Statistical evolution of non-Gaussian initial disturbances under transient growth

Transient growth of disturbances in shear flows is conventionally studied by identifying the optimal initial disturbance that yields maximal energy amplification. While this approach quantifies worst-case amplification, it does not capture the statistics of disturbance growth under realistic disturbance environments. Recent work (Frame & Towne, JFM 2024) has proposed a framework to model the statistics of energy amplification under transient growth, deriving expressions for the expected energy gain that are valid for any initial disturbance distribution with a given correlation matrix. However, determining the full probability density function (p.d.f.) of the evolved disturbance energy requires knowledge of the complete distribution of the initial disturbances. For a multivariate Gaussian initial distribution, they show that an approximate p.d.f. of the energy can be derived analytically; however, extending such approximations to non-Gaussian distributions remains a challenge.

In this work, we extend this statistical framework to non-Gaussian initial disturbances under transient growth. We perform Monte Carlo simulations by sampling disturbances from the true non-Gaussian initial distribution and evolving them using the linearized Navier-Stokes equations to compute the p.d.f. of the evolved disturbance energy. In parallel, we explore the derivation of analytical approximations for the evolved p.d.f. under transient growth, extending existing methods developed for Gaussian initial conditions to selected non-Gaussian distributions. Towards this goal, we investigate using Gaussian mixture models to approximate non-Gaussian initial distributions while retaining analytical tractability. We then compare these approximations with the p.d.f.s obtained from Monte Carlo simulations to assess their accuracy and limitations. This approach enables systematic quantification of how non-Gaussian features in the initial conditions influence transient growth dynamics and their statistical outcomes.

Frame, P. & Towne, A. (2024). Beyond optimal disturbances: a statistical framework for transient growth. Journal of Fluid Mechanics, 983, A2.