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1. On December 17, 1903, in Kitty Hawk, North Carolina, Orville Wright performed the first ever airplane flight and this was the breakthrough to airplane development. The image was downloaded from  . 

1. On December 17, 1903, in Kitty Hawk, North Carolina, Orville Wright performed the first ever airplane flight and this was the breakthrough to airplane development. The image was downloaded from . 

Citations

... These works have shown that, when walls having zero or very low permeability are considered, two-dimensional invariant solutions non-localised in space exist from Re≈3000, whereas localized three-dimensional solutions can be found already at Re around 400 (the Reynolds number Re being based on the displacement thickness). Moreover, it is known that the ASBL can delay by far the growth of linear normal-mode disturbances compared to the Blasius boundary layer flow, the linear stability limit being Re c ≈54 400 for the ASBL [42][43][44], whereas the laminar BBL is unstable to small amplitude perturbations from Re c ≈520 [45], [44]. For small wall permeabilities, refs [39] and [46] have confirmed that wall suction has a stabilising effect on lowamplitude perturbations. ...
... Then, the code has been modified to tackle the case of the asymptotic suction boundary layer flow adding the wall-suction terms. For this case the linear stability of the laminar ASBL has been computed, reproducing the critical Reynolds number Re c ≈54 400 according to refs [42,43,57] and [44] imposing a zero perturbation condition also on the wall-normal component of the velocity. It is worth to point out that this critical Reynolds number decreases considerably when using a small but finite permeability [39]. ...
... The stabilising effects of suction on the flow in a boundary layer constrained on a two-dimensional framework is evident when comparing the linear and nonlinear stability limit in the Reynolds number for the two-dimensional BBL and the ASBL (zero permeability a=0). The linear stability limit is Re BBL c ≈520 [44,45] whereas Re ASBL c ≈54 400 [42][43][44]; the minimum Reynolds number at which 2D nonlinear states are found is Re BBL SN =510, whereas Re ASBL SN =3168 [39,58]. This implies that wall suction delays growth of small amplitude perturbations, as well as the emergence of alternative 2D finite-amplitude solutions of the Navier-Stokes equations, to much higher Re. ...
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This work aims at investigating three-dimensional finite-amplitude traveling wave solutions of the asymptotic suction boundary layer, as well as their role in state space, based on direct numerical simulations. Using a body forcing and allowing wall-normal perturbations at the wall to be nonzero and linked to the pressure gradient through a wall permeability a, nonlinear invariant solutions of the Navier-Stokes equations are found and continued towards low values of the Reynolds number. The solutions, having a reflect symmetry in the spanwise coordinate, set a new threshold in state space for the onset of exact coherent flow structures. The obtained traveling waves emerge from a saddle-node bifurcation at ReSN=225 (based on the displacement thickness) for vanishing wall permeability. Increasing a to small but finite values changes but slightly the value of ReSN. In all cases, the obtained ReSN is below the lowest Reynolds number at which sustained turbulence is observed, namely, Re≈270 according to recent works. The corresponding waves in the Blasius boundary layer flow without suction exist from ReSN=496, meaning that the asymptotic suction boundary layer might be more nonlinearly unstable (although more linearly stable) than the Blasius one. In the interval of Re studied it is found that the traveling waves occupy a region extending from the wall to the log-law region, with maximum root-mean-square velocities approaching those obtained by direct numerical simulations in the turbulent regime, especially for nonvanishing permeability. These solutions are found to be unstable, with steady leading modes, whose growth rate increases with the permeability and with the Reynolds number. When the trajectory escapes from these traveling waves along their unstable directions, transition to turbulence is observed even at Reynolds number as low as Re=240, suggesting that these solutions may play an important role in turbulent transition at low Reynolds numbers.
... To comply with the uniform flow condition we set v (b=0) =0. At the plate, the streamwise disturbance velocity is assumed to vanish, i.e. u ′ (t, x,y=0)=0; however, due to the fact that the wall is porous, it is not obvious that the v ′ -component of the perturbation should be zero, as customarily assumed in most linear and nonlinear studies [16,26,27,[35][36][37][38][39]. For low permeabilities it is acceptable to represent the flow through the porous plate via Darcy's law, neglecting inertia, i.e. ...
... Thus, the critical Reynolds numbers found for a = 0 cannot be compared to those of the Blasius boundary-layer (i. e., Re c =520 and Re g =510 [12,45]), but rather with the ones provided in the literature for the asymptotic suction boundary layer with no-slip boundary conditions for the disturbance velocity, namely, Re c ≈ 54400 and Re g ≈ 3200 [16,36]. Concerning the case with a = 3.85 × 10 −4 , one can notice that the predicted critical Reynolds number and streamwise wavenumber are in a good agreement with the ones provided by Fransson & Alfredsson [33] (i. ...
... The energy E of the wavy part of the nonlinear solutions, see equation 8, mapped out in Re for a streamwise wavenumber of α=0.154 and six values of a. The classical wall condition a=0 corresponds to a bifurcation at Rec≈54000 for αc=0.1555[16,36]. With increasing a, Rec is reduced. ...
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The nonlinear stability of the asymptotic suction boundary layer is studied numerically, searching for finite-amplitude solutions which bifurcate from the laminar flow state. By changing the boundary conditions for disturbances at the plate from the classical no-slip condition to more physically sound ones, the stability characteristics of the flow may change radically, both for the linearised as well as for the nonlinear problem. The new wall boundary condition takes into account the permeability K of the plate; for very low permeability it is acceptable to impose the classical boundary condition ( K=0). This leads to a Reynolds number of approximately Rec=54400 for the onset of linearly unstable waves and close to Reg=3200 for the emergence of nonlinear solutions [F.A. Milinazzo and P.G. Saffman, J. Fluid Mech. 160 (1985); J.H.M. Fransson, PhD thesis, Royal Institute of Technology, KTH, Sweden (2003)]. However, for larger values of the plate's permeability the lower limit for the existence of linear and the nonlinear solutions shifts to significantly lower Reynolds numbers. For the largest permeability studied here the limit values of the Reynolds numbers reduce down to Re_c=796 and Re_g=294. For all cases studied the solutions bifurcate subcritically towards lower Re and this leads to the conjecture that they may be involved in the very first stages of a transition scenario similar to the classical route of the Blasius boundary layer initiated by Tollmien-Schlichting (TS) waves. The stability of these nonlinear solutions is also investigated, showing a low-frequency main unstable mode whose growth rate decreases with increasing permeability and with the Reynolds number, following a power law Re^−ρ , where the value of ρ depends on the permeability coefficient K. The non-linear dynamics of the flow in the vicinity of the computed finite-amplitude solutions is finally investigated by Direct Numerical Simulations, providing a viable scenario for subcritical transition due to TS waves.
... Also, the development of the profile at the evolution region before the asymptotic state is reached have been considered; see for references. The corresponding development of the drag coefficient along the plate is considered in the thesis by Fransson (2003). For the asymptotic state, the drag coefficient is constant and equals 2·Re −1 . ...
Article
Energy thresholds for transition to turbulence in an asymptotic suction boundary layer is calculated by means of temporal direct numerical simulations. The temporal assumption limits the analysis to periodic disturbances with horizontal wave numbers determined by the computational box size. Three well known transition scenarios are investigated: oblique transition, the growth and breakdown of streaks triggered by streamwise vortices, and the development of random noise. Linear disturbance simulations and stability diagnostics are also performed for a base flow consisting of the suction boundary layer and a streak. The scenarios are found to trigger transition by similar mechanisms as obtained for other flows. Transition at the lowest initial energy is provided by the oblique wave scenario for the considered Reynolds numbers 500, 800, and 1200. The Reynolds number dependence on the energy thresholds are determined for each scenario. The threshold scales like Re−2.6 for oblique transition and like Re−2.1 for transition initiated by streamwise vortices and random noise, indicating that oblique transition has the lowest energy threshold also for larger Reynolds numbers.
... Also, the development of the profile at the evolution region before the asymptotic state is reached have been considered; see for references. The corresponding development of the drag coefficient along the plate is considered in the thesis by Fransson (2003). For the asymptotic state, the drag coefficient is constant and equals 2·Re −1 . ...
Article
Full-text available
The effect of high levels of free-stream turbulence on the transition in a Blasius boundary layer is studied by means of direct numerical simulations, where a synthetic turbulent inflow is obtained as superposition of modes of the continuous spectrum of the Orr–Sommerfeld and Squire operators. In the present bypass scenario the flow in the boundary layer develops streamwise elongated regions of high and low streamwise velocity and it is suggested that the breakdown into turbulent spots is related to local instabilities of the strong shear layers associated with these streaks. Flow structures typical of the spot precursors are presented and these show important similarities with the flow structures observed in previous studies on the secondary instability and breakdown of steady symmetric streaks.