Destabilisation and modification of Tollmien–Schlichting disturbances by a three-dimensional surface indentation

Abstract

We consider the influence of a smooth three-dimensional (3-D) indentation on the instability of an incompressible boundary layer by linear and nonlinear analyses. The numerical work was complemented by an experimental study to investigate indentations of approximately $11\unicode[STIX]{x1D6FF}_{99}$ and $22\unicode[STIX]{x1D6FF}_{99}$ width at depths of 45 %, 52 % and 60 % of $\unicode[STIX]{x1D6FF}_{99}$, where $\unicode[STIX]{x1D6FF}_{99}$ indicates 99% boundary layer thickness. For these indentations a separation bubble confined within the indentation arises. Upstream of the indentation, spanwise-uniform Tollmien–Schlichting (TS) waves are assumed to exist, with the objective to investigate how the 3-D surface indentation modifies the 2-D TS disturbance. Numerical corroboration against experimental data reveals good quantitative agreement. Comparing the structure of the 3-D separation bubble to that created by a purely 2-D indentation, there are a number of topological changes particularly in the case of the widest indentation; more rapid amplification and modification of the upstream TS waves along the symmetry plane of the indentation is observed. For the shortest indentations, beyond a certain depth there are then no distinct topological changes of the separation bubbles and hence on flow instability. The destabilising mechanism is found to be due to the confined separation bubble and is attributed to the inflectional instability of the separated shear layer. Finally for the widest width indentation investigated ($22\unicode[STIX]{x1D6FF}_{99}$), results of the linear analysis are compared with direct numerical simulations. A comparison with the traditional criteria of using $N$-factors to assess instability of properly 3-D disturbances reveals that a general indication of flow destabilisation and development of strongly nonlinear behaviour is indicated as $N=6$ values are attained. However $N$-factors, based on linear models, can only be used to provide indications and severity of the destabilisation, since the process of disturbance breakdown to turbulence is inherently nonlinear and dependent on the magnitude and scope of the initial forcing.