Non-Parallel-Flow Effects on Stationary Crossflow Vortices at Their Genesis

The linear stability of a laminar boundary-layer flow over a swept wing is studied for a stationary perturbation with spanwise wavenumber , where β is non-dimensionalised by the chordwise distance L to the leading edge, and Re is the Reynolds number based on L This scaling corresponds to the lower-branch regime of the neutral curve, and hence represents the genesis stage of stationary crossflow vortices. We show that in this stage or regime, the non-parallelism of the flow plays a leading order role in the growth of the instability. Non-parallel-flow effects consist of two parts: the streamwise variation of the base-flow profiles, and the distortion of the eigenfunction. These two contributions to the growth rate are found to be of the same order of magnitude, and both are closely related to the 4th-order vertical derivatives of the base-flow profile on the surface of the wing. The implication of the present finding on the description of stationary vortices using the Orr-Sommerfeld equation is discussed.