13,644 research outputs found
Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble
The stability properties of the wake past an oblate spheroidal bubble held fixed in a uniform stream are studied in the framework of a global linear analysis. In line with previous studies, provided the geometric aspect ratio of the bubble, χ, is large enough, the wake is found to be unstable only within a finite range of Reynolds number, Re. The neutral curves corresponding to the occurrence of the first two unstable modes are determined over a wide range of the (χ, Re) domain and the structure of the modes encountered along the two branches of each neutral curve is discussed. Then, using an adjoint-based approach, a series of sensitivity analyses of the flow past the bubble is carried out in the spirit of recent studies devoted to twodimensionaland axisymmetric rigid bodies. The regions of the flow most sensitiveto an external forcing are found to be concentrated in the core or at the periphery of the standing eddy, as already observed with bluff bodies at the surface of which the flow obeys a no-slip condition. However, since the shear-free condition allows the fluid to slip along the bubble surface, the rear half of this surface turns out to be also significantly sensitive to disturbances originating in the shear stress, a finding which may be related to the well-known influence of surfactants on the structure and stability properties of the flow past bubbles rising in water
Bifurcations and symmetry breaking in the wake of axisymmetric bodies
We consider the generic problem of wake instabilities past fixed axisymmetric bodies, and focus on
the extreme cases of a sphere and a flat disk. Numerical results reveal that the wakes of these two
bodies evolve differently as the Reynolds number is increased. Especially, two new vortex shedding
modes are identified behind a disk. To interpret these results, we introduce a model based on the
theory of mode interactions in presence of O(2) symmetry. This model, which was initially developed for the Taylor–Couette system, allows us to explain the structural differences observed in the evolution of the two types of wakes and to accurately predict the evolution of the lift force
A vector field method for relativistic transport equations with applications
We adapt the vector field method of Klainerman to the study of relativistic
transport equations. First, we prove robust decay estimates for velocity
averages of solutions to the relativistic massive and massless transport
equations, without any compact support requirements (in or ) for the
distribution functions. In the second part of this article, we apply our method
to the study of the massive and massless Vlasov-Nordstr\"om systems. In the
massive case, we prove global existence and (almost) optimal decay estimates
for solutions in dimensions under some smallness assumptions. In the
massless case, the system decouples and we prove optimal decay estimates for
the solutions in dimensions for arbitrarily large data, and in
dimension under some smallness assumptions, exploiting a certain form of
the null condition satisfied by the equations. The -dimensional massive case
requires an extension of our method and will be treated in future work.Comment: 72 pages, 3 figure
The steady oblique path of buoyancy-driven disks and spheres
We consider the steady motion of disks of various thicknesses in a weakly viscous flow, in the case where the angle of incidence (defined as that between the disk axis and its velocity) is small. We derive the structure of the steady flow past the body and the associated hydrodynamic force and torque through a weakly nonlinear expansion of the flow with respect to . When buoyancy drives the body motion, we obtain a solution corresponding to an oblique path with a non-zero incidence by requiring the torque to vanish and the hydrodynamic and net buoyancy forces to balance each other. This oblique solution is shown to arise through a bifurcation at a critical Reynolds number ReSO which does not depend upon the body-to-fluid density ratio and is distinct from the critical Reynolds number ReSS corresponding to the steady bifurcation of the flow past the body held fixed with . We then apply the same approach to the related problem of a sphere that weakly rotates about an axis perpendicular to its path and show that an oblique path sets in at a critical Reynolds number ReSO slightly lower than ReSS , in agreement with available numerical studies
The onset of unsteadiness of two-dimensional bodies falling or rising freely in a viscous fluid: a linear study
We consider the transition between the steady vertical path and the oscillatory path of two-dimensional bodies moving under the effect of buoyancy in a viscous fluid. Linearization of the Navier–Stokes equations governing the flow past the body and of Newton’s equations governing the body dynamics leads to an eigenvalue problem, which is solved numerically. Three different body geometries are then examined in detail, namely a quasi-infinitely thin plate, a plate of rectangular cross-section with an aspect ratio of 8, and a rod with a square cross-section. Two kinds of eigenmodes are observed in the limit of large body-to-fluid mass ratios, namely ‘fluid’ modes identical to those found in the wake of a fixed body, which are responsible for the onset of vortex shedding, and four additional ‘aerodynamic’ modes associated with much longer time scales, which are also predicted using a quasi-static model introduced in a companion paper. The stability thresholds are computed and the nature of the corresponding eigenmodes is investigated throughout the whole possible range of mass ratios. For thin bodies such as a flat plate, the Reynolds number characterizing the threshold of the first instability and the associated Strouhal number are observed to be comparable with those of the corresponding fixed body. Other modes are found to become unstable at larger Reynolds numbers, and complicated branch crossings leading to mode switching are observed. On the other hand, for bluff bodies such as a square rod, two unstable modes are detected in the range of Reynolds number corresponding to wake destabilization. For large enough mass ratios, the leading mode is similar to the vortex shedding mode past a fixed body, while for smaller mass ratios it is of a different nature, with a Strouhal number about half that of the vortex shedding mode and a stronger coupling with the body dynamics
The Stability of the Minkowski space for the Einstein-Vlasov system
We prove the global stability of the Minkowski space viewed as the trivial
solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use
the vector field and modified vector field techniques developed in [FJS15;
FJS17]. In particular, the initial support in the velocity variable does not
need to be compact. To control the effect of the large velocities, we identify
and exploit several structural properties of the Vlasov equation to prove that
the worst non-linear terms in the Vlasov equation either enjoy a form of the
null condition or can be controlled using the wave coordinate gauge. The basic
propagation estimates for the Vlasov field are then obtained using only weak
interior decay for the metric components. Since some of the error terms are not
time-integrable, several hierarchies in the commuted equations are exploited to
close the top order estimates. For the Einstein equations, we use wave
coordinates and the main new difficulty arises from the commutation of the
energy-momentum tensor, which needs to be rewritten using the modified vector
fields.Comment: 139 page
Falling styles of disks
We numerically investigate the dynamics of thin disks falling under gravity in a viscous fluid medium at rest at infinity. Varying independently the density and thickness of the disk reveals the influence of the disk aspect ratio which, contrary to previous belief, is found to be highly significant as it may completely change the route to non-vertical paths as well as the boundaries between the various path regimes. The transition from the straight vertical path to the planar fluttering regime is found to exhibit complex dynamics: a bistable behaviour of the system is detected within some parameter range and several intermediate regimes are observed in which, although the wake is unstable, the path barely deviates from vertical. By varying independently the body-to-fluid inertia ratio and the relative magnitude of inertial and viscous effects over a significant range, we set up a comprehensive map of the corresponding styles of path followed by an infinitely thin disk. We observe the four types of planar regimes already reported in experiments but also identify two additional fully three-dimensional regimes in which the body experiences a slow horizontal precession superimposed onto zigzagging or tumbling motions
A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid
We consider the gravity-driven motion of a heavy two-dimensional rigid body freely falling in a viscous fluid.We introduce a quasi-static linear model of the forces and torques induced by the possible changes in the body velocity,or by the occurrence of a nonzero incidence angle or a spanwise rotation of the body. The coefficients involved in this model are accurately computed over a full range of Reynolds number by numerically resolving the Navier 13Stokes equations, considering three elementary situations where the motion of the body is prescribed. The falling body is found to exhibit three distinct eigenmodes which are always damped in the case of a thin plate with uniform mass loading or a circular cylinder,but may be amplified for other geometries,such as in the case of a square cylinder
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