Focus on Fluids
Swimming in shear
- David Saintillan
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- 12 March 2014, pp. 1-4
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The complex patterns observed in experiments on suspensions of swimming cells undergoing bioconvection have fascinated biologists, physicists and mathematicians alike for over a century. Theoretical models developed over the last few decades have shown a strong similarity with Rayleigh–Bénard thermal convection, albeit with a richer dynamical behaviour due to the orientational degrees of freedom of the cells. In a recent paper, Hwang & Pedley (J. Fluid Mech., vol. 738, 2014, pp. 522–562) revisit previous models for bioconvection to investigate the effects of an external shear flow on pattern formation. In addition to casting light on new mechanisms for instability, their study demonstrates a subtle interplay between shear, swimming motions and bioconvection patterns.
Papers
The non-equilibrium region of grid-generated decaying turbulence
- P. C. Valente, J. C. Vassilicos
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- 13 March 2014, pp. 5-37
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The previously reported non-equilibrium dissipation law is investigated in turbulent flows generated by various regular and fractal square grids. The flows are documented in terms of various turbulent profiles which reveal their differences. In spite of significant inhomogeneity and anisotropy differences, the new non-equilibrium dissipation law is observed in all of these flows. Various transverse and longitudinal integral scales are measured and used to define the dissipation coefficient $C_{\varepsilon }$. It is found that the new non-equilibrium dissipation law is not an artefact of a particular choice of the integral scale and that the usual equilibrium dissipation law can actually coexist with the non-equilibrium law in different regions of the same flow.
On the anisotropy of the turbulent passive scalar in the presence of a mean scalar gradient
- Wouter J. T. Bos
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- 10 March 2014, pp. 38-64
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We investigate the origin of the scalar gradient skewness in isotropic turbulence on which a mean scalar gradient is imposed. The problem of the advection of an anisotropic scalar field is reformulated in terms of the advection of an isotropic vector field. For this field, triadic closure equations are derived. It is shown how the scaling of the scalar gradient skewness depends on the choice of the time scale used for the Lagrangian decorrelation of the vector field. The persistent anisotropy in the small scales for the third-order statistics is shown to be perfectly compatible with Corrsin–Obukhov scaling for second-order quantities, since second- and third-order scalar quantities are governed by a different triad correlation time scale. Whereas the inertial range dynamics of second-order scalar quantities is governed by the Lagrangian velocity correlation time, the third-order quantities remain correlated over a time related to the large-scale dynamics of the scalar field. It is argued that this time is determined by the average time it takes for a fluid particle to travel between ramp-cliff scalar structures.
Patterns of a creeping water-spout flow
- Miguel Herrada, Vladimir Shtern
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- 10 March 2014, pp. 65-88
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This paper explains a mechanism of eddy formation in a slow air–water motion, driven by the rotating top disk, in a vertical sealed cylinder. The numerical simulations reveal nine changes in the flow topology as water volume fraction $H_{w}$ varies from 0 to 1. At $H_{w}$ around 0.8, there are two large regions of the clockwise meridional circulation, one in air and one in water. These regions are separated by two small cells of the anticlockwise circulation adjacent to the interface near the sidewall in water and near the axis in air. The air cell is a thin layer and topologically is a bubble–ring for $0.745 < H_{w} < 0.785$. Alterations of this flow pattern are explored as (i) pressure increases, (ii) the bottom disk co-rotates and (iii) the top-disk rotation speeds up. This paper provides the physical reasoning behind the flow transformations; the results are of fundamental interest and can be utilized in bioreactors.
Spin-up of a two-component superfluid: self-consistent container feedback
- Cornelis A. van Eysden, A. Melatos
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- 10 March 2014, pp. 89-110
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The coupled dynamic response of a rigid container filled with a two-component superfluid undergoing Ekman pumping is calculated self-consistently. The container responds to the back-reaction torque exerted by the viscous component of the superfluid and an arbitrary external torque. The resulting motion is described by a pair of coupled integral equations for which solutions are easily obtained numerically. If the container is initially accelerated impulsively then set free, it relaxes quasi-exponentially to a steady state over multiple time scales, which are a complex combination of the Ekman number, superfluid mutual friction coefficients, the superfluid density fraction, and the varying hydrodynamic torque at different latitudes. The spin-down of containers with relatively small moments of inertia (compared with that of the contained fluid) depends weakly on the above parameters and occurs faster than the Ekman time. When the fluid components are initially differentially rotating, the container can ‘overshoot’ its asymptotic value before increasing again. When a constant external torque is applied, the superfluid components rotate differentially and non-uniformly in the long term. For an oscillating external torque, the amplitude and phase of the oscillation are most sensitive to the driving frequency for containers with relatively small moments of inertia. Applications to superfluid helium experiments and neutron stars are also discussed.
Numerical study on unstable surfaces of oblique detonations
- Hong Hui Teng, Zong Lin Jiang, Hoi Dick Ng
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- 10 March 2014, pp. 111-128
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In this study, the onset of cellular structure on oblique detonation surfaces is investigated numerically using a one-step irreversible Arrhenius reaction kinetic model. Two types of oblique detonations are observed from the simulations. One is weakly unstable characterized by the existence of a planar surface, and the other is strongly unstable characterized by the immediate formation of the cellular structure. It is found that a high degree of overdrive suppresses the formation of cellular structures as confirmed by the results of many previous studies. However, the present investigation demonstrates that cellular structures also appear with degree of overdrive of 2.06 and 2.37, values much higher than ${\sim }$1.8 suggested previously in the literature for the critical value defining the instability boundary of oblique detonations. This contradiction could be explained by the use of differently shaped walls, a straight wall used in this study and a custom-designed curved wedge system so as to induce straight oblique detonations in previous studies. Another possible reason could be due to the low and possibly insufficient resolution used in previously published studies. Hence, simulations with different grid sizes are also performed to examine the effect of resolution on the numerical solutions. Using the present results, analysis also shows that although the characteristic lengths of unstable surfaces are different when the incident Mach number changes, these length scales are proportional to tangential velocities. Hence, the interior time determined by the overdrive degree is identified, and its limitation as the instability parameter is discussed.
A microstructural approach to bed load transport: mean behaviour and fluctuations of particle transport rates
- C. Ancey, J. Heyman
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- 10 March 2014, pp. 129-168
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This paper concerns a model of bed load transport, which describes the advection and dispersion of coarse particles carried by a turbulent water stream. The challenge is to develop a microstructural approach that, on the one hand, yields a parsimonious description of particle transport at the microscopic scale and, on the other hand, leads to averaged equations at the macroscopic scale that can be consistently interpreted in light of the continuum equations used in hydraulics. The cornerstone of the theory is the proper determination of the particle flux fluctuations. Apart from turbulence-induced noise, fluctuations in the particle transport rate are generated by particle exchanges with the bed consisting of particle entrainment and deposition. At the particle scale, the evolution of the number of moving particles can be described probabilistically using a coupled set of reaction–diffusion master equations. Theoretically, this is interesting but impractical, as solving the governing equations is fraught with difficulty. Using the Poisson representation, we show that these multivariate master equations can be converted into Fokker–Planck equations without any simplifying approximations. Thus, in the continuum limit, we end up with a Langevin-like stochastic partial differential equation that governs the time and space variations of the probability density function for the number of moving particles. For steady-state flow conditions and a fixed control volume, the probability distributions of the number of moving particles and the particle flux can be calculated analytically. Taking the average of the microscopic governing equations leads to an average mass conservation equation, which takes the form of the classic Exner equation under certain conditions carefully addressed in the paper. Analysis also highlights the specific part played by a process we refer to as collective entrainment, i.e. a nonlinear feedback process in particle entrainment. In the absence of collective entrainment the fluctuations in the number of moving particles are Poissonian, which implies that at the macroscopic scale they act as white noise that mediates bed evolution. In contrast, when collective entrainment occurs, large non-Poissonian fluctuations arise, with the important consequence that the evolution at the macroscopic scale may depart significantly that predicted by the averaged Exner equation. Comparison with experimental data gives satisfactory results for steady-state flows.
Thermal transpiration flow through a single rectangular channel
- Hiroki Yamaguchi, Marcos Rojas-Cárdenas, Pierre Perrier, Irina Graur, Tomohide Niimi
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- 10 April 2014, pp. 169-182
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A thermal transpiration flow through a single rectangular micro-channel was studied experimentally for various gas species, including all rare gases, in order to investigate the influence of gas species on the flow properties. The final equilibrium flow characteristics and relaxation time of the pressure variation were evaluated as functions of the rarefaction parameter. The thermal molecular pressure difference was well fitted by the log-normal distribution function, and its magnitude was found to be strongly dependent on the gas species: a larger pressure difference was obtained for molecules of smaller diameter. However, for the thermal molecular pressure ratio and the thermal molecular pressure exponent, which are dimensionless quantities, the dependence on the gas species was negligible. The relaxation time of the pressure variation was well normalized by the characteristic time of the system. The influence of the geometry was evaluated by comparing the present results, obtained for the case of a rectangular channel, with already published data obtained for the case of a circular cross-section tube. The comparison showed that these two geometrical configurations influence the fluid flow in equal manner, if appropriate geometrical parameters are used for their representation.
Inertial particle trapping in an open vortical flow
- Jean-Régis Angilella, Rafael D. Vilela, Adilson E. Motter
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- 11 March 2014, pp. 183-216
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Recent numerical results on advection dynamics have shown that particles denser than the fluid can remain trapped indefinitely in a bounded region of an open fluid flow. Here, we investigate this counterintuitive phenomenon both numerically and analytically to establish the conditions under which the underlying particle-trapping attractors can form. We focus on a two-dimensional open flow composed of a pair of vortices and its specular image, which is a system we represent as a vortex pair plus a wall along the symmetry line. Considering particles that are much denser than the fluid, referred to as heavy particles, we show that two attractors form in the neighbourhood of the vortex pair provided that the particle Stokes number is smaller than a critical value of order unity. In the absence of the wall, the attractors are fixed points in the frame rotating with the vortex pair, and the boundaries of their basins of attraction are smooth. When the wall is present, the point attractors describe counter-rotating ellipses in this frame, with a period equal to half the period of one isolated vortex pair. The basin boundaries remain smooth if the distance from the vortex pair to the wall is large. However, these boundaries are shown to become fractal if the distance to the wall is smaller than a critical distance that scales with the inverse square root of the Stokes number. This transformation is related to the breakdown of a separatrix that gives rise to a heteroclinic tangle close to the vortices, which we describe using a Melnikov function. For an even smaller distance to the wall, we demonstrate that a second separatrix breaks down and a new heteroclinic tangle forms farther away from the vortices, at the boundary between the open and closed streamlines. Particles released in the open part of the flow can approach the attractors and be trapped permanently provided that they cross the two separatrices, which can occur under the effect of flow unsteadiness. Furthermore, the trapping of heavy particles from the open flow is shown to be robust to the presence of viscosity, noise and gravity. Navier–Stokes simulations for large flow Reynolds numbers show that viscosity does not destroy the attracting points until vortex merging takes place, while simulation of thermal noise shows that particle trapping persists for extended periods provided that the Péclet number is large. The presence of a gravitational field does not alter the permanent trapping by the attracting points if the settling velocities are not too large. For larger settling velocities, however, gravity can also give rise to a limit-cycle attractor next to the external separatrix and to a new form of trapping from the open flow that is not mediated by a heteroclinic tangle.
Direct numerical simulation of turbulent scalar transport across a flat surface
- H. Herlina, J. G. Wissink
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- 11 March 2014, pp. 217-249
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To elucidate the physical mechanisms that play a role in the interfacial transfer of atmospheric gases into water, a series of direct numerical simulations of mass transfer across the air–water interface driven by isotropic turbulence diffusing from below has been carried out for various turbulent Reynolds numbers ($R_T=84,195,507$). To allow a direct (unbiased) comparison of the instantaneous effects of scalar diffusivity, in each of the DNS up to six scalar advection–diffusion equations with different Schmidt numbers were solved simultaneously. As far as the authors are aware this is the first simulation that is capable to accurately resolve the realistic Schmidt number, $\mathit{Sc}=500$, that is typical for the transport of atmospheric gases such as oxygen in water. For the range of turbulent Reynolds numbers and Schmidt numbers considered, the normalized transfer velocity $K_L$ was found to scale with $R_T^{-{1/2}}$ and $\mathit{Sc}^{-{1/2}}$, which indicates that the largest eddies present in the isotropic turbulent flow introduced at the bottom of the computational domain tend to determine the mass transfer. The $K_L$ results were also found to be in good agreement with the surface divergence model of McCready, Vassiliadou & Hanratty (AIChE J., vol. 32, 1986, pp. 1108–1115) when using a constant of proportionality of 0.525. Although close to the surface large eddies are responsible for the bulk of the gas transfer, it was also observed that for higher $R_T$ the influence of smaller eddies becomes more important.
Numerical study of the effect of surface wave on turbulence underneath. Part 2. Eulerian and Lagrangian properties of turbulence kinetic energy
- Xin Guo, Lian Shen
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- 11 March 2014, pp. 250-272
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The effect of the rapid distortion of a surface wave on the kinetic energy of turbulence underneath is studied based on the simulation data of Part 1 (Guo & Shen, J. Fluid Mech., vol. 733, 2013, pp. 558–587). In the Eulerian frame, Reynolds normal stresses, which contribute to turbulence kinetic energy, are found to vary with the wave phase. An analysis of their budgets shows that their variation is dominated not only by the normal production term representing the wave straining effect on wave–turbulence energy exchange, but also by pressure effects including the pressure–strain correlation and pressure transport terms. In the Lagrangian frame, the net energy transfer from the wave to turbulence is analysed. It is found to be mainly contributed by the mean Lagrangian effect and the correlation between the Lagrangian fluctuations of the wave and turbulence; the former plays a major role in the overall wave energy dissipation, while the latter is associated with the viscous effect of the wave surface and is appreciable in the near-surface region. Models for various terms in wave–turbulence energy flux are discussed. The decay time scale of swells in oceans estimated from our simulations compares well with the results in the literature.
Scaling arguments for the fluxes in turbulent miscible fountains
- H. C. Burridge, G. R. Hunt
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- 11 March 2014, pp. 273-285
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For established axisymmetric turbulent miscible Boussinesq fountains in quiescent uniform environments, expressions are developed for the fluxes of volume, momentum and buoyancy at the outflow from the fountain: the outflow referring to the counterflow at the horizontal plane of the source. The fluxes are expressed in terms of the fountain source conditions and two dimensionless functions of the source Froude number, ${\rm Fr}_{0}$: a radial function (relating a horizontal scale of the outflow to the source radius) and a volume flux function (relating the outflow and source volume fluxes). The forms taken by these two functions at low ${\rm Fr}_{0}$ and high ${\rm Fr}_{0}$ are deduced, thereby providing the outflow fluxes and outflow Froude number solely in terms of the source conditions. For high ${\rm Fr}_{0}$, the outflow Froude number, ${\rm Fr}_{out}$, is shown to be invariant, indicating (by analogy with plumes for which the ‘far-field’ Froude number is invariant with source Froude number) that the outflow may be regarded as ‘far-field’ since the fluxes within the fountain have adjusted to attain a balance which is independent of the source conditions. Based on ${\rm Fr}_{out}$, the fluxes in the plume that forms beyond the fountain outflow are deduced. Finally, from the results of previously published studies, we show that the scalings deduced for fountains are valid for $0.0025 \lesssim {\rm Fr}_{0} \lesssim 1.0 $ for low ${\rm Fr}_{0}$ and $ {\rm Fr}_{0} \gtrsim 3.0 $ for high ${\rm Fr}_{0}$.
Vorticity generation due to cross-sea
- M. Postacchini, M. Brocchini, L. Soldini
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- 11 March 2014, pp. 286-309
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Similarly to shore-parallel waves interacting with submerged obstacles, two wave trains, approaching the shore with different angles, generate breakers of finite cross-flow length and an intense vorticity at their edges. The dynamics of crossing wave trains in shallow waters is studied by means of a simple theoretical approach that is used to inspect the flow characteristics at breaking. The post-breaking dynamics, with specific focus on the vorticity generation and evolution processes, is described on the basis of the analytical results of Brocchini et al. (J. Fluid Mech., vol. 507, 2004, pp. 289–307). Ad hoc numerical simulations, performed by means of a nonlinear shallow-water equations (NSWE) solver, are used to support the analytical findings and detail the post-breaking flow evolution. Comparisons between numerical and analytical findings confirm that: (i) the cross-sea theory successfully predicts the breaking position when a finite-length breaker stems from two crossing wave trains and (ii) the dynamics induced by such a breaking (i.e. vorticity generation, mutual-advection and self-advection mechanisms) is similar to that occurring after the breaking event of a shore-parallel wave over a submerged obstacle: vortices generated at the breaker edges are first subjected to wave forcing and self-advection, these pushing the vortices shoreward; then, oppositely-signed vortices pair and the mutual interaction enables them to invert the motion and move seaward. Useful relationships have been found to describe the main features of such a dynamics (i.e. breaker length, vortex trajectories, etc.).
The virtual power principle in fluid mechanics
- Yongliang Yu
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- 11 March 2014, pp. 310-328
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A conceptual framework on analytical mechanics for continuous fluid medium, which connects the fluid motion and all of the (internal and external) forces with mechanical power, is proposed by using the virtual power and the virtual velocity. Based on this framework, it is found that the internal virtual power is equal to the external virtual power in fluid dynamics, which is called the virtual power principle. This framework is also proved to be equivalent to the vector dynamics (Cauchy’s equation or Navier–Stokes equation). Furthermore, based on the virtual power principle, a theorem is introduced for continuous fluid medium, which indicates the relationship between the force (or torque) acting on a body immersed in a fluid and the specified virtual power. Subsequently, according to Galilean invariance, the detailed relationship for Newtonian fluids in incompressible flows is derived and used to illustrate the mechanisms on instantaneous forces: the added inertial effects, the boundary energy flux and dissipation effects, the vortex contribution, and the explicit body force contribution. As an application of the principle, the advantage of the V formation flight of geese is preliminarily discussed in the view of aerodynamics. Specifically, the total drag of the flock is reduced by contrast with the simple sum of the drag in solo fight and the optimal angle of V ranges from $60^{\circ }$ to $120^{\circ }$. The principle could be a useful approach to reveal the contributions of the flow structures and the moving or deforming boundaries to the force and torque acting on a body, especially in a multibody system.
Large-amplitude acoustic streaming
- G. P. Chini, Z. Malecha, T. D. Dreeben
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- 12 March 2014, pp. 329-351
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A mechanism is proposed for the generation of large-amplitude acoustically-driven streaming flows in which time-mean flow speeds are comparable to the instantaneous speed of fluid particles in a high-frequency sound-wave field. Motivated by streaming observed in high-intensity discharge (HID) lamps, two-dimensional flow of a density-stratified ideal gas in a channel geometry is analysed in the asymptotic limit of high-frequency acoustic-wave forcing. Predictions of streaming flow magnitudes based on classical arguments invoking Reynolds stress divergences originating in viscous boundary layers are orders of magnitude too small to account for the observed mean flows. Moreover, classical ‘Rayleigh streaming’ theory cannot account for the direction of the cellular mean flows often observed in HID lamps. In contrast, the mechanism proposed here, which invokes fluctuating baroclinic torques away from viscous boundary layers and thus is largely independent of viscous effects, can account both for the magnitude and the orientation of the observed streaming flows.
The initial transient and approach to self-similarity of a very viscous buoyant thermal
- Gunnar G. Peng, John R. Lister
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- 12 March 2014, pp. 352-375
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An isolated buoyant thermal in very viscous fluid has been shown to attain a self-similar form at large times which grows as $t^{1/2}$ (Whittaker & Lister J. Fluid Mech., vol. 606, 2008, pp. 295–324). For large values of the Rayleigh number $\mathit{Ra}$ (based on the conserved total buoyancy), the similarity solution is slender with a roughly spherical head at the top and a long tail that contains most of the buoyancy and extends down to the origin. We investigate the time-dependent behaviour of the thermal numerically; both the long-time behaviour in terms of perturbations to the similarity solution and the short-time evolution from a spherical initial condition. Using a spectral method, we find the growth rates of the linear perturbations and their spatial structure in similarity space. All eigenmodes decay monotonically for $\mathit{Ra}\lesssim 360$, while for larger $\mathit{Ra}$ the dominant (slowest decaying or fastest growing) eigenmodes are oscillatory with waves propagating up the tail. Above a critical value $\mathit{Ra}_c \approx 10\, 000$, the steady solution becomes unstable to a limit cycle. A one-dimensional reduction to horizontally integrated quantities hints at a theoretical explanation for the oscillatory behaviour, but does not reproduce the loss of stability. Investigation of the initial transient at large $\mathit{Ra}$ reveals that an initially spherical thermal can rise $O(100)$ times its initial diameter before approaching its final self-similar shape. The presence of a rigid horizontal floor below the thermal makes a quantitative difference of around 10 % to the rate of rise at large $\mathit{Ra}$.
On the onset of wake meandering for an axial flow turbine in a turbulent open channel flow
- Seokkoo Kang, Xiaolei Yang, Fotis Sotiropoulos
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- 12 March 2014, pp. 376-403
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Laboratory experiments have yielded evidence suggestive of large-scale meandering motions in the wake of an axial flow hydrokinetic turbine in a turbulent open channel flow (Chamorro et al., J. Fluid Mech., vol. 716, 2013, pp. 658–670). We carry out a large-eddy simulation (LES) of the experimental flow to investigate the structure of turbulence in the wake of the turbine and elucidate the mechanism that gives rise to wake meandering. All geometrical details of the turbine structure are taken into account in the simulation using the curvilinear immersed boundary LES method with wall modelling (Kang et al., Adv. Water Resour., vol. 34(1), 2011, pp. 98–113). The simulated flow fields are in good agreement with the experimental measurements and confirm the theoretical model of turbine wakes (Joukowski, Tr. Otdel. Fizich. Nauk Obshch. Lyub. Estestv., vol. 16, 1912, no. 1), yielding a near-turbine wake that consists of two layers: the tip vortex (or outer) shear layer that rotates in the same direction as the rotor; and the inner layer counter-rotating hub vortex. Analysis of the calculated instantaneous flow fields reveals that the hub vortex undergoes spiral vortex breakdown and precesses slowly in the direction opposite to the turbine rotation. The precessing vortex core remains coherent for three to four rotor diameters, expands radially outwards, and intercepts the outer shear layer at approximately the location where wake meandering is initiated. The wake meandering manifests itself in terms of an elongated region of increased turbulence kinetic energy and Reynolds shear stress across the top tip wake boundary. The interaction of the outer region of the flow with the precessing hub vortex also causes the rotational component of the wake to decay completely at approximately the location where the wake begins to meander (four rotor diameters downstream of the turbine). To further investigate the importance of turbine geometry on far-wake dynamics, we carry out LES under the same flow conditions but using actuator disk and actuator line parametrizations of the turbine. While both actuator approaches yield a meandering wake, the actuator line model yields results that are in better overall agreement with the measurements. However, comparisons between the actuator line and the turbine-resolving LES reveal significant differences. Namely, in the actuator line LES model: (i) the hub vortex does not develop spiral instability and remains stable and columnar without ever intercepting the outer shear layer; (ii) wake rotation persists for much longer distance downstream than in the turbine-resolving LES; and (iii) the level of turbulence kinetic energy within and the overall size of the far-wake meandering region are considerably smaller (this discrepancy is even more pronounced for the actuator disk LES case) compared with the turbine-resolving LES. Our results identify for the first time the instability mechanism that amplified wake meandering in the experiment of Chamorro et al., show that computational models that do not take into account the geometrical details of the turbine cannot capture such phenomena, and point to the potential significance of the near-hub rotor design as a means for suppressing the instability of the hub vortex and diminishing the extent and intensity of the far-wake meandering region.
Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization
- Anand U. Oza, Daniel M. Harris, Rodolfo R. Rosales, John W. M. Bush
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- 13 March 2014, pp. 404-429
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We present the results of a theoretical investigation of droplets walking on a rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an integro-differential equation that incorporates the influence of its propulsive wave force. Predictions for the dependence of the orbital radius on the bath’s rotation rate compare favourably with experimental data and capture the progression from continuous to quantized orbits as the vibrational acceleration is increased. The orbital quantization is rationalized by assessing the stability of the orbital solutions, and may be understood as resulting directly from the dynamic constraint imposed on the drop by its monochromatic guiding wave. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing.
Extended Squire’s transformation and its consequences for transient growth in a confined shear flow
- J. John Soundar Jerome, Jean-Marc Chomaz
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- 13 March 2014, pp. 430-456
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The classical Squire transformation is extended to the entire eigenfunction structure of both Orr–Sommerfeld and Squire modes. For arbitrary Reynolds numbers $\mathit{Re}$, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number $\mathit{Re}_{2D}$. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal gain at all optimization times $t$ with ${t}/{\mathit{Re}}$ finite or large. This result is an extension of the well-known scaling laws first obtained by Gustavsson (J. Fluid Mech., vol. 224, 1991, pp. 241–260) and Reddy & Henningson (J. Fluid Mech., vol. 252, 1993, pp. 209–238) for arbitrary $\alpha \mathit{Re}$, where $\alpha $ is the streamwise wavenumber. The Squire transformation is also extended to the adjoint problem and, hence, the adjoint Orr–Sommerfeld and Squire modes. It is, thus, demonstrated that the long-time optimal growth of 3D perturbations as given by the exponential growth (or decay) of the leading eigenmode times an extra gain representing its receptivity, may be decomposed as a product of the gains arising from purely 2D mechanisms and an analytical contribution representing 3D growth mechanisms equal to $1+ \left (\beta \mathit{Re}/\mathit{Re}_{2D}\right )^2 \mathcal{G}$, where $\beta $ is the spanwise wavenumber and $\mathcal{G}$ is a known expression. For example, when the leading eigenmode is an Orr–Sommerfeld mode, it is given by the product of respective gains from the 2D Orr mechanism and an analytical expression representing the 3D lift-up mechanism. Whereas if the leading eigenmode is a Squire mode, the extra gain is shown to be solely due to the 3D lift-up mechanism. Direct numerical solutions of the optimal gain for plane Poiseuille and plane Couette flow confirm the novel predictions of the Squire transformation extended to the initial-value problem. These results are also extended to confined shear flows in the presence of a temperature gradient.
Rapids
Satellite formation during bubble transition through an interface between immiscible liquids
- E. Q. Li, S. A. Al-Otaibi, I. U. Vakarelski, S. T. Thoroddsen
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- 12 March 2014, R1
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When a bubble rises to an interface between two immiscible liquids, it can pass through the interface, if this is energetically favourable, i.e. the bubble preferring the side of the interface with the lower air–liquid surface tension. Once the intermediate film between the bubble and the interface has drained sufficiently, the bubble makes contact with the interface, forming a triple-line and producing strong capillary waves which travel around the bubble and can pinch off a satellite on the opposite side, akin to the dynamics in the coalescence cascade. We identify the critical Ohnesorge numbers where such satellites are produced and characterize their sizes. The total transition time scales with the bubble size and differential surface tension, while the satellite pinch-off time scales with the capillary-inertial time of the pool liquid, which originally surrounds the bubble. We also use high-speed video imaging to study the motion of the neck of the contact. For low viscosity we show that it grows in time with a power-law exponent between 0.44 and 0.50, with a prefactor modified by the net sum of the three interfacial tensions. Increasing the viscosity of the receiving liquid drop drastically slows down the motion of the triple-line, when the Ohnesorge number exceeds ${\sim }$0.08. This differs qualitatively from the coalescence of two miscible drops of different viscosities, where the lower viscosity sets the coalescence speed. We thereby propose a strong resistance from the triple-line.