Transient energy growth of optimal streaks in parallel round jets
Résumé
We present a linear optimal perturbation analysis of streamwise invariant disturbances evolving in parallel round jets. The potential for transient energy growth of perturbations with azimuthal wavenumber $m\geq1$ is analyzed for different values of Reynolds number Re. Two families of steady (frozen) and unsteady (diffusing) base flow velocity profiles have been used, for different aspect ratios $\alpha=R/\theta$, where $R$ is the jet radius and $\theta$ the shear layer momentum thickness. Optimal initial conditions correspond to infinitesimal streamwise vortices, which evolve transiently to produce axial velocity streaks, whose spatial structure and intensity depend on base flow and perturbation parameters. Their dynamics can be characterized by a maximum optimal value of the energy gain $G_{opt}$, reached at an optimal time $\tau_{opt}$ after which the perturbations eventually decays. Optimal energy gain and time are shown to be respectively proportional to Re^2 and Re, regardless of the frozen or diffusing nature of the base flow. Besides, it is found that the optimal gain scales like $G_{opt}\propto 1/m^3$ for all $m$ except $m=1$. This quantitative difference for azimuthal wavenumber $m=1$ is shown to be based on the nature of transient mechanisms. For $m = 1$ perturbations, the shift-up effect [J. I. Jiménez-González et al., “Modal and non-modal evolution of perturbations for parallel round jets,” Phys. Fluids 27, 044105-1–044105-19 (2015)] is active: an initial streamwise vorticity dipole induces a nearly uniform velocity flow in the jet core, which shifts the whole jet radially. By contrast, optimal perturbations with $m\geq2$, are concentrated along the shear layer, in a way that resembles the classical lift-up mechanism in wall-shear flows. The $m=1$ shift-up effect is more energetic than the $m\geq2$ lift-up, but it is slower, with optimal times considerably shorter in the case of $m\geq2$ disturbances. This suggests that these perturbations may emerge very quickly in the flow when injected as initial conditions. When the base flow diffuses, the large timescale for $m=1$ disturbances allows the shear layer to spread and the jet core velocity to decrease substantially, thus lowering the values of corresponding optimal gain and time. For $m\geq2$, results are less affected, since the shorter transient dynamics does not leave room for significant modifications of the base flow velocity profiles, and the scaling laws obtained in the frozen case are recovered. Nevertheless, base flow diffusion hinders the transient growth, as a consequence of a weaker component-wise non-normality and a smoother, radially spread structure of optimal disturbances.
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