Transient growth of a wake vortex and its initiation via inertial particles

Abstract

The transient dynamics of a wake vortex, modelled as a strong swirling $q$-vortex, are investigated with a focus on optimal transient growth driven by continuous eigenmodes associated with continuous spectra. The pivotal contribution of viscous critical-layer eigenmodes (Lee & Marcus, J. Fluid Mech., vol. 967) amongst the entire eigenmode families to optimal perturbations is numerically confirmed, utilising a spectral collocation method for a radially unbounded domain that ensures correct analyticity and far-field behaviour. The consistency of the numerical method across different sensitivity tests supports the reliability of the results and provides flexibility for tuning. Both axisymmetric and helical perturbations with axial wavenumbers of order unity or less are examined through linearised theory and non-linear simulations, yielding results that align with existing literature on energy growth curves and optimal perturbation structures. The initiation process of transient growth is also explored, highlighting its practical relevance. Inspired by ice crystals in contrails, the backward influence of inertial particles on the vortex flow, particularly through particle drag, is emphasised. In the pursuit of optimal transient growth, particles are initially distributed at the periphery of the vortex core to disturb the flow. Two-way coupled vortex-particle simulations reveal clear evidence of optimal transient growth during ongoing vortex-particle interactions, reinforcing the robustness and significance of transient growth in the original non-linear vortex system over finite time periods.