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 is analyzed for different values of Reynolds number . Two families of steady (frozen) and unsteady (diffusing) base flow velocity profiles have been used, for different aspect ratios α = R/θ, where R is the jet radius and θ is 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 Gopt, reached at an optimal time τopt after which the perturbations eventually decay. Optimal energy gain and time are shown to be, respectively, proportional to and , regardless of the frozen or diffusing nature of the base flow. Besides, it is found that the optimal gain scales like 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 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 lift-up, but it is slower, with optimal times considerably shorter in the case of 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 time scale 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 , 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.
As underlined by one referee, it should be noted that slightly larger growths are observed at for the diffusing base flow, which is counter-intuitive. The diffusion of the base flow is associated with the diffusion of the jet momentum thickness and therefore a decrease of the aspect ratio α. As a first approximation, one could extrapolate the analysis done in the frozen case by applying these results to an effective aspect ratio, which is smaller than the initial one due to diffusion. In terms of energy gain, this approach would predict a wrong trend, namely, a lower growth, since the gain decreases for decreasing α [see Fig. 3(a)]. Nevertheless, it turns out that the optimal perturbation in the diffusing case has a radial extension which is larger than in the frozen case [data not shown for but this trend is already visible for m = 2, see Figs. 12(i) and 12(l)]. As suggested by the referee, it is as if the optimal perturbation anticipates the diffusion of the shear layer and adapts itself with a wider spatial extent in order to resist better to its own viscous diffusion. As a consequence, the optimal perturbation is expected to be less affected by viscous diffusion than in the frozen case and therefore to better benefit from the energy growth mechanism at play in the shear layer, hence an eventual higher growth.