This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.

Direct numerical simulation (DNS) and large-eddy simulation (LES) are carried out to investigate the frequency effect of zero-net-mass-flux forcing (synthetic jet) on a generic separated flow. The selected test case is a rounded ramp at a Reynolds number based on the step height of 28 275. The incoming boundary layer is fully turbulent withRθ=1410. The whole flow in the synthetic jet cavity is computed to ensure an accurate description of the actuator effect on the flow field. In a first step, DNS is used to validate LES of this particular flow. In a second step, the effect of a synthetic jet at two reduced frequencies of 0.5 and 4 (based on the separation length of the uncontrolled case and the free-stream velocity) is investigated using LES. It is demonstrated that, with a proper choice of the oscillating frequency, separation can be drastically reduced for a velocity ratio between the jet and the flow lower than one. The low frequency is close to the natural vortex shedding frequency. Two different modes of the synthetic jet have been identified. Avorticity-dominated modeis observed in the low-frequency forcing case for which the separation length is reduced by 54%, while anacoustic-dominated modeis identified in the high-frequency forcing case for which the separation length is increased by 43%. The decrease of the separation length in the low-frequency forcing case is correlated with an increase of the turbulent kinetic energy level and consequently with an increase of the entrainment in the separated zone. A linear inviscid stability analysis shows that the increase of the separation length in the high-frequency forcing case is due to a modification of the mean velocity profile suggested by Stanek and coworkers. The result is a lower amplification of the perturbations and consequently, a lower entrainment into the mixing layer. To our knowledge, it is the first time that Stanek's hypothesis has been assessed, thanks to numerical simulations of fully turbulent flow.

The objective of this paper is to analyze the 3D buffet phenomenon which appears on a swept wing at a high Mach number and/or high angle of attack. This aerodynamic instability induces strong wall pressure fluctuations and as such limits aircraft envelope. Consequently, it is interesting to understand this phenomenon in order to not only improve aircraft performance but also to provide more flexibility during the design phase. Results from two wind tunnel tests on a 3D half wing-body configuration are presented for several freestream Mach numbers (0.78–0.86) and Reynolds numbers (2.83 × 106–8.49 × 106, based on the aerodynamic mean chord). The buffet phenomenon is characterized using steady and unsteady wall pressure measurements. By opposition to the 2D buffet which exhibits rather a well marked peak in the pressure spectra, the 3D buffet is characterized by a broadband bump at a much higher Strouhal number (between 4 and 7 times higher). It is also observed that two different instabilities coexist on the suction side of the wing: the 3D buffet phenomenon (with Strouhal numbers ranging between 0.2 and 0.6) and the Kelvin-Helmholtz instability (with Strouhal numbers ranging between 1 and 4). Each phenomenon has a different Strouhal number but also different convection velocities and propagation directions.

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