Frequency response analysis of jet–vortex interaction for variable relative spacing

This work presents frequency response analysis of the Lamb–Oseen vortex with a single-point injection (SPI) acting in the axial direction. In particular, we address the vortex response-mode selection as the radial location, af, of the SPI is varied. We interpret these results by complementary linear stability and resolvent analyses. Our results demonstrate a way to excite certain modal behavior of the optimal response using SPI. Energy gain maps of the frequency response as a function of the streamwise wavenumber and the location and frequency of the SPI show that the modes with azimuthal wavenumbers |m|=0 and |m|=1 have the highest gain and correspond to different known families of Kelvin vortex waves. The value of af strongly influences the occurrence and importance of the dominant vortex response modes. In particular, for relatively low af values, the axisymmetric mode dominates. However, as af is increased, the dominant modes shift to helical modes. Additionally, we identify non-normal effects in specific areas of the parameter space, highlighting two particular regions: (i) an area explained only by the viscous critical layer characterized by a concentrated ring and (ii) a region given by the resolvent analysis for short-wavelength perturbations that exhibits a combination of the ring and a distinctive helical mode near the vortex core. Finally, we prove that flow patterns obtained at the maximum gain points of the frequency response exhibit a similar behavior to the optimal response, thus confirming the SPI as a powerful candidate to promote the optimal response in practical applications.