Linear transient growth analysis is commonly used to suggest the structure of disturbances which are particularly efficient in triggering transition to turbulence in shear flows. We demonstrate that the addition of nonlinearity to the analysis can substantially change the prediction made in pipe flow from simple two-dimensional streamwise rolls to a spanwise and cross-stream localized three-dimensional state. This new nonlinear optimal is demonstrably more efficient in triggering turbulence than the linear optimal indicating that there are better ways to design perturbations to achieve transition.
Pringle, C., & Kerswell, R. R. (2010). Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Physical Review Letters, 105(15), . https://doi.org/10.1103/PhysRevLett.105.154502