Data-based transient growth analysis

Transient growth analysis has been extensively studied in asymptotically stable flows to identify their short-term amplification of perturbations. Generally, in global transient growth analyses, matrix-free methods are adopted, requiring the construction of adjoint equations, either in the discrete or continuous form. This paper introduces a data-driven algorithm that circumvents the adjoint equations by extracting the optimal initial perturbation and its energy growth over a specified time horizon from transient snapshots of perturbations. This method is validated using data from the linearised complex Ginzburg–Landau equation, backward-facing step flow and the Batchelor vortex. Unlike model-based methods, which require $S$ sets of integrations of the linearised governing equation and its adjoint for $S$ time horizons, the proposed approach collects the snapshots of $S$ time horizons in one integration of the linearised equation. Furthermore, this study provides a robust framework for utilising proper orthogonal decomposition modes to synthesise optimal modes. The developed capacity to conduct transient growth analyses without solving the adjoint equations is expected to significantly reduce the barriers to transient dynamics research.