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Effects of circulation on the evolution of vortex tubes and the associated response of near-wall flows in the shear of laminar boundary-layer flows are investigated using a model proposed by Hon and Walker (Hon, T.L. & Walker, J.D.A, Computers & Fluids, 20(3), pp. 343–358, 1991). Direct numerical simulations with freestream Mach number of 0.5 are conducted. Firstly, the dynamics of single hairpin vortex is investigated. Numerous secondary hairpin vortices, much more than previously reported, which are regularly aligned in the streamwise direction are allowed to be newly generated according to the shear-layer instability of the legs of an initial hairpin vortex. Small-scale turbulence is then produced when the circulation is sufficiently large. Secondly, a straight vortex tube model is investigated. Sinuous deformation of a shear layer, which leads to the generation of discrete hairpin vortices, becomes obvious especially near the upper region of the vortex tube. In order to quantify the initial instability triggering the generation of the secondary hairpin vortices, quasi-linear stability analysis is conducted. While only one unstable mode appears when the circulation is small, two modes, that is, off-wall mode and near-wall mode, appear when the circulation is large. The cases of circulation where the two modes appear correspond to those of circulation where the production of small-scale turbulence is observed in the simulations of the single hairpin vortex.
boundary layer, direct numerical simulation, hairpin vortex, laminar-turbulent transition, stability, turbulence, vortex tube
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