Onset of secondary instabilities on the zigzag instability in stratified fluids (original) (raw)
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Journal of Fluid Mechanics, 2008
We present high-resolution direct numerical simulations of the nonlinear evolution of a pair of counter-rotating vertical vortices in a stratified fluid for various high Reynolds numbers Re and low Froude numbers F h . The vortices are bent by the zigzag instability producing high vertical shear. There is no nonlinear saturation so that the exponential growth is stopped only when the viscous dissipation by vertical shear is of the same order as the horizontal transport, i.e. when Z h
Journal of Fluid Mechanics, 2006
This paper investigates the three-dimensional stability of a pair of co-rotating vertical vortices in a rotating strongly stratified fluid. In a companion paper , we have shown that such a basic flow in a strongly stratified fluid is affected by a zigzag instability which bends the two vortices symmetrically. In the non-rotating flow, the most unstable wavelength of this instability scales as the buoyancy length and its growth rate scales as the external strain that each vortex induces on the other one. Here, we show that the zigzag instability remains active whatever the magnitude of the planetary rotation and is therefore connected to the tallcolumn instability in quasi-geostrophic fluids. Its growth rate is almost independent of the Rossby number. The most amplified wavelength follows the universal scaling λ = 2πF h b γ 1 /Ro 2 + γ 2 /Ro + γ 3 , where b is the separation distance between the two vortices, (γ 1 , γ 2 , γ 3 ) are constants, F h is the horizontal Froude number and Ro the Rossby number (F h = Γ /πa 2 N, Ro = Γ /πa 2 f , where Γ is the circulation of each vortex, a the vortex radius, N the Brunt-Väisälä frequency and f the Coriolis parameter). When Ro = ∞, the scaling λ ∝ F h b found in the companion paper Otheguy et al. is recovered. When Ro → 0, λ ∝ bf/N in agreement with the quasi-geostrophic theory. In contrast to previous results, the wavelength is found to depend on the separation distance between the two vortices b, and not on the vortex radius a.
Journal of Fluid Mechanics, 2000
This paper shows that a long vertical columnar vortex pair created by a double flap apparatus in a strongly stratified fluid is subjected to an instability distinct from the Crow and short-wavelength instabilities known to occur in homogeneous fluid. This new instability, which we name zigzag instability, is antisymmetric with respect to the plane separating the vortices. It is characterized by a vertically modulated twisting and bending of the whole vortex pair with almost no change of the dipole's cross- sectional structure. No saturation is observed and, ultimately, the vortex pair is sliced into thin horizontal layers of independent pancake dipoles. For the largest Brunt–Väisälä frequency N = 1.75 rad s−1 that may be achieved in the experiments, the zigzag instability is observed only in the range of Froude numbers: 0.13 < Fh0 < 0.21 (Fh0 = U0/NR, where U0 and R are the initial dipole travelling velocity and radius). When Fh0 > 0.21, the elliptic instability develop...
Fluid Dynamics Research, 1997
The linear instability of a steady elliptical vortex in a stably stratified rotating fluid is investigated, using the quasi-geostrophic, f-plane approximation. The vortex is embedded in a uniform background straining field e with uniform vorticity 27 (the Moore-Saffman vortices). An elliptical vortex in an irrotational strain field (7 = 0) is shown to be unstable to long wave quasi-three-dimensional disturbances of azimuthal wavenumber m = 1 (bending wave). The long wave instability has a two-dimensional origin. An elliptical vortex in a simple shear, whose major axis is parallel to the shear streamlines (e = 7), is stable against any disturbance. In contrast, an ellipse in a simple shear, whose major axis is perpendicular to the shear streamlines (e =-7), is unstable to quasi-three-dimensional bending modes irrespective of a/b. Short wave quasi-three-dimensional disturbances grow faster than two-dimensional instability modes in the parameter range-0.5 < 7 < 0. The origin of short wave instability is attributed to resonance between inertial waves and the imposed strain field e.
Three-dimensional stability of a vertical columnar vortex pair in a stratified fluid
Journal of Fluid Mechanics, 2000
This paper investigates the three-dimensional stability of a Lamb–Chaplygin columnar vertical vortex pair as a function of the vertical wavenumber kz, horizontal Froude number Fh, Reynolds number Re and Schmidt number Sc. The horizontal Froude number Fh (Fh = U/NR, where U is the dipole travelling velocity, R the dipole radius and N the Brunt–Väisälä frequency) is varied in the range [0.033, ∞[ and three set of Reynolds-Schmidt numbers are investigated: {Re = 10 000, Sc = 1}, Re = 1000, Sc = 1}, {Re = 200, Sc = 637}. In the whole range of Fh and Re, the dominant mode is always antisymmetric with respect to the middle plane between the vortices but its physical nature and properties change when Fh is varied. An elliptic instability prevails for Fh > 0.25, independently of the Reynolds number. It manifests itself by the bending of each vortex core in the opposite direction to the vortex periphery. The growth rate of the elliptic instability is reduced by stratification effects but ...
5th AIAA Theoretical Fluid Mechanics Conference, 2008
An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investigations on the three-dimensionnal instabilities and transient growth of such dipole are performed. By means of a three-dimensionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic modes but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of perturbations on this dipole, investigated by computing the optimal linear perturbations with a direct-adjoint technique, demonstrates the crucial role of the region of maximal strain at short time and of the hyperbolic point at intermediate time. Investigations on the three-dimensionnal dynamics of trailing vortices in stratified fluids are performed. The elliptic instability is almost unaffected by weak and moderate stratifications. Résumé : Le sillage d'un avion est constitué d'un paire de tourbillons contra-rotatifs et est affectée par une instabilité à grande (Crow) et à petite (elliptique) longueur d'onde. On réalise des études numériques sur les instabilités tridimensionnelles d'un tel dipole. Par une étude de stabilité linéaire tridimensionnelle, on retrouve les bandes d'instabilité correspondant aux modes de Crow et elliptiques mais on observe également des modes oscillants moins instables avec des pics très larges. Les croissances transitoires des perturbations sur ce dipole, qui sont étudiées en calculant les perturbations optimales linéaires par une technique direct-adjoint, démontrent le rôle crucial de la région où l'étirement est maximal aux temps courts et du point hyperbolique aux temps intermédiaires. Des études sur la dynamique tridimensionnelle des tourbillons de sillage d'avion en fluide stratifié sont réalisées. L'instabilité elliptique n'est pratiquement pas affectée par des stratifications faible et modérée.