Abstract

By analog with the linear approaches, and specifically the frequency-response analysis, we model transition from laminar to turbulent flow as a stationary process where disturbances are continually supplied to the system from the environment, i.e. we consider the nonlinear receptivity problem in the frequency domain. In order to calculate finite-amplitude perturbations in the frequency domain, we seek self-consistent solutions of the Navier-Stokes equations under the form of an expansion consisting of a mean-flow solution, a fundamental mode and N harmonics of the fundamental. We call the resulting equations Harmonic-Balanced Navier-Stokes (HBNS). Subsequently, optimal nonlinear forcing mechanisms that lead to turbulence and maximise the skin-friction coefficient (drag) are calculated using a variational method and direct-adjoint looping. The proposed nonlinear input/output analysis method reveals the most dangerous nonlinear mechanisms that lead to turbulent dynamics in the frequency domain, which can be viewed as the minimal forcing seed in the frequency domain.