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The aim of this paper is to investigate the linear and weakly nonlinear dynamics in flow over a flat-plate with leading edge. Linear optimal and suboptimal inflow perturbations are obtained using a Lagrangian multiplier technique. In particular, the suboptimal inflow conditions and the corresponding downstream responses are investigated in detail for the first time. Unlike the suboptimal dynamics reported in other canonical cases such as the backward-facing step flow, the growth rate of the suboptimal perturbation is in the same order as the optimal one, and both of them depend on the lift-up mechanism even though they are orthogonal. The suboptimal mode has an additional layer of vorticity that penetrates into the boundary layer farther downstream, generating a second patch of high- and low-speed streaks. The farther suboptimal ones spread to the free-stream without entering the boundary layer. The weakly nonlinear dynamics are examined by decomposing the flow field into multiple orders of perturbations using the Volterra series. Small structures in the higher order perturbations mainly concentrate in the region farther away from wall, suggesting a mechanism of outward perturbation developments, which is opposite with the well reported inward development of perturbations, i.e., from free-stream to boundary layer. The significance of these modes is then demonstrated through a prediction of flow field from the inflow condition by exploiting the orthogonality of the modes. [ABSTRACT FROM AUTHOR]

The development of free-stream disturbances in flow over a vertically vibrating flat plate with a slender leading edge is investigated. The evolution of the optimal inflow perturbation that results in the maximum amplification is computed to investigate the effect of the plate vibration on the development of free-stream disturbance, secondary instability of streaks and subsequently the bypass transition to turbulence. It is observed that the plate vibration leads to periodic change of the angle of attack, shifting the free-stream disturbance to the upper or lower side of the plate. Therefore, the development of steady inflow perturbations, which receive the largest amplification, is interrupted by the vibration, and the perturbation amplification via the lift-up mechanism is weakened. The vibration brings a second peak of perturbation growth at the vibration frequency, leading to high-frequency free-stream perturbations penetrating into the base boundary layer, which is not observed in flow over a stationary plate owing to the sheltering mechanism. This resonance of the flow perturbation and the vibrating plate is explained by the staggering effect of the leading edge. Further, the direct numerical simulations with the optimal inflow perturbation imposed on the inflow boundary show that the vertical vibration of the plate leads to streamwise periodic vorticity near the edge of the boundary layer. This inhomogeneity of the streamwise vorticity brings about streamwisely localized distortion of the low-speed streaks and, thus, an intermittent secondary instability. Therefore, before the streaks break down to turbulence, they undergo several rounds of secondary instabilities, resulting in an elongated bypass transition process. [ABSTRACT FROM AUTHOR]

Sensitivity of drag to tangential surface motion is calculated in flow past a circular cylinder in both two- and three-dimensional conditions at Reynolds number Re≤1000. The magnitude of the sensitivity maximises in the region slightly upstream of the separation points where the contour lines of spanwise vorticity are normal to the cylinder surface. A control to reduce drag can be obtained by (negatively) scaling the sensitivity. The high correlation of sensitivities of controlled and uncontrolled flow indicates that the scaled sensitivity is a good approximation of the nonlinear optimal control. It is validated through direct numerical simulations that the linear range of the steady control is much higher than the unsteady control, which synchronises the vortex shedding and induces lock-in effects. The steady control injects angular momentum into the separating boundary layer, stabilises the flow and increases the base pressure significantly. At Re=100, when the maximum tangential motion reaches 50% of the free-stream velocity, the vortex shedding, boundary-layer separation and recirculation bubbles are eliminated and 32% of the drag is reduced. When the maximum tangential motion reaches 2.5 times of the free-stream velocity, thrust is generated and the power savings ratio, defined as the ratio of the reduced drag power to the control input power, reaches 19.6. The mechanism of drag reduction is attributed to the change of the radial gradient of spanwise vorticity (∂rζ^) and the subsequent accelerated pressure recovery from the uncontrolled separation points to the rear stagnation point. [ABSTRACT FROM AUTHOR]

The interaction of a vortex and a pipe flow, modelled as the Lamb-Oseen vortex and the Poiseuille flow, respectively, is investigated by means of stability analyses and direct numerical simulations (DNS). From the distribution of the most unstable mode, it is observed that the instability is induced by the combination of the radial gradients of the base azimuthal and axial velocity components, e.g. an axial (or azimuthal) vorticity perturbation acts on the axial (or azimuthal) base velocity via a lift-up effect to generate axial (or azimuthal) velocity streaks, which are further stretched by the base azimuthal (or axial) velocity to create azimuthal (or axial) vorticity. This lift-up-stretch mechanism is confirmed in DNS of the model base flow initially perturbed by the most unstable mode. After nonlinear saturation, the perturbations decay since the flow no longer supports instability after sufficient radial mixing induced by the lift-up of the azimuthal and axial velocity components. These observations suggest that the vorticity outside the vortex core can be suppressed by instabilities if a streamwise boundary layer flow exists outside the core. [ABSTRACT FROM AUTHOR]

The spectra of the Batchelor vortex are obtained by discretizing its linearized evolution operator using a modified Chebyshev polynomial approximation at a Reynolds number of 1000 and zero azimuthal wavenumber. Three types of eigenmodes are identified from the spectra: discrete modes, potential modes and free-stream modes. The discrete modes have been extensively documented but the last two modes have received little attention. A convergence study of the spectra and pseudospectra supports the classification that discrete modes correspond to discrete spectra while the other two modes correspond to continuous spectra. Free-stream modes have finite amplitude in the far field whilst potential modes decay to zero in the far field. The free-stream modes are therefore a limiting form of the potential modes when the radial decay rate of velocity components reduces to zero. The radial form of the free-stream modes with axial and radial wavenumbers is investigated and the penetration of the free-stream mode into the vortex core highlights the possibility of interaction between the potential region and the vortex core. A wavepacket pseudomode study confirms the existence of continuous spectra and predicts the locations and radial wavenumbers of the eigenmodes. The pseudomodes corresponding to the potential modes are observed to be in the form of one or two wavepackets while the free-stream modes are not observed to be in the form of wavepackets. [ABSTRACT FROM PUBLISHER]