Sabine Ortiz

Spatial Holmboe instability

In mixing-layers between two parallel streams of different densities, shear and gravity effects interplay; buoyancy acts as a restoring force and the Kelvin-Helmholtz mode is known to be stabilized by the stratification. If the density interface is sharp enough, two new instability modes, known as Holmboe modes, appear, propagating in opposite directions. This mechanism has been studied in the temporal instability framework. The present paper analyzes the associated spatial instability problem. It considers, in the Boussinesq approximation, two immiscible inviscid fluids with a piecewise linear broken-line velocity profile. We show how the classical scenario for transition between absolute and convective instability should be modified due to the presence of propagating waves. In the convective region, the spatial theory is relevant and the slowest propagating wave is shown to be the most spatially amplified, as suggested by intuition. Predictions of spatial linear theory are compared with mixing-layer [C.G. Koop and F.K. Browand, J. Fluid Mech. 93, 135 (1979)] and exchange flow [G. Pawlak and L. Armi, J. Fluid Mech. 376, 1 (1999)] experiments. The physical mechanism for Holmboe mode destabilization is analyzed via an asymptotic expansion that predicts the absolute instability domain at large Richardson number.

Three-dimensional instabilities and transient growth of a counter-rotating vortex pair

This paper investigates the three-dimensional instabilities and the transient growth of perturbations on a counter-rotating vortex pair. The two dimensional base flow is obtained by a direct numerical simulation initialized by two Lamb–Oseen vortices that quickly adjust to a flow with elliptic vortices. In the present study, the Reynolds number, ReΓ=Γ/ν, with Γ the circulation of one vortex and ν the kinematic viscosity, is taken large enough for the quasi steady assumption to be valid. Both the direct linearized Navier–Stokes equation and its adjoint are solved numerically and used to investigate transient and long time dynamics. The transient dynamics is led by different regions of the flow, depending on the optimal time considered. At very short times compared to the advection time of the dipole, the dynamics is concentrated on the points of maximal strain of the base flow, located at the periphery of the vortex core. At intermediate times, depending on the symmetry of the perturbation, one of the hype...

Transient growth of secondary instabilities in parallel wakes: Anti lift-up mechanism and hyperbolic instability

This paper investigates the three-dimensional temporal instabilities and the transient growth of perturbations on a Von Karman vortex street, issuing from the development of the primary instability of a parallel Bickley velocity profile typical of a wake forming behind a thin flat plate. By solving iteratively the linearized direct Navier Stokes equations and its adjoint equations, we compute the optimal perturbations that exhibit the largest transient growth of energy between the initial instant and different time horizons. At short time horizons, optimal initial perturbations are concentrated on the points of maximal strain of the base flow. The optimal gain of energy and the mechanism of instability are well predicted by local theories that describe the lagrangian evolution of a perturbation wave packet. At time of order unity, hyperbolic region leads the dynamics. Only at large time (t ? 20), the growth is led by the most amplified eigenmode. This eigenmode evolves, when the wavenumber increases, from perturbation centred in the core of the vortices, to perturbations localised on the stretching manifold of the hyperbolic points. At finite and large time, the gain in energy is initially associated with a mechanism reminiscent to the anti lift-up mechanism described by Antkowiak and Brancher [J. Fluid Mech. 578, 295 (2007)] in the context of an axisymmetric vortex. Presently, the optimal initial condition (the adjoint modes at large time) corresponding to streamwise streaks localised on the contracting manifold of the hyperbolic point induces streamwise vortices aligned with the stretching manifold of the hyperbolic point (the direct modes). The localisation on distinct manifolds of direct and adjoint eigenmodes is more pronounced when the Reynolds number is increased. An interpretation is proposed based on a balance between diffusion and stretching effects that predicts the thickness of the energy containing region for the adjoint and the direct mode decreasing as 1/?Re. The extra gain of energy due to non normal effects grows, since direct and adjoint modes are localised in different regions of space, i.e., the stretching and contracting manifold, a novel effect of the so called convective non normality associated with the transport of the perturbation by the base flow.

Stability of Quasi Two-Dimensional Vortices

Large-scale coherent vortices are ubiquitous features of geophysical flows. They have been observed as well at the surface of the ocean as a result of meandering of surface currents but also in the deep ocean where, for example, water flowing out of the Mediterranean sea sinks to about 1000 m deep into the Atlantic ocean and forms long-lived vortices named Meddies (Mediterranean eddies).

Three-dimensional instabilities and optimal perturbations of a counter-rotating vortex pair in stratified flows

This paper investigates the three-dimensional instabilities and the optimal perturbations on a pair of horizontal counter-rotating Lamb-Oseen vortices in a vertically stably stratified flow. Two-dimensional (2D) simulations are first performed, showing that while the dipole moves vertically against the stratification the vortex parameters: the radius a ∗, the separation distance b ∗, and the circulation Γ∗ are solely function of the time rescaled by the Brunt-Vaisala frequency N, independently of the Froude number, when the Reynolds number is large enough. Here, the Froude number is Fr = W 0/Nb 0 with W 0 the initial advection velocity of the dipole and b 0 the initial separation distance between the two vortices. For weak and moderate stratifications (large Fr), the stratification acts on a long time scale compared to the advection time of the dipole implying that the 2D flow can be considered as quasi-steady. In that case, when three dimensional instabilities are added, a linear stability analysis of this 2D flow at different instants retrieves the instability peaks corresponding to the Crow instability for the long wavelengths and to the elliptic instability for the short wavelengths showing that the dynamics is almost unaffected by buoyancy effects. The Crow and elliptic instabilities scale with the instantaneous dipole parameters showing in particular that stratification promotes instability by reducing the distance b ∗ between vortices [K. K. Nomura et al., “Short-wavelength instability and decay of a vortex pair in stratified fluid,” J. Fluid Mech. 553, 283-322 (2006)]. For strong stratifications (Froude numbers of order unity or smaller), the quasi-steady approximation is not valid, and the question of stability should be formulated in a different way, by, for example, searching for the transient growth of the energy of perturbation that may be computed for steady or unsteady base flow. Then, for each time horizon τ, we should determine the critical perturbation leading to the largest energy growth by the time τ. Presently, we compute the optimal perturbations at two time horizons τ = 4 and τ = 10 dimensionalized by with a direct-adjoint technique which takes into account the evolution of the base flow. In the homogeneous case, this technique allows to investigate the effect of the weak unsteadiness of the flow due to viscous diffusion which induces a growth of the vortex core radius a ∗. Both Crow and elliptic instabilities are retrieved in the optimal response and in the energy gain curves. Even if very slow, the viscous diffusion is found to increase the gain of the antisymmetric elliptic perturbation compared to the symmetric one. When the fluid is stratified, peaks at small wavenumber and at wavenumber of the order of the vortex core size are found for all Froude numbers with optimal responses strongly resembling, respectively, the Crow and the elliptic modes with optimal gains corresponding to mean growth rates larger than in the homogeneous case for both modes. However, as the strength of stratification increases (Froude numbers smaller than 2), optimal perturbations start departing from their homogeneous counterpart with large perturbation in the wake of the dipole associated with density effects. 2πb20/Γ0 © 2015 AIP Publishing LLC

Transient evolution and high stratification scaling in horizontal mixing layers

Mixing layers (sheared flows in homogeneous or stratified fluid) are present in many geophysical contexts and may lead to turbulence and mixing. In several cases, mixing layers are known to exhibit the Kelvin-Helmholtz instability leading to the roll-up of spanwise vortices, the Kelvin-Helmholtz (KH) billows. This is an essentially two-dimensional (2D) process. In fact, in the homogeneous cases the Squire’s theorem implies that the most unstable mode is 2D. However, Squire’s theorem applies only for the exponentially growing perturbations that control the large time dynamics and is not valid for the transient dynamics at short time. Indeed, Iams et al.[1] have shown that, in the non-stratified case, the most amplified optimal perturbations for short times are three-dimensional (3D) and result from a cooperation between the lift-up and Orr mechanisms[2]. This provides a finite time mechanism for spanwise scale selection, scale that may persist at later times if nonlinearities are strong enough.

Inviscid Transient Growth on Horizontal Shear Layers with Strong Vertical Stratification

We report an investigation of the three-dimensional stability of an horizontal free shear layer in an inviscid fluid with strong, vertical and constant density stratification. We compute the optimal perturbations for different optimization times and wavenumbers. The results allow comparing the potential for perturbation energy amplification of the free shear layer instability and the different mechanisms of transient growth. We quantify the internal wave energy content of the perturbations and identify different types of optimal perturbations. Intense excitation of gravity waves due to transient growth of perturbations is found in a broad region of the wavevector plane. Those gravity waves are eventually emitted away from the shear layer.

Instabilities and transient growth of trailing vortices in stratified fluid

The wake, which forms behind an aircraft due to its lift, is a pair of horizontal counter-rotating vortices propagating downwards. Depending on the atmospheric conditions, such dipole can persist over a long time or be rapidely destroyed. This vortex pair, in homogeneous fluids, is unstable with respect to three-dimensional perturbations. Crow [1] has discovered a long-wavelength instability, symmetric with respect to the plane separating the two vortices. The existence of a short-wavelength elliptic instability has been revealed by Tsai & Widnall [2], Moore & Saffman [3] and numerous articles ever since for both symmetric and antisymmetric modes. Recently Donnadieu et al. [4] observed a novel oscillatory instability, less unstable than Crow and elliptic odes, for large Reynolds numbers. However, in many atmospheric situations, as such dipoles propagate downwards, they evolve under the influence of the stable stratification of the atmosphere. The three-dimensional dynamics of this vortex pair in stratified flow, has received much less attention: still Robins & Delisi [5] and Garten et al. [6] have discussed persistance of the Crow instability and Nomura et al. [7] of the short-wavelength instability.

Three-dimensionnal instabilities and transient growth of trailing vortices in homogeneous and stratified flows

An aircraft wake is made of counter-rotating vortices and isknown to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investig ations on the three-dimensionnal instabilities and transi ent growth of such dipole are performed. By means of a three-dime nsionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic mo des but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of perturbation s on this dipole, investigated by computing the optimal linear perturbations with a direct-adjoint technique, dem onstrates the crucial role of the region of maximal strain at short time and of the hyperbolic point at intermediate tim e . 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.