Direct Numerical Simulations of Tollmien-Schlichting Disturbances in the Presence of Surface Irregularities

Kurzfassung

A realistic wing is not expected to be an ideal smooth surface: the influence of junctions or wing-panels must be taken into account in the laminar wing design. The presence of these spanwise invariant two-dimensional surface perturbations amplify the streamwise instabilities existing on a smooth surface, potentially causing an early transition to turbulence. In the present work, the effect of surface irregularities such as steps, gaps and humps on the development of Tollmien-Schlichting (TS) waves in an incompressible boundary layer is investigated using direct numerical simulations (DNS). The calculations are performed with the open source spectral element code NEK5000 and consist of a DNS of the steady laminar two-dimensional base flows and succeeding DNS for the disturbance propagation. A Blasius profile is specified at the inflow boundary and when the base flow reaches its steady state the TS waves are artificially excited. This is done either by superimposing the eigensolutions of the Orr-Sommerfeld equations onto the Blasius profiles, or by periodic blowing and suction through a narrow strip at the wall. Arbitrary two- or three-dimensional (oblique) disturbance waves can be generated. For the oblique waves, the two-dimensional base flow is extruded in the spanwise direction over a spanwise width that is fixed to the spanwise wavelength of the introduced disturbance. After a transient phase where the perturbations propagate through the entire domain, a time-periodic flow field in the unsteady DNS is reached. This enables a Fourier analysis in time of the streamwise component of the disturbance velocity to obtain the perturbation amplitudes at different streamwise locations. The N-factor is then used to assess the growth of the disturbances. The amplitudes of the introduced disturbances are chosen to be small enough to grant a purely linear development throughout the domain. The geometric irregularities are directly meshed or defined by analytical mapping of the Gauss-Lobatto-Legendre (GLL) points, used to build the velocity Lagrange polynomial interpolants in each spectral element. The latter procedure requires the irregularities to be localised and smooth. Depending on their size, the surface imperfections give rise to a local separation bubble which interacts with the oncoming TS waves. Results show, as expected, that these surface perturbations have a destabilizing effect when compared to the case of the smooth flat plate. Detailed results on the amplification of two-dimensional and oblique TS waves encountering steps, gaps, humps and indentations of different heights and widths will be presented.