UC Berkeley

The early-time growth of an individual vortex in a wake vortex pair is explored, focusing on its linear stability and transient growth, which are meaningful for understanding the lifespan of vortices and mitigating hazards from wake turbulence. This study synthesises two parts: linear stability analysis and transient growth analysis. The linear stability of wake vortices is examined in both inviscid and viscous contexts, using the Batchelor or Lamb-Oseen vortex as a base vortex model. Linear stability analysis determines the growth rates of infinitesimal perturbations in the vortex flow by linearising the incompressible Navier-Stokes or Euler equations around the base vortex, leading to an eigenvalue problem that discloses the vortex's stability properties. Inviscid analysis, with zero viscosity, successfully reveals a continuous spectrum associated with critical-layer singularities, causing discontinuity in perturbation velocity in the inviscid limit and requiring careful numerical parameter tuning to avoid under-resolved, or so-called spurious, solutions. Non-zero viscosity alters both the discrete and continuous spectra, necessitating numerical resolution conforming to the Reynolds number to the one-third power scaling law to resolve the newly discovered viscous critical-layer eigenmode family. The transient growth phenomenon is then explored through non-modal analysis to understand how initial perturbations, as combinations of numerous eigenmodes, can experience significant transient growth even in linearly stable vortices. The transient growth formalism is applied to the linearised Navier-Stokes equations. Optimal perturbations that maximise energy amplification over finite time intervals are found, to which the viscous critical-layer eigenmode family serves as the main contributor. Numerical simulations quantify the growth of perturbations considering nonlinearity, showing that while nonlinearity may slightly alter transient dynamics, overall trends align with formalism predictions. Lastly, motivated by contrails around wake vortices, the research investigates particle-initiated transient growth, where perturbations evolve through particle-vortex interactions. Two-way coupled particle-vortex simulations demonstrate continual perturbation development, presenting primary growth patterns as captured in the transient growth formalism. This investigation implies the practicability of the transient growth phenomenon.