Optimal disturbances in round submerged jets

In this paper, optimal growth analysis of spatial disturbances in round submerged jets is performed. Optimal energy growth is studied for various Reynolds numbers Re and frequencies ω. Different velocity profiles proposed by Michalke [“Survey on jet instability theory,” Prog. Aerosp. Sci. 21, 159–199 (1984)] are investigated by varying the shear layer momentum thickness δ, while the jet radius R is considered fixed (as characteristic length). In the case of stationary disturbances, there are no amplified eigenmodes, so eventually the energy of such disturbances decays downstream. Such disturbances are characterized by optimal energy growth as a function of streamwise coordinate z. At certain location downstream zmax this value reaches its maximum Gmax and then decays, although not monotonically, unlike the result of the temporary setting [Jimenez-Gonzalez et al., “Modal and non-modal evolution of perturbations for parallel round jets,” Phys. Fluids 27, 044105 (2015); Jimenez-Gonzalez and Brancher, “Transient energy growth of optimal streaks in parallel round jets,” Phys. Fluids 29, 114101 (2017)]. The well-known scaling Gmax∝Re2 and zmax∝Re is confirmed for stationary disturbances. Conversely, for nonstationary disturbances there is a range of frequencies in which an amplified mode is present. In this case, optimal perturbations far downstream consist only of the amplified mode. The main effect of non-modality here is “the energy pumping” of the amplified mode, so the disturbances are characterized by the “energy ratio” G/Em, i.e., the ratio of the energy of an arbitrary perturbation to the energy of the amplified mode. The influence of the shear layer momentum thickness on optimal disturbances is studied; increasing the shear layer momentum thickness of the base flow “smears” the area occupied by the optimal disturbance, although it does not dramatically affect energy gain values, slightly decreasing maximum energy as shear layer momentum thickness increases. Spatial oscillations in the energy of stationary optimal disturbances in jet flows are discovered. The spectrum structure examination shows that the frequency of these oscillations corresponds to the frequencies of the two least damped discrete eigenmodes.