Transition induced by linear and nonlinear perturbation growth in flow past a compressor blade

Abstract

Flow past a NACA 65 blade at chord-based Reynolds number 138 500 is studied using stability analysis, generalized (spatially weighted) transient growth analysis and direct numerical simulations (DNS). The mechanisms of transition on various sections of the blade observed in previous work by Zaki et al. (J. Fluid Mech., vol. 665, 2010, pp. 57-98) are examined, with a focus on the pressure side around the leading edge. In this region, the linearly most energetic perturbation has spanwise wavenumber 40π (five boundary-layer thicknesses) and is tilted against the mean shear to take advantage of the Orr mechanism. In a DNS, the nonlinear development of this optimal perturbation induces Λ structures, which are further stretched to hairpin vortices before breaking down to turbulence. At higher spanwise wavenumber, e.g. 120π, a free-stream optimal perturbation is obtained upstream of the leading edge, in the form of streamwise vortices. During its nonlinear evolution, this optimal perturbation tilts the mean shear and generates spanwise periodic high-and low-speed streaks. Then through a nonlinear lift-up mechanism, the low-speed streaks are lifted above the high-speed ones. This layout of streaks generates a mean shear with two inflectional points and activates secondary instabilities, namely inner and outer instabilities previously reported in the literature.

Laminar–turbulent transition in channel flow with superhydrophobic surfaces modelled as a partial slip wall

“…Sweep and ejection events are attenuated when the slip length is increased from L s = 0.0 to L s = 0.01 (compare figure 11a with 11b), whereas the u + and u − velocity profiles in figure 11(d,e), representing the wake of hairpin legs and heads respectively, remain qualitatively similar. In the case L s = 0.02, Q4 events are strongly inhibited and the probability density function is mostly dominated by Q2-Q3 events (see figure 11c), indicating that the flow is characterised by stronger low-speed streaks whose peak value moves upwards in the wall-normal direction (see figure 11c, bottom), further evidence of the development of nonlinear streaks (Mao et al 2017). Very similar u-v distribution is found further from the wall (not shown), confirming that the reduced shear succeeds in inhibiting Q4 events everywhere in the flow.…”

“…Once A s reaches a threshold amplitude, nonlinear effects set in (according to Brandt et al (2003), A s ≈ 26 % of the free-stream velocity for sinuous instability), and lead to a saturation of both Re τ and A s , while the region of high amplitude ω z departs from the wall. This indicates that the initial 'linear' streaks are deformed in the wall-normal direction due to nonlinear effects (Mao et al 2017). Finally, at t > 400, abrupt secondary instability of these nonlinearly saturated flows occurs and breakdown to turbulence is finally reached.…”

Low-frequency selectivity in flat-plate boundary layer with elliptic leading edge

“…was explained in terms of a nonlinear lift-up mechanism, and it was shown to render the flow unstable to high-frequency perturbations (Mao et al 2017). In order to illustrate how the inflow streamwise and vertical vorticity induce the streaky boundary-layer response, we examine the leading-edge region.…”

“…At larger x, the base flow becomes more unstable even though the streak amplitude, as defined above in (6.1) and illustrated in figure 15, is slightly reduced. This further destabilization is therefore associated with the change in the streak shape, which becomes non-periodic in the spanwise direction due to the nonlinear lift-up (Mao et al 2017). The fastest-amplifying instability has a streamwise wavenumber α ≈ 1.…”

Bypass transition in flow over a vibrating flat plate

“…After the formation and amplification of streaks via the linear lift-up mechanism, the next stage of bypass transition is the nonlinear deformation of streaks and their secondary instabilities. The interaction between the streaks and the streamwise vorticity lifts the low-(or high-)speed streaks away from (or towards) the wall via a nonlinear lift-up mechanism, resulting in a mean shear profile with an inflection point, which is prone to shear flow instabilities (Mao et al 2017). The lifted low-speed streaks are also exposed to the high-frequency free-stream disturbances near the top of boundary layer, activating secondary instabilities of streaks (Jacobs & Durbin 2001;Brandt, Schlatter & Henningson 2004;Zaki & Durbin 2005).…”