On the relationship between the wall-shear-stress and transient-growth disturbances in a laminar boundary layer

The present study is motivated by the development of wall-based flow-state estimators for feedback control of transient-growth disturbances in a laminar boundary layer. Such estimators have been successfully demonstrated in recent numerical studies, but they rely on information that, while available in simulations, is generally not accessible in the laboratory. Here, some physical aspects of transient-growth modes and their relationship to the unsteady wall shear stress are examined to guide the development of estimation methods that perform satisfactorily yet are practical to implement. The usefulness of the resulting physical understanding for flow estimation is demonstrated using a proper orthogonal decomposition mode estimator, applied to a direct numerical simulation of boundary layer transition beneath free-stream turbulence. The results shed light on the strengths and weaknesses of employing each of the streamwise and spanwise wall-shear-stress components in estimating the disturbance wall-normal velocity and vorticity fields. It is also found that the streamwise-elongated nature of transient-growth disturbances allows coarse sampling of the wall-shear-stress information used in the estimation without substantially deteriorating the estimation accuracy. This is particularly true for estimates based on the streamwise shear stress.