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

Linear and nonlinear dynamics of axisymmetric waves in the hollow core vortex

The dynamics of trailing vortices are under constant investigation during last decades since it is of considerable interest to reduce aircraft wakes and associated hazards to forthcoming planes. The isolated axisymmetric vortex is the commonly used simplest elementary model when considering such issue. Although asymptotically stable, recent studies have revealed its sensitiveness to specific perturbations, leading in some cases to considerable gains of energy. Albeit of evident interest, the underlying mechanisms of energy growth are considered in the linear regime. The nonlinear dynamics of such vortices need also to be considered in order to complete the picture. Rather than performing direct numerical simulations3, an interesting way to investigate it is to consider the nonlinear interactions of waves. This approach is motivated by the possible existence of resonance between wave components. For this purpose, the base flow model is simplified by considering the hollow core vortex. Arising naturally when a tank is drained (bath-tube vortex), it presents simpler dynamics than the Lamb-Oseen vortex as it only possesses two families of waves. This point is of crucial importance for the tractability of the problem. In this work, the nonlinear temporal evolution of axisymmetric waves are investigated through numerical integration when the flow is submitted to various initial conditions (travelling or standing wave, pinching of the free surface, wave trains). We focus on wave trains as important energy exchanges between the main component and its sideband waves are observed. This phenomenon is related to the Benjamin-Feir instability4 (triadic resonance) occurring for wave trains on deep water

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

Stochastic forcing of the Lamb–Oseen vortex

The aim of the present paper is to analyse the dynamics of the Lamb–Oseen vortex when continuously forced by a random excitation. Stochastic forcing is classically used to mimic external perturbations in realistic configurations, such as variations of atmospheric conditions, weak compressibility effects, wing-generated turbulence injected in aircraft wake, or free-stream turbulence in wind tunnel experiments. The linear response of the Lamb–Oseen vortex to stochastic forcing can be decomposed in relation to the azimuthal symmetry of the perturbation given by the azimuthal wavenumber m. In the axisymmetric case m = 0, we find that the response is characterised by the generation of vortex rings at the outer periphery of the vortex core. This result is consistent with recurrent observations of such dynamics in the study of vortex-turbulence interaction. When considering helical perturbations m = 1, the response at large axial wavelengths consists of a global translation of the vortex, a feature very similar to the phenomenon of vortex meandering (or wandering) observed experimentally, corresponding to an erratic displacement of the vortex core. At smaller wavelengths, we find that stochastic forcing can excite specific oscillating modes of the Lamb–Oseen vortex. More precisely, damped critical-layer modes can emerge via a resonance mechanism. For perturbations with higher azimuthal wavenumber m > 2, we find no structure that clearly dominates the response of the vortex

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

3D optimal perturbations developing in homogeneous mixing layers in presence of subharmonic vortex-pairing

Many experimental and numerical studies have found that the pairing of primary Klevin-Helmholtz (KH) vortices in mixing layers generally inhibits the growth of infinintesimal three-dimensional disturbances, delaying the transition to turbulence. In this work, we investigate the existence of 3D perturbations that grow fast enough to survive the subharmonic merging instability. For this purpose, we perform a numerical study of the transient linear evolution of 3D perturbations emerging in a homogeneous time-evolving mixing layer which undergoes pairing. We look at the optimal perturbation that yields the largest gain of energy at a specific time horizon, by use of an optimization method which solves iteratively the linearized direct and adjoint Navier-Stokes equations. In particular, we consider the influence of the time horizon relative to the saturation time of both the primary KH and subharmonic pairing instabilities

The combined Lagrangian advection method

We present and test a new hybrid numerical method for simulating layerwise-two-dimensional geophysical flows. The method radically extends the original Contour-Advective Semi-Lagrangian (CASL) algorithm by combining three computational elements for the advection of general tracers (e.g. potential vorticity, water vapor, etc.): (1) a pseudospectral method for large scales, (2) Lagrangian contours for intermediate to small scales, and (3) Lagrangian particles for the representation of general forcing and dissipation. The pseudo-spectral method is both efficient and highly accurate at large scales, while contour advection is efficient and accurate at small scales, allowing one to simulate extremely finescale structure well below the basic grid scale used to represent the velocity field. The particles allow one to efficiently incorporate general forcing and dissipation

Vortical control of forced two-dimensional turbulence

A new numerical technique for the simulation of forced two-dimensional turbulence (Dritschel and Fontane, 2010) is used to examine the validity of Kraichnan-Batchelor scaling laws at higher Reynolds number than previously accessible with classical pseudo-spectral methods,making use of large simulation ensembles to allow a detailed consideration of the inverse cascade in a quasi-steady state. Our results support the recent finding of Scott (2007), namely that when a direct enstrophy cascading range is well-represented numerically, a steeper energy spectrum proportional to k^(−2) is obtained in place of the classical k^(−5/3) prediction. It is further shown that this steep spectrum is associated with a faster growth of energy at large scales, scaling like t^(−1) rather than Kraichnan’s prediction of t^(−3/2). The deviation from Kraichnan’s theory is related to the emergence of a population of vortices that dominate the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum. An analytical model is proposed which closely matches the numerical spectra between the large scales and the forcing scale