New SubmissionTitle: Hairpin modeling of turbulent boundary layers at extremely high Reynolds numbers

How the wall alters the three dimensional (3D) vorticity (ω) of turbulent boundary layers with Reynolds (Re) and Mach (M) numbers is considered. It is shown that, in the dynamical part of the viscous sublayer, wall proximity suppresses the nonlinearities, reducing the vorticity equation to a 1D equation of bistable solution in agreement with measurements, standing harmonically in axisymmetric bodies at high Re, M, but jittering in flat plates at low Re, M, eluding detection. At high Re, M, the hairpin vortices, maximally stretched along the principal plane, break, converting centrifugal energy to circulation Γ squared pressure radiated in the freestream. Freestream Γ² fronts at 18◦ and 27◦ correlate with upstream slants of aligned and staggered hairpin arrays. The Γ² pressure impulse of hairpin break travels down the two broken hairpin legs summing at the wall to stand, as per the 1D wave equation, to two types of wall pressure signatures seen in measurements. Low Re visualization shows that the aligned hairpins make the rotational bulges large with deep excursions, while the staggered hairpins produce uniform smaller bulges; abrupt forcing by favorable pressure gradient unambiguously visualizes the hairpin subharmonic and harmonic instabilities.