36 research outputs found
A flow-pattern map for phase separation using the Navier-Stokes Cahn-Hilliard model
We use the Navier-Stokes-Cahn-Hilliard model equations to simulate phase
separation with flow. We study coarsening - the growth of extended domains
wherein the binary mixture phase separates into its component parts. The
coarsening is characterized by two competing effects: flow, and the
Cahn-Hilliard diffusion term, which drives the phase separation. Based on
extensive two-dimensional direct numerical simulations, we construct a
flow-pattern map outlining the relative strength of these effects in different
parts of the parameter space. The map reveals large regions of parameter space
where a standard theory applies, and where the domains grow algebraically in
time. However, there are significant parts of the parameter space where the
standard theory does not apply. In one region, corresponding to low values of
viscosity and diffusion, the coarsening is accelerated compared to the standard
theory. Previous studies involving Stokes flow report on this phenomenon; we
complete the picture by demonstrating that this anomalous regime occurs not
only for Stokes flow, but also, for flows dominated by inertia. In a second
region, corresponding to arbitrary viscosities and high Cahn-Hilliard
diffusion, the diffusion overwhelms the hydrodynamics altogether, and the
latter can effectively be ignored, in contrast to the prediction of the
standard scaling theory. Based on further high-resolution simulations in three
dimensions, we find that broadly speaking, the above description holds there
also, although the formation of the anomalous domains in the
low-viscosity-low-diffusion part of the parameter space is delayed in three
dimensions compared to two.Comment: 17 pages, 13 figure
Introduction of longitudinal and transverse Lagrangian velocity increments in homogeneous and isotropic turbulence
Based on geometric considerations, longitudinal and transverse Lagrangian
velocity increments are introduced as components along, and perpendicular to,
the displacement of fluid particles during a time scale {\tau}. It is argued
that these two increments probe preferentially the stretching and spinning of
material fluid elements, respectively. This property is confirmed (in the limit
of vanishing {\tau}) by examining the variances of these increments conditioned
on the local topology of the flow. Interestingly, these longitudinal and
transverse Lagrangian increments are found to share some qualitative features
with their Eulerian counterparts. In particular, direct numerical simulations
at turbulent Reynolds number up to 300 show that the distributions of the
longitudinal increment are negatively skewed at all {\tau}, which is a
signature of time irreversibility of turbulence in the Lagrangian framework.
Transverse increments are found more intermittent than longitudinal increments,
as quantified by the comparison of their respective flatnesses and scaling
laws. Although different in nature, standard Lagrangian increments (projected
on fixed axis) exhibit scaling properties that are very close to transverse
Lagrangian increments
Multiscale analysis of the structure of homogeneous rotating turbulence
International audienceThe structure of homogeneous rotating turbulence at moderate Reynolds number is investigated by analyzing the instantaneous statistics of the scale-dependent velocity gradient tensor perceived by a set of four fluid elements equally spaced. The relative orien-tations between dynamical vectors such as vorticity, rate-of-strain eigenframe, and vortex stretching vector, together with their orientations with the rotating frame, are measured by direct numerical simulation at different rotation rates. Measurements are performed in the entire inertial range of scales. The preferential orientation of turbulence with the rotating frame is found to be maximal at the scale of the horizontal large structures of the flow. The relative orientations between dynamical vectors exhibit a continuous and monotonic evolution with scale. Overall, the orientation properties reflect the Gaussianization and two-dimensionalization of turbulence under the effect of rotation. In particular, rotation suppresses some alignment properties valid in isotropic turbulence, which in turn induces a strong decrease of the enstrophy production and strain production rates. These results are found to be valid at all scales
Small-scale anisotropy induced by spectral forcing and by rotation in non-helical and helical turbulence
We study the effect of large-scale spectral forcing on the scale-dependent
anisotropy of the velocity field in direct numerical simulations of homogeneous
incompressible turbulence. Two forcing methods are considered: the steady ABC
single wavenumber scheme and the unsteady non-helical or helical Euler scheme.
The results are also compared with high resolution data obtained with the
negative viscosity scheme. A fine-grained characterization of anisotropy,
consisting in measuring some quantities related to the two-point velocity
correlations, is used: we perform a modal decomposition of the spectral
velocity tensor into energy, helicity and polarization spectra. Moreover, we
include the explicit dependence of these three spectra on the wavevector
direction. The conditions that allow anisotropy to develop in the small scales
due to forcing alone are clearly identified. It is shown that, in turbulent
flows expected to be isotropic, the ABC forcing yields significant energy and
helicity directional anisotropy down to the smallest resolved scales, like the
helical Euler scheme when an unfavourable forcing scale is used. The
direction-and scale-dependent anisotropy is then studied in rotating
turbulence. It is first shown that, in the ABC-forced simulations the slope of
the energy spectrum is altered and the level of anisotropy is similar to that
obtained at lower Rossby number in Euler-forced runs, a result due both to the
nature of the forcing itself and to the fact that it allows an inverse cascade
to develop. Second, we show that, even at low rotation rate, the natural
anisotropy induced by the Coriolis force is visible at all scales. Finally, we
identify two different wavenumber ranges in which anisotropy behaves
differently, and show that if the Rossby number is not too low the
characteristic lenghscale separating them is the one at which rotation and
dissipation effects balance
Flow-parametric regulation of shear-driven phase separation in two and three dimensions
The Cahn-Hilliard equation with an externally-prescribed chaotic shear flow
is studied in two and three dimensions. The main goal is to compare and
contrast the phase separation in two and three dimensions, using
high-resolution numerical simulation as the basis for the study. The model flow
is parametrized by its amplitudes (thereby admitting the possibility of
anisotropy), lengthscales, and multiple time scales, and the outcome of the
phase separation is investigated as a function of these parameters as well as
the dimensionality. In this way, a parameter regime is identified wherein the
phase separation and the associated coarsening phenomenon are not only arrested
but in fact the concentration variance decays, thereby opening up the
possibility of describing the dynamics of the concentration field using the
theories of advection diffusion. This parameter regime corresponds to long flow
correlation times, large flow amplitudes and small diffusivities. The onset of
this hyperdiffusive regime is interpreted by introducing Batchelor
lengthscales. A key result is that in the hyperdiffusive regime, the
distribution of concentration (in particular, the frequency of extreme values
of concentration) depends strongly on the dimensionality. Anisotropic scenarios
are also investigated: for scenarios wherein the variance saturates
(corresponding to coarsening arrest), the direction in which the domains align
depends on the flow correlation time. Thus, for correlation times comparable to
the inverse of the mean shear rate, the domains align in the direction of
maximum flow amplitude, while for short correlation times, the domains
initially align in the opposite direction. However, at very late times (after
the passage of thousands of correlation times), the fate of the domains is the
same regardless of correlation time, namely alignment in the direction of
maximum flow amplitude.Comment: 27 pages, 14 figure
Buoyancy-driven bubbly flows: ordered and free rise at small and intermediate volume fraction
International audienceVarious expressions have been proposed previously for the rise velocity of gas bubbles for homogeneous steady bubbly flows, generally a monotonically decreasing function of the bubble volume fraction. For suspensions of freely moving bubbles, some of these are of the form expected for ordered arrays of bubbles, and vice versa, as they do not reduce to the behaviour expected theoretically in the dilute limit. The microstructure of weakly inhomogeneous bubbly flows not being known generally, the effect of microstructure is an important consideration. We revisit this problem here for bubbly flows at small to moderate Reynolds number values for deformable bubbles, using direct numerical simulation and analysis. For ordered suspensions, the rise velocity is demonstrated not to be monotonically decreasing with volume fraction due to cooperative wake interactions. The fore-and-aft asymmetry of an isolated ellipsoidal bubble is reversed upon increasing the volume fraction, and the bubble aspect ratio approaches unity. Recent work on rising bubble pairs is used to explain most of these results; the present work therefore forms a platform of extending the former to suspensions of many bubbles. We adopt this new strategy also to support the existence of the oblique rise of ordered suspensions, the possibility of which is also demonstrated analytically. Finally, we demonstrate that most of the trends observed in ordered systems also appear in freely evolving suspensions. These similarities are supported by prior experimental measurements, and attributed to the fact that free bubbles keep the same neighbours for extended periods of time
The interaction between a solid particle and a turbulent flow \ud
The interaction between a fixed solid spherical particle and stationary turbulence with zero mean flow is investigated numerically. The object diameter, D, lies in the inertial range (Dâ0.6Lâ0.9λâ8η, where L, λ and η, respectively, denote the integral scale, the Taylor microscale and the Kolmogorov length) and the particle Reynolds number is close to 20. It is found that the turbulence statistics at different distances from the solid/fluid interface are modified by the presence of the object in a region that extends more than 10 times further than the viscous layer. This estimate is confirmed by the analysis of the correlation between the force and torque on the particle and the force and torque on spherical surfaces surrounding the particle, although the torque decorrelates somewhat faster with increasing distance from the object surface. The angular slip velocity of the particle, a quantity of crucial importance for the modeling of the turbulent transport of large objects, is also characterized.\ud
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Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined
A simplified thermodynamic approach of the incompressible axisymmetric Euler
equations is considered based on the conservation of helicity, angular momentum
and microscopic energy. Statistical equilibrium states are obtained by
maximizing the Boltzmann entropy under these sole constraints. We assume that
these constraints are selected by the properties of forcing and dissipation.
The fluctuations are found to be Gaussian while the mean flow is in a Beltrami
state. Furthermore, we show that the maximization of entropy at fixed helicity,
angular momentum and microscopic energy is equivalent to the minimization of
macroscopic energy at fixed helicity and angular momentum. This provides a
justification of this selective decay principle from statistical mechanics.
These theoretical predictions are in good agreement with experiments of a von
Karman turbulent flow and provide a way to measure the temperature of
turbulence and check Fluctuation-Dissipation Relations (FDR). Relaxation
equations are derived that could provide an effective description of the
dynamics towards the Beltrami state and the progressive emergence of a Gaussian
distribution. They can also provide a numerical algorithm to determine maximum
entropy states or minimum energy states.Comment: 25 pages, 2 figure
Direct numerical simulation of axisymmetric turbulence
International audienceThe dynamics of decaying strictly axisymmetric, incompressible turbulence is investigated using direct numerical simulations. It is found that the angular momentum is a robust invariant of the system. It is further shown that long-lived coherent structures are generated by the flow, associated with stationary solutions of the Euler equations. The structures obey relations in agreement with predictions from selective decay principles, compatible with the decay laws of the system. Two different types of decay scenarios are highlighted. The first case results in a quasi-two-dimensional flow with a dynamical behaviour in the poloidal plane similar to freely decaying two-dimensional turbulence. In a second regime, the long-time dynamics is dominated by a single three-dimensional mode