A blow-up mechanism in boundary layer transition

Abstract

Laminar fluid flows typically undergo transition to turbulence as flow speed increases. This is a problem of fundamental importance in fluid mechanics and yet, despite research over many decades, laminar-turbulent transition is still not well understood. This thesis presents new results indicating how small but finite amplitude disturbances in a laminar boundary layer flow can experience rapid amplification potentially leading quickly to turbulence. It is well known that when the freestream disturbance level is low enough, linear stability theory predicts exponential growth of boundary layer disturbances. However, in many flow structures these growth rates are relatively weak. Furthermore, linear theories do not predict amplitude thresholds for breakdown to turbulence; they only give growth factors. Wind tunnel experiments have shown that transition involves nonlinear interaction of wavy disturbances, and that resonant mechanisms are particularly important. Weakly nonlinear theory provides the framework for studying these interactions. Previous theories have been developed in the large Reynolds number limit, but moderate Reynolds numbers are more relevant to practical applications. It is shown here that in the latter case, the interaction coefficients take a qualitatively different form such that rapid growth may be expected when disturbances exceed a critical amplitude. The behaviour is shown to be prevalent at low amplitude thresholds even for subcritical Reynolds numbers, meaning that finite, but numerically small perturbations tend to ‘blow-up’ even if the flow is linearly stable. The scenario agrees with experiments, and so may provide a dominant mechanism for laminar-turbulent transition.