.13 s-1).
 
Article
In small vessels blood is usually treated as a Newtonian fluid down to diameters of ~200 μm. We investigate the flow of red blood cell (RBC) suspensions driven through small tubes (diameters 10-150 μm) in the range marking the transition from arterioles and venules to the largest capillary vessels. The results of the simulations combined with previous simulations of uniform shear flow and experimental data show that for diameters less than ~100 μm the suspension's stress cannot be described as a continuum, even a heterogeneous one. We employ the dissipative particle dynamics (DPD) model, which has been successfully used to predict human blood bulk viscosity in homogeneous shear flow. In tube flow the cross-stream stress gradient induces an inhomogeneous distribution of RBCs featuring a centreline cell density peak, and a cell-free layer (CFL) next to the wall. For a neutrally buoyant suspension the imposed linear shear-stress distribution together with the differentiable velocity distribution allow the calculation of the local viscosity across the tube section. The viscosity across the section as a function of the strain rate is found to be essentially independent of tube size for the larger diameters and is determined by the local haematocrit (H) and shear rate. Other RBC properties such as asphericity, deformation, and cell-flow orientation exhibit similar dependence for the larger tube diameters. As the tube size decreases below ~100 μm in diameter, the viscosity in the central region departs from the large-tube similarity function of the shear rate, since H increases significantly towards the centreline. The dependence of shear stress on tube size, in addition to the expected local shear rate and local haematocrit, implies that blood flow in small tubes cannot be described as a heterogeneous continuum. Based on the analysis of the DPD simulations and on available experimental results, we propose a simple velocity-slip model that can be used in conjunction with continuum-based simulations.
 
Article
To understand the fluid dynamics of a biologically inspired unsteady low-aspect-ratio propulsor, unsteady pressure distributions were measured and compared with time-averaged thrust performance and wake visualizations. The experiments were performed on rigid rectangular panels with different aspect ratios, pitching in a uniform flow. Panel aspect ratio and pitching amplitude were shown to affect the magnitude and time dependence of the pressure distribution on the panel surface, the vorticity generation on the panel, and thrust production. A new scaling is proposed that includes these parameters and collapses the oscillating pressure magnitude and the thrust coefficient.
 
Article
Disease states characterized by airway fluid occlusion and pulmonary surfactant insufficiency, such as respiratory distress syndrome, have a high mortality rate. Understanding the mechanics of airway reopening, particularly involving surfactant transport, may provide an avenue to increase patient survival via optimized mechanical ventilation waveforms. We model the occluded airway as a liquid-filled rigid tube with the fluid phase displaced by a finger of air that propagates with both mean and sinusoidal velocity components. Finite-time Lyapunov exponent (FTLE) fields are employed to analyse the convective transport characteristics, taking note of Lagrangian coherent structures (LCSs) and their effects on transport. The Lagrangian perspective of these techniques reveals flow characteristics that are not readily apparent by observing Eulerian measures. These analysis techniques are applied to surfactant-free velocity fields determined computationally, with the boundary element method, and measured experimentally with micro particle image velocimetry (μ-PIV). We find that the LCS divides the fluid into two regimes, one advected upstream (into the thin residual film) and the other downstream ahead of the advancing bubble. At higher oscillatory frequencies particles originating immediately inside the LCS experience long residence times at the air-liquid interface, which may be conducive to surfactant transport. At high frequencies a well-mixed attractor region is identified; this volume of fluid cyclically travels along the interface and into the bulk fluid. The Lagrangian analysis is applied to velocity data measured with 0.01 mg ml(-1) of the clinical pulmonary surfactant Infasurf in the bulk fluid, demonstrating flow field modifications with respect to the surfactant-free system that were not visible in the Eulerian frame.
 
Article
New exact solutions of the Navier-Stokes equations are obtained for the unbounded and bounded oscillatory and impulsive tangential edgewise motion of touching half-infinite plates in their own plane. In contrast to Stokes classical solutions for the harmonic and impulsive motion of an infinite plane wall, where the solutions are separable or have a simple similarity form, the present solutions have a two-dimensional structure in the near region of the contact between the half-infinite plates. Nevertheless, it is possible to obtain relatively simple closed-form solutions for the flow field in each case by defining new variables which greatly simplify the r- and theta- dependence of the solutions in the vicinity of the contact region. These solutions for flow in a half-infinite space are then extended to bounded flows in a channel using an image superposition technique. The impulsive motion has application to the motion near geophysical faults, whereas the oscillatory motion has arisen in the design of a novel oscillating half-plate flow chamber for examining the effect of fluid shear stress on cultured cell monolayers.
 
Article
A two-dimensional model is used to simulate the motion and deformation of a single mammalian red blood cell (RBC) flowing close to the wall of a microvessel, taking into account the effects of a porous endothelial surface layer (ESL) lining the vessel wall. Migration of RBCs away from the wall leads to the formation of a cell-depleted layer near the wall, which has a large effect on the resistance to blood flow in microvessels. The objective is to examine the mechanical factors causing this migration, including the effects of the ESL. The vessel is represented as a straight parallel-sided channel. The RBC is represented as a set of interconnected viscoelastic elements, suspended in plasma, a Newtonian fluid. The ESL is represented as a porous medium, and plasma flow in the layer is computed using the Brinkman approximation. It is shown that an initially circular cell positioned close to the ESL in a shear flow is deformed into an asymmetric shape. This breaking of symmetry leads to migration away from the wall. With increasing hydraulic resistivity of the layer, the rate of lateral migration increases. It is concluded that mechanical interactions of RBCs flowing in microvessels with a porous wall layer may reduce the rate of lateral migration and hence reduce the width of the cell-depleted zone external to the ESL, relative to the cell-depleted zone that would be formed if the interface between the ESL and free-flowing plasma were replaced by an impermeable boundary.
 
The conditional Nusselt number NuSRS computed only from data taken while the system was in the SRS. The results were normalised by Nu and are shown as a function of the tilt angle. They are for Ra = 7.2 × 10 10. The dashed horizontal line marks NuSRS/Nu = 1.
The probability-density functions of the normalised LSC strength δ k /δ k for the top (bullets, blue online), the middle (diamonds, green online) and the bottom (squares, red online) thermistor row. Shown are data for (a) β = 0 and (b) β = 0.035 rad. The solid lines (green online) are Gaussian fits to the right side of the peak of δm/δm. The experiments were done at Ra = 7.2 × 10 10 .  
Article
We report measurements of properties of turbulent thermal convection of a fluid with a Prandtl number $\Pra=4.38$ in a cylindrical cell with an aspect ratio $\Gamma=0.50$. The rotational symmetry was broken by a small tilt of the sample axis relative to gravity. Measurements of the heat transport (as expressed by the Nusselt number \Nu), as well as of large-scale-circulation (LSC) properties by means of temperature measurements along the sidewall, are presented. In contradistinction to similar experiments using containers of aspect ratio $\Gamma=1.00$ \cite[]{ABN06} and $\Gamma=0.50$ \cite[]{CRCC04,SXX05,RGKS10}, we see a very small increase of the heat transport for tilt angles up to about 0.1 rad. Based on measurements of properties of the LSC we explain this increase by a stabilization of the single-roll state (SRS) of the LSC and a de-stabilization of the double-roll state (DRS) (it is known from previous work that the SRS has a slightly larger heat transport than the DRS). Further, we present quantitative measurements of the strength of the LSC, its orientation, and its torsional oscillation as a function of the tilt angle.
 
Article
We report on experimental determinations of the temperature field in the interior (bulk) of turbulent Rayleigh-Benard convection for a cylindrical sample with aspect ratio (diameter over height) of 0.50, both in the classical and in the ultimate state. The Prandtl number was close to 0.8. We find a "logarithmic layer" in which the temperature varies as A*ln(z/L) + B with the distance z from the bottom plate of the sample. The amplitude A varies with radial position r. In the classical state these results are in good agreement with direct numerical simulations (DNS); in the ultimate state there are as yet no DNS. A close analogy between the temperature field in the classical state and the "Law of the Wall" for the time-averaged down-stream velocity in shear flow is discussed.
 
Article
The mechanisms of sound generation in a Mach 0.9, Reynolds number 3600 turbulent jet are investigated by direct numerical simulation. Details of the numerical method are briefly outlined and results are validated against an experiment at the same flow conditions. Lighthill's theory is used to define a nominal acoustic source in the jet, and a numerical solution of Lighthill's equation is compared to the simulation to verify the computational procedures. The acoustic source is Fourier transformed in the axial coordinate and time and then filtered in order to identify and separate components capable of radiating to the far field. This procedure indicates that the peak radiating component of the source is coincident with neither the peak of the full unfiltered source nor that of the turbulent kinetic energy. The phase velocities of significant components range from approximately 5% to 50% of the ambient sound speed which calls into question the commonly made assumption that the noise sources convect at a single velocity. Space-time correlations demonstrate that the sources are not acoustically compact in the streamwise direction and that the portion of the source that radiates at angles greater than 45 deg. is stationary. Filtering non-radiating wavenumber components of the source at single frequencies reveals that a simple modulated wave forms for the source, as might be predicted by linear stability analysis. At small angles from the jet axis the noise from these modes is highly directional, better described by an exponential than a standard Doppler factor.
 
Dependence of (solid) the characteristic length scale Λ =  
Schematic illustration of the change of the structure of the local maximizers in the staggered arrangement of the vortex cells as P 0 increases (dashed lines represent the principal directions of stretching and compression).
Article
In this study we investigate the vortex structures which lead to the maximum possible growth of palinstrophy in two-dimensional incompressible flows on a periodic domain. It is shown that these questions are related to a broader research program concerning the problem of the finite-time singularity formation in the three-dimensional Navier-Stokes system. Such extreme vortex events are found systematically via numerical solution of suitable variational optimization problems. We identify several families of maximizing vortex states parameterized by their palinstrophy, palinstrophy and energy and palinstrophy and enstrophy. Evidence is shown that some of these families saturate estimates for the instantaneous rate of growth of palinstrophy obtained using rigorous methods of mathematical analysis, thereby demonstrating that this analysis is in fact sharp. In the limit of small palinstrophies the optimal vortex states are found analytically, whereas for large palinstrophies they exhibit a self-similar multipolar structure. It is also shown that the time evolution obtained using some families of the instantaneously optimal states as the initial conditions saturates the theoretical upper bound for the maximum growth of palinstrophy in finite time. Possible consequences of this finding for the study of extreme events in fluid flows are discussed.
 
Article
Kolmogorov flow in two dimensions - the two-dimensional Navier-Stokes equations with a sinusoidal body force - is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimicks the forcing at small forcing am- plitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn-Hilliard-type equation and as a result coarsening dynam- ics are observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially-localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dy- namics which is captured by a system of 5 (1-space and 1-time) PDEs based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions - a kink and antikink - which connect two steady one-dimensional flow regions of essen- tially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour.
 
Article
We explore the direct modification of the pseudo-spectral truncation of 2D, incompressible fluid dynamics to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the purpose of matching simulation statistics to given information, arising from observations, theoretical prediction or high fidelity simulation. In the scheme outlined here, Nos\'e-Hoover thermostats, commonly used in molecular dynamics, are introduced as feedback controls applied to energy shells of the Fourier-discretized Navier-Stokes equations. As we demonstrate in numerical experiments, the dynamical properties (quantified using autocorrelation functions) are only modestly perturbed by our device, while ensemble dispersion is significantly enhanced in comparison with simulations of a corresponding truncation incorporating hyperviscosity.
 
Article
High resolution direct numerical simulations of two-dimensional turbulence in stationary conditions are presented. The development of an energy-enstrophy double cascade is studied and found to be compatible with the classical Kraichnan theory in the limit of extended inertial ranges. The analysis of the joint distribution of energy and enstrophy fluxes in physical space reveals a small value of cross correlation. This result supports many experimental and numerical studies where only one cascade is generated.
 
Article
The torque in turbulent Taylor-Couette flows for shear Reynolds numbers Re_S up to 3x10^4 at various mean rotations is studied by means of direct numerical simulations for a radius ratio of \eta=0.71. Convergence of simulations is tested using three criteria of which the agreement of dissipation values estimated from the torque and from the volume dissipation rate turns out to be most demanding. We evaluate the influence of Taylor vortex heights on the torque for a stationary outer cylinder and select a value of the aspect ratio of \Gamma=2, close to the torque maximum. The connection between the torque and the transverse current J^\omega of azimuthal motion which can be computed from the velocity field enables us to investigate the local transport resulting in the torque. The typical spatial distribution of the individual convective and viscous contributions to the local current is analysed for a turbulent flow case. To characterise the turbulent statistics of the transport, PDF's of local current fluctuations are compared to experimental wall shear stress measurements. PDF's of instantaneous torques reveal a fluctuation enhancement in the outer region for strong counter-rotation. Moreover, we find for simulations realising the same shear Re_S>=2x10^4 the formation of a torque maximum for moderate counter-rotation with angular velocities \omega_o\approx-0.4\omega_i. In contrast, for Re_S<=4x10^3 the torque features a maximum for a stationary outer cylinder. In addition, the effective torque scaling exponent is shown to also depend on the mean rotation state. Finally, we evaluate a close connection between boundary-layer thicknesses and the torque.
 
Article
Any realistic disturbance in a flow field is of limited spatial extent. During the last 10-15 years considerable progress has been made in the understanding of the physics of locally excited disturbances in unstable flows. Theoretical, experimental and numerical work has contributed to this progress. First, the aim of the colloquium was to summarize the state of the art knowledge in the field of wave packet dynamics including the three-dimensional compressible, and non-linear theory and extensions to nonlinearity as well as experiments and numerical simulations. Further the colloquium was intended to show how the developed theory of localizeddisturbances can be used in applied engineering problems. The role these disturbances play in the framework of transition to turbulence, active flow control etc. were to be discussed.
 
First panel : The initial condition is characterized by long anti-parallel vortices with a localized perturbation for the 
Plots versus time t of the total kinetic energy (first panel, black curve), the enstrophy Z (first panel, blue curve), the normalized enstrophy-production rate −Su (first panel, red curve), Dm for 2 ≤ m ≤ 9 (second panel, blue to brown curves), and D∞ (second panel, dark green curve) for our DNS of decaying, 3D Navier-Stokes turbulence ; the value of D1 is very high, so it is omitted. The third panel is of statistically steady forced turbulence at constant Grashof number. The mean values of Dm in the statistically steady state are as follows: D1 = 3.1 × 10 11 , D2 = 5.5 × 10 4 , D3 = 1.1 × 10 4 , D4 = 6.6 × 10 3 , D5 = 5.2 × 10 3 , D6 = 4.5 × 10 3 , D7 = 4.1 × 10 3 , D8 = 3.9 × 10 3 , D9 = 3.7 × 10 3 , and D∞ = 3.0 × 10 3. Zooming in to the right panel makes it clear that Dm+1 < Dm for all values of m considered.
Scaling of the Ωm as a function of Re λ for forced stationary isotropic turbulence with resolutions up to 4096 3. Lines are for m = 1 (circles), 2 (squares), 3 (triangles), 4 (stars), 5 (left triangles), 6 (diamonds). Open and closed symbols correspond to EP and FEK forcing respectively. Dashed line is ∼ Re 6 λ (see text). Note that for Re λ ≈ 650 at 4096 3 with FEK forcing, moments up to fourth order (instead of sixth) are available from our database.
Article
The issue of intermittency in numerical solutions of the 3D Navier-Stokes equations on a periodic box $[0,\,L]^{3}$ is addressed through four sets of numerical simulations that calculate a new set of variables defined by $D_{m}(t) = \left(\varpi_{0}^{-1}\Omega_{m}\right)^{\alpha_{m}}$ for $1 \leq m \leq \infty$ where $\alpha_{m}= \frac{2m}{4m-3}$ and $\left[\Omega_{m}(t)\right]^{2m} = L^{-3}\I |\bom|^{2m}dV$ with $\varpi_{0} = \nu L^{-2}$. All four simulations unexpectedly show that the $D_{m}$ are ordered for $m = 1\,,...,\,9$ such that $D_{m+1} < D_{m}$. Moreover, the $D_{m}$ squeeze together such that $D_{m+1}/D_{m}\nearrow 1$ as $m$ increases. The first simulation is of very anisotropic decaying turbulence\,; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at constant Grashof number respectively\,; the fourth is of very high Reynolds number forced, stationary, isotropic turbulence at up to resolutions of $4096^{3}$.
 
Singular/non singular behaviour of system (3.1)-(3.3) depending on the value of the model's parameter λ. The limiting case λ = −1 (not analysed in this paper) gives an infinite-time singularity.
Article
Motivated by the work on stagnation-point-type exact solutions (with infinite energy) of 3D Euler fluid equations by Gibbon et al. ( Physica D, vol. 132 (4), 1999, pp. 497–510) and the subsequent demonstration of finite-time blowup by Constantin ( Int. Math. Res. Not. IMRN , vol. 9, 2000, pp. 455–465) we introduce a one-parameter family of models of the 3D Euler fluid equations on a 2D symmetry plane. Our models are seen as a deformation of the 3D Euler equations which respects the variational structure of the original equations so that explicit solutions can be found for the supremum norms of the basic fields: vorticity and stretching rate of vorticity. In particular, the value of the model’s parameter determines whether or not there is finite-time blowup, and the singularity time can be computed explicitly in terms of the initial conditions and the model’s parameter. We use a representative of this family of models, whose solution blows up at a finite time, as a benchmark for the systematic study of errors in numerical simulations. Using a high-order pseudospectral method, we compare the numerical integration of our ‘original’ model equations against a ‘mapped’ version of these equations. The mapped version is a globally regular (in time) system of equations, obtained via a bijective nonlinear mapping of time and fields from the original model equations. The mapping can be constructed explicitly whenever a Beale–Kato–Majda type of theorem is available therefore it is applicable to the 3D Euler equations (Bustamante, Physica D, vol. 240 (13), 2011, pp. 1092–1099). We show that the mapped system’s numerical solution leads to more accurate (by three orders of magnitude) estimates of supremum norms and singularity time compared with the original system. The numerical integration of the mapped equations is demonstrated to entail only a small extra computational cost. We study the Fourier spectrum of the model’s numerical solution and find that the analyticity strip width (a measure of the solution’s analyticity) tends to zero as a power law in a finite time. This is in agreement with the finite-time blowup of the fields’ supremum norms, in the light of rigorous bounds stemming from the bridge (Bustamante & Brachet, Phys. Rev. E, vol. 86 (6), 2012, 066302) between the analyticity-strip method and the Beale–Kato–Majda type of theorems. We conclude by discussing the implications of this research on the analysis of numerical solutions to the 3D Euler fluid equations.
 
Article
This paper analyzes the adjoint solution of the Navier-Stokes equation. We focus on flow across a circular cylinder at three Reynolds numbers, Re_D=20, 100 and 500. The quantity of interest in the adjoint formulation is the drag on the cylinder. We use classical fluid mechanics approaches to analyze the adjoint solution, which is a vector field similar to a flow field. Production and dissipation of kinetic energy of the adjoint field is discussed. We also derive the evolution of circulation of the adjoint field along a closed material contour. These analytical results are used to explain three numerical solutions of the adjoint equations presented in this paper. The adjoint solution at Re_D=20, a viscous steady state flow, exhibits a downstream suction and an upstream jet, opposite of the expected behavior of a flow field. The adjoint solution at Re_D=100, a periodic 2D unsteady flow, exhibits periodic, bean shaped circulation in the near wake region. The adjoint solution at Re_D=500, a turbulent 3D unsteady flow, has complex dynamics created by the shear layer in the near wake. The magnitude of the adjoint solution increases exponentially at the rate of the first Lyapunov exponent. These numerical results correlate well with the theoretical analysis presented in this paper.
 
Article
A direct numerical simulation of incompressible channel flow at a friction Reynolds number (Reτ) of 5186 has been performed, and the flow exhibits a number of the characteristics of high-Reynolds-number wall-bounded turbulent flows. For example, a region where the mean velocity has a logarithmic variation is observed, with von Kármán constant κ=0.384±0.004. There is also a logarithmic dependence of the variance of the spanwise velocity component, though not the streamwise component. A distinct separation of scales exists between the large outer-layer structures and small inner-layer structures. At intermediate distances from the wall, the one-dimensional spectrum of the streamwise velocity fluctuation in both the streamwise and spanwise directions exhibits k−1 dependence over a short range in wavenumber (k). Further, consistent with previous experimental observations, when these spectra are multiplied by k (premultiplied spectra), they have a bimodal structure with local peaks located at wavenumbers on either side of the k−1 range.
 
Article
We investigate the instability and nonlinear saturation of temperature-stratified Taylor-Couette flows in a finite height cylindrical gap and calculate angular-momentum transport in the nonlinear regime. The model is based on an incompressible fluid in Boussinesq approximation with a positive axial temperature gradient applied. While both ingredients itself, the differential rotation as well as the stratification due to the temperature gradient, are stable, together the system becomes subject of the stratorotational instability and nonaxisymmetric flow pattern evolve. This flow configuration transports angular momentum outwards and will therefor be relevant for astrophysical applications. The belonging viscosity $\alpha$ coefficient is of the order of unity if the results are adapted to the size of an accretion disc. The strength of the stratification, the fluids Prandtl number and the boundary conditions applied in the simulations are well-suited too for a laboratory experiment using water and a small temperature gradient below five Kelvin. With such a rather easy realizable set-up the SRI and its angular momentum transport could be measured in an experiment. Comment: 10 pages, 6 figures, revised version appeared in J. Fluid Mech. (2009), vol. 623, pp. 375--385
 
Article
Oceanic geostrophic turbulence is mostly forced at the surface, yet strong bottom-trapped flows are commonly observed along topographic anomalies. Here we consider the case of a freely evolving, initially surface-intensified velocity field above a topographic bump, and show that the self-organization into a bottom-trapped current can result from its turbulent dynamics. Using equilibrium statistical mechanics, we explain this phenomenon as the most probable outcome of turbulent stirring. We compute explicitly a class of solutions characterized by a linear relation between potential vorticity and streamfunction, and predict when the bottom intensification is expected. Using direct numerical simulations, we provide an illustration of this phenomenon that agrees qualitatively with theory, although the ergodicity hypothesis is not strictly fulfilled.
 
Article
We present a detailed analysis of temperature statistics in an oceanographic observational dataset. The data are collected using a moored array of thermistors, $100~\text{m}$ tall and starting $5~\text{m}$ above the bottom, deployed during four months above the slopes of a Seamount in the north-eastern Atlantic Ocean. Turbulence at this location is strongly affected by the semidiurnal tidal wave. Mean stratification is stable in the entire dataset. We compute structure functions, of order up to 10, of the distributions of temperature increments. Strong intermittency is observed, in particular, during the downslope phase of the tide, and farther from the solid bottom. In the lower half of the mooring during the upslope phase, the temperature statistics are consistent with those of a passive scalar. In the upper half of the mooring, the temperature statistics deviate from those of a passive scalar, and evidence of turbulent convective activity is found. The downslope phase is generally thought to be more shear-dominated, but our results suggest on the other hand that convective activity is present. High-order moments also show that the turbulence scaling behaviour breaks at a well-defined scale (of the order of the buoyancy length scale), which is however dependent on the flow state (tidal phase, height above the bottom). At larger scales, wave motions are dominant. We suggest that our results could provide an important reference for laboratory and numerical studies of mixing in geophysical flows.
 
Article
Detailed measurements of the time-dependent velocities induced inside and outside the opening of an acoustically excited, two-dimensional Helmholtz resonator imbedded in a grazing flow are presented. The remarkably clear structure of the perturbation field which evokes a pulsating source and a coherently pulsating vortex-image pair is described.
 
A definition sketch of the submerged cross-section of a cylinder. 
Article
The coupled motion is investigated for a mechanical system consisting of water and a body freely floating in it. Water occupies either a half-space or a layer of constant depth into which an infinitely long surface-piercing cylinder is immersed, thus allowing us to study two-dimensional modes. Under the assumption that the motion is of small amplitude near equilibrium, a linear setting is applicable and for the time-harmonic oscillations it reduces to a spectral problem with the frequency of oscillations as the spectral parameter. It is essential that one of the problem's relations is linear with respect to the parameter, whereas two others are quadratic with respect to it. Within this framework, it is shown that the total energy of the water motion is finite and the equipartition of energy holds for the whole system. On this basis, it is proved that no wave modes can be trapped provided their frequencies exceed a bound depending on cylinder's properties, whereas its geometry is subject to some restrictions and, in some cases, certain restrictions are imposed on the type of mode.
 
Article
We study two-phase stratified flow where the bottom layer is a thin laminar liquid and the upper layer is a fully-developed gas flow. The gas flow can be laminar or turbulent. To determine the boundary between convective and absolute instability, we use Orr--Sommerfeld stability theory, and a combination of linear modal analysis and ray analysis. For turbulent gas flow, and for the density ratio r=1000, we find large regions of parameter space that produce absolute instability. These parameter regimes involve viscosity ratios of direct relevance to oil/gas flows. If, instead, the gas layer is laminar, absolute instability persists for the density ratio r=1000, although the convective/absolute stability boundary occurs at a viscosity ratio that is an order of magnitude smaller than in the turbulent case. Two further unstable temporal modes exist in both the laminar and the turbulent cases, one of which can exclude absolute instability. We compare our results with an experimentally-determined flow-regime map, and discuss the potential application of the present method to non-linear analyses.
 
The locus of points, for various values of Pe, (solid lines) where the concentration c attains its maximum along the streamlines (dashed lines) around a circular disk for desorption into an unconcentrated fluid. (In the equivalent problem of absorption from a concentrated fluid, the solid curves give the minimum concentration along streamlines for different Pe. ) 
Article
We perform an exhaustive study of the simplest, nontrivial problem in advection-diffusion -- a finite absorber of arbitrary cross section in a steady two-dimensional potential flow of concentrated fluid. This classical problem has been studied extensively in the theory of solidification from a flowing melt, and it also arises in Advection-Diffusion-Limited Aggregation. In both cases, the fundamental object is the flux to a circular disk, obtained by conformal mapping from more complicated shapes. We construct the first accurate numerical solution using an efficient new method, which involves mapping to the interior of the disk and using a spectral method in polar coordinates. Our method also combines exact asymptotics and an adaptive mesh to handle boundary layers. Starting from a well-known integral equation in streamline coordinates, we also derive new, high-order asymptotic expansions for high and low P\'eclet numbers ($\Pe$). Remarkably, the `high' $\Pe$ expansion remains accurate even for such low $\Pe$ as $10^{-3}$. The two expansions overlap well near $\Pe = 0.1$, allowing the construction of an analytical connection formula that is uniformly accurate for all $\Pe$ and angles on the disk with a maximum relative error of 1.75%. We also obtain an analytical formula for the Nusselt number ($\Nu$) as a function of the P\'eclet number with a maximum relative error of 0.53% for all possible geometries. Because our finite-plate problem can be conformally mapped to other geometries, the general problem of two-dimensional advection-diffusion past an arbitrary finite absorber in a potential flow can be considered effectively solved. Comment: 29 pages, 12 figs (mostly in color)
 
Article
This paper considers the large-amplitude symmetric and asymmetric irrota-tional motion of an inviscid incompressible fluid with a liquid—vapour interface in an accelerating container of revolution. A combined analytical—numerical method which involves no linearizations in the hydrodynamical equations and applies to all but surface-tension dominated motions is used to compute a variety of such motions. One important aspect of this non-linear method is that it accurately determines the initial development of surface instabilities such as breakers near the wall of the container.
 
Article
The modelling of fluid particle accelerations in homogeneous, isotropic turbulence in terms of second-order stochastic models for the Lagrangian velocity is considered. The basis for the Reynolds model (A. M. Reynolds, \textit{Phys. Rev. Lett.} $\mathbf{91}(8)$, 084503 (2003)) is reviewed and examined by reference to DNS data. In particular, we show DNS data that support stochastic modelling of the logarithm of pseudo-dissipation as an Ornstein-Uhlenbeck process (Pope and Chen 1990) and reveal non-Gaussianity of the conditional acceleration PDF. The DNS data are used to construct a simple stochastic model that is exactly consistent with Gaussian velocity and conditionally cubic-Gaussian acceleration statistics. This model captures the effects of intermittency of dissipation on acceleration and the conditional dependence of acceleration on pseudo-dissipation (which differs from that predicted by the refined Kolmogorov (1962) hypotheses). Non-Gaussianity of the conditional acceleration PDF is accounted for in terms of model nonlinearity. The diffusion coefficient for the new model is chosen based on DNS data for conditional two-time velocity statistics. The resulting model predictions for conditional and unconditional velocity statistics and timescales are shown to be in good agreement with DNS data.
 
Article
We present the results of direct numerical simulations of heavy particle transport in homogeneous, isotropic, fully developed turbulence, up to resolution $512^3$ ($R_\lambda\approx 185$). Following the trajectories of up to 120 million particles with Stokes numbers, $St$, in the range from 0.16 to 3.5 we are able to characterize in full detail the statistics of particle acceleration. We show that: ({\it i}) The root-mean-squared acceleration $a_{\rm rms}$ sharply falls off from the fluid tracer value already at quite small Stokes numbers; ({\it ii}) At a given $St$ the normalised acceleration $a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$ increases with $R_\lambda$ consistently with the trend observed for fluid tracers; ({\it iii}) The tails of the probability density function of the normalised acceleration $a/a_{\rm rms}$ decrease with $St$. Two concurrent mechanisms lead to the above results: preferential concentration of particles, very effective at small $St$, and filtering induced by the particle response time, that takes over at larger $St$. Comment: 10 pages, 3 figs, 2 tables. A section with new results has been added. Revised version accepted for pubblication on Journal of Fluid Mechanics
 
A sketch of the deformation of the simulation domain under straining. The mean flow U = (−2Sx, Sy, Sz) corresponds to an ideal flow onto a flat plate. The deforming domain is initially elongated in the x-direction but becomes wider in y and z directions with time. Arrows indicate the directions of the stream lines of the induced mean flow. 
Article
The dynamics of inertial particles in turbulence is modelled and investigated by means of direct numerical simulation of an axisymmetrically expanding homogeneous turbulent strained flow. This flow can mimic the dynamics of particles close to stagnation points. The influence of mean straining flow is explored by varying the dimensionless strain rate parameter $Sk_0/\epsilon_0$ from 0.2 to 20. We report results relative to the acceleration variances and probability density functions for both passive and inertial particles. A high mean strain is found to have a significant effect on the acceleration variance both directly, through an increase in wave number magnitude, and indirectly, through the coupling of the fluctuating velocity and the mean flow field. The influence of the strain on normalized particle acceleration pdfs is more subtle. For the case of passive particle we can approximate the acceleration variance with the aid of rapid distortion theory and obtain good agreement with simulation data. For the case of inertial particles we can write a formal expressions for the accelerations. The magnitude changes in the inertial particle acceleration variance and the effect on the probability density function are then discussed in a wider context for comparable flows, where the effects of the mean flow geometry and of the anisotropy at the small scales are present.
 
Article
The variances of the fluid-particle acceleration and of the pressure-gradient and viscous force are given. The scaling parameters for these variances are velocity statistics measureable with a single-wire anemometer. For both high and low Reynolds numbers, asymptotic scaling formulas are given; these agree quantitatively with DNS data. Thus, the scaling can be presumed known for all Reynolds numbers. Fluid-particle acceleration variance does not obey K41 scaling at any Reynolds number; this is consistent with recent experimental data. The non-dimensional pressure-gradient variance named lambda-sub{T} /lambda-sub{P} is shown to be obsolete.
 
Article
We use silicon strip detectors (originally developed for the CLEO III high energy particle physics experiment) to measure fluid particle trajectories in turbulence with temporal resolution of up to 70,000 frames per second. This high frame rate allows the Kolmogorov time scale of a turbulent water flow to be fully resolved for 140 <= R_lambda <= 970. Particle trajectories exhibiting accelerations up to 16,000 m\s^2 (40 times the rms value) are routinely observed. The probability density function of the acceleration is found to have Reynolds number dependent stretched exponential tails. The moments of the acceleration distribution are calculated. The scaling of the acceleration component variance with the energy dissipation is found to be consistent with the results for low Reynolds number direct numerical simulations, and with the K41 based Heisenberg-Yaglom prediction for R_lambda >= 500. The acceleration flatness is found to increase with Reynolds number, and to exceed 60 at R_lambda = 970. The coupling of the acceleration to the large scale anisotropy is found to be large at low Reynolds number and to decrease as the Reynolds number increases, but to persist at all Reynolds numbers measured. The dependence of the acceleration variance on the size and density of the tracer particles is measured. The autocorrelation function of an acceleration component is measured, and is found to scale with the Kolmogorov time tau_eta. Comment: 35 pages. 34 figures corrected typos and incorrect figure. added figure
 
Article
Two incompressible viscous fluids with different densities meet at a planar interface. The fluids are subject to an externally imposed oscillating acceleration directed normal to the interface. The resulting basic-state flow is motionless with an internal pressure oscillation. We discuss the linear evolution of perturbations to this basic state. General viscosities and densities for the two fluids are considered but a Boussinesq equal-viscosity approximation is discussed in particular detail. For this case we show that the linear evolution of a perturbation to the interface subject to an arbitrary oscillating acceleration is governed by a single integro-differential equation. We apply a Floquet analysis to the fluid system for the case of sinusoidal forcing. Parameter regions of subharmonic, harmonic, and untuned modes are delineated. The critical Stokes-Reynolds number is found as a function of the surface tension and the difference in density and viscosity between the two fluids. The most unstable perturbation wavelengths are determined. For zero surface tension these are found to be short, on the order of a small multiple of the Stokes viscous lengthscale. The critical Stokes-Reynolds number and the most unstable perturbation wavelengths are found to be insensitive to the degree of density and viscosity differences between the two fluids.
 
Article
We discuss the inviscid 2D instability recently uncovered by Ilin & Morgulis (2013) in the context of irrotational Taylor-Couette flow with a radial flow imposed. By finding a simplier rectilinear example of the instability - the sheared half plane, the minimal ingredients for the instability are identified and the destabilizing/stabilizing effect of the inflow/outflow boundaries clarified. The instability - christened `boundary inflow instability' here - is of critical layer type where this layer is either at the inflow wall and the growth rate is $O(\eta^{1/2})$ (as found by Ilin & Morgulis 2013), or in the interior of the flow and the growth rate is $O(\eta \log(1/\eta) )$ where $\eta$ measures the (small) inflow-to-tangential-flow ratio. The instability is robust to changes in the rotation profile even to those which are very Rayleigh-stable and the addition of further physics such as viscosity, 3-dimensionality and compressibility but is sensitive to the boundary condition imposed on the tangential velocity field at the inflow boundary. Both the primary bifurcation to 2D states and secondary bifurcations to 3D states are found to be supercritical. Assuming an accretion flow driven by molecular viscosity only so $\eta=O(Re^{-1})$, the instability is not immediately relevant for accretion disks since the critical threshold is $O(Re^{-2/3})$ and the inflow boundary conditions are more likely to be stress-free than non-slip. However, the analysis presented here does highlight the potential for mass entering a disk to disrupt the orbiting flow if this mass flux possesses vorticity.
 
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