Linear response to harmonic forcing of the non-parallel Batchelor vortex

The Batchelor vortex1 is a slowly developing asymptotic solution of the flow of a trailing line vortex. This flow has been comprehensively studied in its parallel approximation. In particular, the convective/absolute nature of the instability has been investigated2. In the non-parallel case, the Batchelor vortex can be whether stable or unstable, depending on the inlet conditions. For the globally unstable flow, linear stability analysis has been carried out through modal analysis3. In contrast, when the flow is stable, all global modes are damped and the most suitable technique to investigate the dynamics of coherent structures is to compute the response to an external forcing4. In the present work, we investigate the linear response to harmonic forcing of the non-parallel Batchelor vortex through global resolvent calculations, considering both boundary and volume forcing. The base-flow is directly calculated by axisymmetric DNS in cylindrical coordinates. At the inlet, a Batchelor vortex profile is imposed, with swirl and wake defect parameters chosen in order to obtain a globally stable flow and enhance mode competition. The Reynolds number is set to 1000. Figure 1 a) reports the energy gain as a function of the frequency for all amplified helical modes, for prescribed inlet forcing. Single and double helical modes are the most amplified in the low frequency range, although at higher forcing frequency, higher wavenumber modes become more amplified. Figure 1 b) and c) depict the shape of harmonic responses at the frequencies corresponding to the maximum gain for single and double helical mode, respectively. This problem has been also assessed through WKB analysis in the framework of weakly-non-parallel flows. Remarkably, local and global analyses predict practically the same amplification factors and mode selection (figure 1). The validity of this predicted linear mode selection is confirmed through the computation of the nonlinear evolution of the response, via 3D-DNS simulations, harmonically forced in time at the inlet. The nonlinear response is seen to be dominated by different helical numbers depending on the forcing frequency, in accordance with the linear analysis.