Transition to Turbulence, Small Disturbances, and Sensitivity Analysis II: The Navier-Stokes Equations
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Recent research has shown that small disturbances in the linearized Navier-Stokes equations cause large energy growth in solutions. Although many researchers believe that this interaction triggers transition to turbulence in flow systems, the role of the nonlinearity in this process has not been thoroughly investigated. This paper is the second of a two part work in which sensitivity analysis is used to study the effects of small disturbances on the transition process. In the first part, sensitivity analysis was used to predict the effects of a small disturbance on solutions of a motivating problem, a highly sensitive one dimensional Burgers' equation. In this paper, we extend the analysis to study the effects of small disturbances on transition to turbulence in the three dimensional Navier-Stokes equations. We show that the change in a laminar flow with respect to small variations in the initial flow or small forcing acting on the system is large when the linearized operator is stable yet non-normal. In this case, the solution of the disturbed problem can be very large (and potentially turbulent) even if the disturbances are extremely small. We also give bounds on the disturbed flow in terms of certain constants associated with the linearized operator.