Malek Abid

Decaying Kolmogorov turbulence in a model of superflow

Superfluid turbulence is studied using numerical simulations of the nonlinear Schrodinger equation (NLSE), which is the correct equation of motion for superflows at low temperatures. This equation depends on two parameters: the sound velocity and the coherence length. It naturally contains nonsingular quantized vortex lines. The NLSE mass, momentum, and energy conservation relations are derived in hydrodynamic form. The total energy is decomposed into an incompressible kinetic part, and other parts that correspond to acoustic excitations. The corresponding energy spectra are defined and computed numerically in the case of the two-dimensional vortex solution. A preparation method, generating initial data reproducing the vorticity dynamics of any three-dimensional flow with Clebsch representation is given and is applied to the Taylor–Green (TG) vortex. The NLSE TG vortex is studied with resolutions up to 5123. The energetics of the flow is found to be remarkably similar to that of the viscous TG vortex. The...

DIRECT NUMERICAL SIMULATIONS OF THE BATCHELOR TRAILING VORTEX BY A SPECTRAL METHOD

The nonlinear vorticity dynamics of the Batchelor trailing vortex is presented. Direct numerical simulations of the three-dimensional, incompressible, Navier-Stokes equations by a spectral method are used. It is shown that the initial vortex core is subjected to three transformations: a twisting phase, a lateral expansion of its cross section and formation of a spiral structure. These transformations are accompanied by a gradual deceleration of axial velocity. Mean kinetic energy is then transferred to radial velocity. When the transfer is maximum a secondary instability is initiated leading to the spiral structure. These transformations are in agreement with vortex breakdown observed in recent experiments.

Nonlinear mode selection in a model of trailing line vortices

Nonlinear mode selection, from initial random Gaussian field perturbations, in a model of trailing line vortices (swirling jets), in the breakdown regime, is addressed by direct numerical simulations with a Reynolds number equal to 1000. A new concept of mode activity in the nonlinear evolution is introduced. The selected modes, according to their activities, are reported and related to strain eigenvectors (with maximum eigenvalues) of the basic flow corresponding to the trailing line vortex under consideration. The selected modes are also related to results from the linear eigenmode (exponential growth) instability theory using the concept of dispersion relation envelope. It is found that the global mode hypothesis of the linear eigenmode theory is violated near the flow axis when the swirl number increases. However, far from the flow axis the linear eigenmode theory is in good agreement with the nonlinear evolution in the breakdown regime. The discrepancy between the nonlinear evolution and the linear eigenmode theory is related to the transient growth of optimal perturbations resulting from the non-normality of the linearized Navier-Stokes equations about shear flows. A clear distinction between an eigenmode, an optimal perturbation (non-modal) and a direct numerical simulation (DNS) mode is made. It is shown that the algebraic (transient) growth contributions from the inviscid continuous spectrum could trigger nonlinearities near the flow axis. The DNS mode selected in the nonlinear regime coincides with the long-wave eigenmode benefiting from the algebraic growth in the linear regime. This eigenmode is different from the short-wave eigenmode with the absolute maximum exponential growth. Although it is promoted by transients, in the linear regime, the long-wave component is selected nonlinearly.

Numerical characterization of the dynamics of vortex filaments in round jets

The dynamics of streamwise vorticity in axisymmetric jets is studied by direct numerical integration of the Navier–Stokes equations coupled with a passive scalar. Consistently with recent experiments and inviscid numerical simulations, the present viscous simulations show the appearance of pairs of axially counter‐rotating vortex filaments. After their formation the filaments move away from the jet, dragging the tracer into finger‐shaped structures. The three‐dimensional topology of the rings, filaments, and fingers is described, well beyond the time of filament formation. Finally, visualizations of the pressure gradient field are presented, which suggest that the filaments can be directly observed experimentally by seeding the flow with microbubbles.

Stability of a vortex sheet roll-up

The stability of vortex sheet roll-up is studied using a Lagrangian vortex method. We consider an initially unstable (Kelvin–Helmholtz) vortex sheet. During its nonlinear evolution, a perturbation is added to test it for a secondary instability. The growth of the perturbation depends on its phase and on the local strain rate. In the linear stage of this secondary instability, the dispersion relation is calculated. It is found that the growth rate and the cutoff wave number are fixed by the regularization parameter of the Birkhoff–Rott equation.

Direct numerical simulations of variable-density plane jets

Two-dimensional incompressible and heterogeneous free jets are studied numerically. They are characterized by two linearly unstable modes: the sinuous mode and the varicose mode. When the density is constant, the sinuous mode dominates, in a sense that it has a maximum growth rate nearly three times greater than that of the varicose mode, both in the temporal and spatial stability frameworks. This suggests that jet evolution at the linear and nonlinear regimes, at least at early stage, would be sinuous with the frequency and wavenumber of the most unstable mode

Energy spectra in a helical vortex breakdown

Using spectrally accurate, direct numerical simulations, we show that vortices with axial flows (without an imposed strain) generate small scales, i.e., breakdown, when they are randomly perturbed. We show that the breakdown that occurs without the necessity of a stagnation point in the flow is nonlinear and that energy spectra of the breakdown present a range of a k−5/3 scaling (k being a wavenumber). These features are the same for centrifugally stable and unstable azimuthal velocities. However, centrifugally unstable azimuthal velocities promote mixing in vortices with an axial flow. We also find that vortex breakdown could be a physical mechanism against a finite time singularity in Navier–Stokes equations. Contrarily, we do not observe a breakdown using spectrally truncated Euler equations. For these equations we find a scaling of the energy spectra compatible with the existence of a finite time singularity.

Experimental and numerical investigations of low-temperature superfluid turbulence

Low-temperature superfluid turbulence is studied experimentally in a Helium swirling flow and numerically with the Gross-Pitaevskii equation in the geometry of the Taylor-Green (TG) vortex flow. Numerically, it was found in Nore et al. (1997a, b) that the kinetic energy transfer in the superfluid TG vortex is comparable to that of the viscous TG vortex and that the energy spectrum of the superflow is compatible with Kolmogorovs scaling. The vorticity dynamics of the superflow are similar to that of the viscous flow. In both cases, many vortex reconnection events happen throughout the flow. Experimentally, power measurements and pressure fluctuation spectra show very little difference above and far below the superfluid transition temperature, where the normal-fluid component of Helium is negligible (less than 5% in mass at T = 1.2 K).

Experimental and numerical investigation of a variable density swirling-jet stability

Linear stability of a variable density, and incompressible, swirling jet is studied at moderate Reynolds numbers, both experimentally and numerically. Mean flow characteristics are obtained experimentally. They show a correlation between axial velocity and density profiles. For low density ratios, s, the presence of self-sustained oscillations of the flow (global modes) is highlighted. Using direct numerical simulations (at low azimuthal wavenumbers m = 0, −1, −2) to study the local convective/absolute transition at the jet exit-nozzle, it is shown that frequencies of the local absolute-instability are in good agreement with those of the experimentally obtained global modes. Furthermore, it is found that increasing the swirl number q enhances the absolute instability for helical modes m = −1 and −2. Finally, it is also shown that the unstable selected mode, experimentally obtained, has the spatio-temporal properties of the absolutely unstable one obtained numerically.

Oscillating structures in a stretched–compressed vortex

The dynamics of a vortex subject to a localized stretching is numerically investigated. The structure of the flow is analysed in the case of an initially two-dimensional vortex surrounded by a periodic array of vortex rings localized far from its core. Amplified oscillations of both the axial vorticity and the stretching are found, in strong contrast with Burgers-like vortices. The resulting dynamics is the appearance, around the vortex, of successive vortical structures of smaller and smaller radius and alternate sign embedded in the previous vortical rings. The frequency scaling of the oscillations is recovered by linear analysis (Kelvin modes) but not the amplification nor the shape of the successive tori. An inviscid model based on structures is presented, which compares better with the numerical computations. These results suggest that the formalism of Kelvin waves is not sufficient to describe the full dynamics, which is instead related to the feedback of rotation on stretching and more conveniently described in terms of localized structures. We finally discuss the relative timescales of vortex stretching and of vortex reaction. The Burgers-like vortices, where there is no such reaction, turn out to correspond to a nearly pure strain field, slightly disturbed by rotation.