Decay of semilinear damped wave equations:cases without geometric control condition

We consider the semilinear damped wave equation $\partial_{tt}^2 u(x,t)+\gamma(x)\partial_t u(x,t)=\Delta u(x,t)-\alpha u(x,t)-f(x,u(x,t))$. In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $\gamma$ does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that $\|e^{At}A^{-1}\|\leq h(t)$ for some function $h$ with $h(t)\rightarrow 0$ when $t\rightarrow +\infty$. We provide general tools to deal with the semilinear stabilization problem in the case where $h(t)$ has a sufficiently fast decay

The stability of the variable-density Kelvin-Helmholtz billow

We perform a three-dimensional stability analysis of the Kelvin-Helmholtz billow, developing in a shear-layer between two fluids with different density. We begin with two-dimensional simulations of the temporally evolving mixing-layer yielding the unsteady base flow fields. The Reynolds number is 1500 while the Schmidt and Froude numbers are infinite. Then exponentially unstable modes are extracted from a linear stability analysis performed at the saturation of the primary mode kinetic energy. The spectrum of the least stable modes exhibits two main classes. The first class comprises three-dimensional core-centred and braid-centred modes already present in the homogeneous case. The baroclinic vorticity concentration in the braid lying on the light side of the KH-billow turns the flow into a sharp vorticity ridge holding high shear levels. The hyperbolic modes benefit from the enhanced level of shear in the braid while elliptic ones remain quite insensitive to the modifications of the base flow. In the second class, we found typical two-dimensional modes resulting from a shear instability of the curved vorticity-enhanced braid. For a density contrast of 0.5, the wavelength of the two-dimensional instability is about ten times shorter than the one of the primary wave. Its amplification rate competes well against the ones of the hyperbolic three-dimensional modes. The vorticity-enhanced braid thus becomes the preferred location for the development of secondary instabilities. This stands as the key feature of the transition of the variable-density mixing layer. We carry out a fully resolved numerical continuation of the nonlinear development of the two-dimensional braid-mode. Secondary roll-ups due to a small-scale Kelvin-Helmholtz mechanism are promoted by the underlying strain field and develop rapidly in the compression part of the braid. Originally analysed in Reinaud et al. (2000) from two-dimensional non-viscous numerical simulations, this instability is shown to substantially increase the mixing

The baroclinic forcing of the shear-layer three-dimensional instability

It has been demonstrated that, within the context of variable-density shear flows, the generation-destruction of vorticity by the baroclinic torque may substantially alter the transition dynamics of shear flows. The focus of the present contribution is on baroclinic effects beyond the Boussinesq approximation but uncorrelated to compressibility. The baroclinic torque results from the inertial component of the pressure gradient only. The vorticity evolves within a quasi-solenoidal velocity field without suffering from strong dilatationnal effects that scale with any relevant Mach number. This purely inertial influence of density variations is likely to occur in high Reynolds number mixing of fluids of different densities or in thermal mixing. The vorticity is redistributed to the benefits of the light-side vorticity braid, the other being vorticity depleted in a first stage and feeded with an opposite sign vorticity afterwards, as stressed by Reinaud et al. (1999). These two opposite-sign vorticity sheets are lying around the vanishing primary structure core, still figuring the center of this two-layers system. In three-dimensions the vorticity dynamics is also affected by the vortex stretching mechanism that enable circulation to travel among vorticity components through 3D instability modes. The consequences of the baroclinic redistribution of spanwise vorticity on the development of three-dimensionnal modes is the focus point of the present proposition. The interference with the pairing process and further subharmonics emergence is not yet considered

Density fluctuation correlations in free turbulent binary mixing

This paper is devoted to the analysis of the turbulent mass flux and, more generally, of the density fluctuation correlation (d.f.c.) effects in variable-density fluid motion. The situation is restricted to the free turbulent binary mixing of an inhomogeneous round jet discharging into a quiescent atmosphere. Based on conventional (Reynolds) averaging, a ternary regrouping of the correlations occurring in the statistical averaging of the open equations is first introduced. Then an exact algebraic relationship between the d.f.c. terms and the second-order moments is demonstrated. Some consequences of this result on the global behaviour of variable-density jets are analytically discussed. The effects of the d.f.c. terms are shown to give a qualitative explanation of the influence of the ratio of the densities of the inlet jet and ambient fluid on the centerline decay rates of mean velocity and mass fraction, the entrainment rate and the restructuring of the jet. Finally, the sensitivity of second-order modelling to the d.f.c. terms is illustrated and it is suggested that such terms should be considered as independent variables in the closing procedure

Fractal Kelvin-Helmholtz breakups

The Kelvin–Helmholtz billow developing in an infinite- Schmidt number mixing layer at Re=1500 between two density-contrasted fluids experiences a two-dimensional shear instability. Secondary Kelvin–Helmholtz billows are seen to emerge on the light side of the primary structure, and then are advected towards the core of the main billow as the wave overturns. Due to the inertial baroclinic vorticity production, the braid region turns into a sharp vorticity ridge holding high shear levels and is thus sensitized to the Kelvin–Helmholtz instability. We carry out numerical simulations of the temporal development of the secondary mode when the flow is seeded at t=18 with the perturbation obtained from a linear stability analysis of the primary billow

The Rayleigh–Taylor instability of two-dimensional high-density vortices

We investigate the stability of variable-density two-dimensional isolated vortices in the frame of incompressible mixing under negligible gravity. The focus on a single vortex flow stands as a first step towards vortex interactions and turbulent mixing. From heuristic arguments developed on a perturbed barotropic vortex, we find that highdensity vortices are subject to a Rayleigh–Taylor instability. The basic mechanism relies on baroclinic vorticity generation when the density gradient is misaligned with the centripetal acceleration field. For Gaussian radial distributions of vorticity and density, the intensity of the baroclinic torque due to isopycnic deformation is shown to increase with the ratio δ/δρ of the vorticity radius to the density radius. Concentration of mass near the vortex core is confirmed to promote the instability by the use of an inviscid linear stability analysis. We measure the amplification rate for the favoured azimuthal wavenumbers m=2, 3 on the whole range of positive density contrasts between the core and the surroundings. The separate influence of the density-contrast and the radius ratio is detailed for modes up to m=6. For growing azimuthal wavenumbers, the two-dimensional structure of the eigenmode concentrates on a ring of narrowing radial extent centred on the radius of maximum density gradient. The instability of the isolated high-density vortex is then explored beyond the linear stage based on high-Reynolds-number numerical simulations for modes m=2,3 and a moderate density contrast Cρ =0.5. Secondary roll-ups are seen to emerge from the nonlinear evolution of the vorticity and density fields. The transition towards m smaller vortices involves vorticity exchange between initially-rotating dense fluid particles and the irrotational less-dense medium. It is shown that baroclinic enstrophy production is associated with the centrifugal mass ejection away from the vortex centre

The structure of some variable-density shear flows

The following discussion focuses on the influence of density contrasts on the development of basic shear flows. It is expected, as usual, that what is learnt from these prototype flows can be transposed, at least locally, to more complex geometries. The specific features of these variable-density flows are best accounted for as seen from their vorticity dynamics. The baroclinic torque, connecting misaligned pressure and density gradients, reorganizes the vorticity field according to the fluid inertia. It gives rise to thinner vorticity patches thus shifting enstrophy toward higher wave numbers. It also changes the stability characteristics and the transition to turbulence of laminar shear flows. While frequently commented in geophysical flows under the Boussinesq approximation, it nonetheless deserves a particular attention in the context of inertia-dominated flows encountered in high Reynolds number mixing situations. The baroclinic torque is introduced after a short literature survey. Then the particular cases of the mixing layer and the jet are examined. The two-dimensional and some three-dimensional aspects are documented based on temporally and spatially developing numerical simulations

Large scale simulation of turbulence using a hybrid spectral/finite difference solver

Performing Direct Numerical Simulation (DNS) of turbulence on large-scale systems (offering more than 1024 cores) has become a challenge in high performance computing. The computer power increase allows now to solve flow problems on large grids (with close to 10^9 nodes). Moreover these large scale simulations can be performed on non-homogeneous turbulent flows. A reasonable amount of time is needed to converge statistics if the large grid size is combined with a large number of cores. To this end we developed a Navier-Stokes solver, dedicated to situations where only one direction is heterogeneous, and particularly suitable for massive parallel architecture. Based on an hybrid approach spectral/finite-difference, we use a volumetric decomposition of the domain to extend the FFTs computation to a large number of cores. Scalability tests using up to 32K cores as well as preliminary results of a full simulation are presented