Numerical simulation of boundary-layer transition

The transition to turbulence in boundary layers was investigated by direct numerical solution of the nonlinear, three-dimensional, incompressible Navier-Stokes equations in the half-infinite domain over a flat plate. Periodicity was imposed in the streamwise and spanwise directions. A body force was applied to approximate the effect of a nonparallel mean flow. The numerical method was spectra, based on Fourier series and Jacobi polynomials, and used divergence-free basis functions. Extremely rapid convergence was obtained when solving the linear Orr-Sommerfeld equation. The early nonlinear and three-dimensional stages of transition, in a boundary layer disturbed by a vibrating ribbon, were successfully simulated. Excellent qualitative agreement was observed with either experiments or weakly nonlinear theories. In particular, the breakdown pattern was staggered or nonstaggered depending on the disturbance amplitude.