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The stability and sensitivity of two- and three-dimensional global modes developing on steady spanwise-homogeneous laminar separated flows around NACA 4412 swept wings are numerically investigated for different Reynolds numbers ${\textit {Re}}$ and angles of attack $\alpha$. The wake dynamics is driven by the two-dimensional von Kármán mode whose emergence threshold in the $\alpha \unicode{x2013}{\textit {Re}}$ plane is computed with that of the three-dimensional centrifugal mode. At the critical Reynolds number, the Strouhal number, the streamwise wavenumber of the von Kármán mode and the spanwise wavenumber of the leading three-dimensional centrifugal mode scale as a power law of $\alpha$. The introduction of a sweep angle attenuates the growth of all unstable modes and entails a Doppler effect in the leading modes' dynamics and a shift towards non-zero frequencies of the three-dimensional centrifugal modes. These are found to be non-dispersive as opposed to the von Kármán modes. The sensitivity of the leading global modes is investigated in the vicinity of the critical conditions through adjoint-based methods. The growth-rate sensitivity map displays a region on the suction side of the wing, wherein a streamwise-oriented force has a net stabilising effect, comparable to what could have been obtained inside the recirculation bubble. In agreement with the predictions of the sensitivity analysis, a spanwise-homogeneous force suppresses the Hopf bifurcation and stabilises the entire branch of von Kármán modes. In the limit of small amplitudes, passive control via spanwise-wavy forcing produces a stabilising effect similar to that of a spanwise-homogeneous control and is more effective than localised spherical forces. [ABSTRACT FROM AUTHOR]

Resolvent analysis is used to study the low-frequency behaviour of the laminar oblique shock wave/boundary layer interaction (SWBLI). It is shown that the computed optimal gain, which can be seen as a transfer function of the system, follows a first-order low-pass filter equation, recovering the results of Touber & Sandham (J. Fluid Mech., vol. 671, 2011, pp. 417-465). This behaviour is understood as proceeding from the excitation of a single stable, steady global mode whose damping rate sets the time scale of the filter. Different Mach and Reynolds numbers are studied, covering different recirculation lengths L. This damping rate is found to scale as 1/L, leading to a constant Strouhal number StL as observed in the literature. It is associated with a breathing motion of the recirculation bubble. This analysis furthermore supports the idea that the low-frequency dynamics of the SWBLI is a forced dynamics, in which background perturbations continuously excite the flow. The investigation is then carried out for three-dimensional perturbations for which two regimes are identified. At low wavenumbers of the order of L, a modal mechanism similar to that of two-dimensional perturbations is found and exhibits larger values of the optimal gain. At larger wavenumbers, of the order of the boundary layer thickness, the growth of streaks, which results from a non-modal mechanism, is detected. No interaction with the recirculation region is observed. Based on these results, the potential prevalence of three-dimensional effects in the low-frequency dynamics of the SWBLI is discussed. [ABSTRACT FROM AUTHOR]

In this work, nonlinear variational optimization is used for obtaining minimal seeds for the formation of turbulent bands in channel flow. Using nonlinear optimization together with energy bisection, we have found that the minimal energy threshold for obtaining spatially patterned turbulence scales with Re-8.5 for Re > 1000. The minimal seed, which is different to that found in a much smaller domain, is characterized by a spot-like structure surrounded by a low-amplitude large-scale quadrupolar flow filling the whole domain. This minimal-energy perturbation of the laminar flow has dominant wavelengths close to 4 in the streamwise direction and 1 in the spanwise direction, and is characterized by a spatial localization increasing with the Reynolds number. At Re ≲ 1200, the minimal seed evolves in time, creating an isolated oblique band, whereas for Re ≲ 1200, a quasi-spanwise-symmetric evolution is observed, giving rise to two distinct bands. A similar evolution is found also at low Re for non-minimal optimal perturbations. This highlights two different mechanisms of formation of turbulent bands in channel flow, depending on the Reynolds number and initial energy of the perturbation. The selection of one of these two mechanisms appears to be dependent on the probability of decay of the newly created stripe, which increases with time, but decreases with the Reynolds number. [ABSTRACT FROM AUTHOR]

The dynamics of a fully three-dimensional lid-driven cubical cavity (3D-LDC) flow at several post-critical conditions, i.e., beyond the first bifurcation, are elucidated using both linear and nonlinear analyses. When the Reynolds number is increased beyond the critical value, symmetry breaks down intermittently with subsequent gradual growth in spanwise inhomogeneity. This results in crossflow as well as pronounced secondary flow due to enhanced imbalance between centrifugal and pressure forces. Thus, while a stable solution is obtained at Re = 1900 (Reynolds number based on lid velocity and cavity side length), nonlinear analysis identifies intermittent and nearly saturated regimes at Re = 2100 and Re = 3000 , respectively. These changes in the regime are examined by considering five basic states at different Reynolds numbers starting from Re = 1900 . At the lowest Reynolds number, linear analysis yields only symmetric modes, characterized by Taylor–Görtler-like (TGL) vortices. However, at Re = 2100 , the intermittent breakdown of symmetry results in the emergence of an antisymmetric low-frequency mode apart from primary high-frequency mode. The frequencies of both these modes are numerically close to those obtained from corresponding nonlinear simulations. When the Reynolds number is increased even further, the TGL structures drift under the influence of the crossflow to occupy the previously structureless region near the wall. The frequency of each mode decreases with increase in Re ; between 1900 and 3000, the frequency of the primary mode changes by more than 20%. Furthermore, the spatial support of each mode becomes larger within the cavity. Both primary and secondary modes are increasingly destabilized with Re ; however, the secondary antisymmetric mode is destabilized at a higher rate. The current study thus provides a comprehensive picture of the overall dynamics of 3D-LDC flows in pre- and post-bifurcation regimes in an extended Re range not considered hitherto. [ABSTRACT FROM AUTHOR]

A numerical study is conducted to unveil the large scale dynamics of a high Reynolds number axisymmetric separating/reattaching flow at M∞ = 0.7. The numerical simulation allows us to acquire a high rate sampled unsteady volumetric dataset. This huge amount of spatial and temporal information is exploited in the Fourier space to visualize for the first time in physical space and at such a high Reynolds number (ReD = 1.2 × 106) the statistical signature of the helical structure related to the antisymmetric mode (m = 1) at StD = 0.18. The main hydrodynamic mechanisms are identified through the spatial distribution of the most energetic frequencies, i.e., StD = 0.18 and StD ≥ 3.0 corresponding to the vortex-shedding and Kelvin-Helmholtz instability phenomena, respectively. In particular, the dynamics related to the dimensionless shedding frequency is shown to become dominant for 0.35 ≤ x/D ≤ 0.75 in the whole radial direction as it passes through the shear layer. The spatial distribution of the coherence function for the most significant modes as well as a three-dimensional Fourier decomposition suggests the global features of the flow mechanisms. More specifically, the novelty of this study lies in the evidence of the flow dynamics through the use of cross-correlation maps plotted with a frequency selection guided by the characteristic Strouhal number formerly identified in a local manner in the flow field or at the wall. Moreover and for the first time, the understanding of the scales at stake is supported both by a Fourier analysis and a dynamic mode decomposition in the complete three-dimensional space surrounding the afterbody zone. [ABSTRACT FROM AUTHOR]

Attached sheet cavitation is usually observed in turbulent water flows within small laminar separation bubbles which can provide favorable conditions for inception and attachment of cavities. In the present study, viscous silicone oils are used within a small scale Venturi geometry to investigate attached cavitation into laminar separated flows for Reynolds numbers from 346 to 2188. Numerical simulations about single phase flows are performed with steady simulations for a Reynolds number range Re ∈ [50; 1400] and with unsteady simulations for Re ∈ [1000; 2000]. They reveal the emergence of two large laminar boundary layer separations downstream of the Venturi throat in addition to low pressure zones which can possibly induce both degassing or cavitation features. Experiments are performed with high-speed photography, and several multiphase dynamics are observed in these viscous flows, which are considered as quasisteady flows at low Reynolds numbers Re ≤ 1400. Degassing phenomenon with air bubble recirculation has been first observed at pressures far above liquid vapor pressure whereas typical attached cavities have been identified for low pressure conditions as "band" and "tadpole" cavities into the different separations of the laminar flows. For higher Reynolds numbers, a flow regime transition can be noticed in the wake of well-developed gas structures, characterized by wake instabilities, causing vortex cavitation above a critical Reynolds number associated with the bubble width R e c b ≃ 616. This regime transition can possibly occur either quasicontinuously in the wake of an attached "band" vapor cavity or intermittently behind a recirculating air bubble generated with degassing. This last phenomenon is associated in our study to classical "patch" cavitation. [ABSTRACT FROM AUTHOR]

A fully three-dimensional linear stability analysis is carried out to investigate the unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper technique is used to compute both equilibrium states and leading eigenmodes. In agreement with previous studies, the numerical results reveal a regular bifurcation under the action of a steady mode and a supercritical Hopf bifurcation that causes the onset of unsteadiness but also illustrate the limitations of previous linear approaches, based on parallel and axisymmetric base flow assumptions, or weakly nonlinear theories. The evolution of the unstable bifurcations is investigated up to low-supersonic speeds. For increasing Mach numbers, the thresholds move towards higher Reynolds numbers. The unsteady fluctuations are weakened and an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds number, the regular bifurcation disappears and the flow directly passes from an unsteady planar-symmetric solution to a stationary axisymmetric stable one when the Mach number is increased. A stability map is drawn by tracking the bifurcation boundaries for different Reynolds and Mach numbers. When supersonic conditions are reached, the flow becomes globally stable and switches to a noise-amplifier system. A continuous Gaussian white noise forcing is applied in front of the shock to examine the convective nature of the flow. A Fourier analysis and a dynamic mode decomposition show a modal response that recalls that of the incompressible unsteady cases. Although transition in the wake does not occur for the chosen Reynolds number and forcing amplitude, this suggests a link between subsonic and supersonic dynamics. [ABSTRACT FROM AUTHOR]

The onset of unsteadiness in a boundary-layer flow past a cylindrical roughness element is investigated for three flow configurations at subcritical Reynolds numbers, both experimentally and numerically. On the one hand, a quasi-periodic shedding of hairpin vortices is observed for all configurations in the experiment. On the other hand, global stability analyses have revealed the existence of a varicose isolated mode, as well as of a sinuous one, both being linearly stable. Nonetheless, the isolated stable varicose modes are highly sensitive, as ascertained by pseudospectrum analysis. To investigate how these modes might influence the dynamics of the flow, an optimal forcing analysis is performed. The optimal response consists of a varicose perturbation closely related to the least stable varicose isolated eigenmode and induces dynamics similar to that observed experimentally. The quasi-resonance of such a global mode to external forcing might thus be responsible for the onset of unsteadiness at subcritical Reynolds numbers, hence providing a simple explanation for the experimental observations. [ABSTRACT FROM AUTHOR]

The effect of a constant homogeneous wall suction on the nonlinear transient growth of localized finite amplitude perturbations in a boundary-layer flow is investigated. Using a variational technique, nonlinear optimal disturbances are computed for the asymptotic suction boundary layer (ASBL) flow, defined as those finite amplitude disturbances yielding the largest energy growth at a given target time T. It is found that homogeneous wall suction remarkably reduces the optimal energy gain in the nonlinear case. Furthermore, mirror-symmetry breaking of the shape of the optimal perturbation appears when decreasing the Reynolds number from 10000 to 5000, whereas spanwise mirror-symmetry was a robust feature of the nonlinear optimal perturbations found in the Blasius boundary-layer flow. Direct numerical simulations show that the different evolutions of the symmetric and of the non-symmetric initial perturbations are linked to different mechanisms of transport and tilting of the vortices by the mean flow. By bisecting the initial energy of the nonlinear optimal perturbations, minimal energy thresholds for subcritical transition to turbulence have been obtained. These energy thresholds are found to be 1-4 orders of magnitude smaller than those provided in the literature for other transition scenarios. For low to moderate Reynolds numbers, the energy thresholds are found to scale with Re-2, suggesting a new scaling law for transition in the ASBL. [ABSTRACT FROM AUTHOR]

This paper describes a scenario of transition from laminar to turbulent flow in a spatially developing boundary layer over a flat plate. The base flow is the Blasius non-parallel flow solution; it is perturbed by optimal disturbances yielding the largest energy growth over a short time interval. Such perturbations are computed by a nonlinear global optimization approach based on a Lagrange multiplier technique. The results show that nonlinear optimal perturbations are characterized by a localized basic building block, called the minimal seed, defined as the smallest flow structure which maximizes the energy growth over short times. It is formed by vortices inclined in the streamwise direction surrounding a region of intense streamwise disturbance velocity. Such a basic structure appears to be a robust feature of the base flow since it is practically invariant with respect to the initial energy of the perturbation, the target time, the Reynolds number and the dimensions of the computational domain. The minimal seed grows very rapidly in time while spreading, and it triggers nonlinear effects which bring the flow to turbulence in a very efficient manner, through the formation of a turbulence spot. This evolution of the initial optimal disturbance has been studied in detail by direct numerical simulations. Using a perturbative formulation of the Navier–Stokes equations, each linear and nonlinear convective term of the equations has been analysed. The results show the fundamental role of the streamwise inclination of the vortices in the process. The nonlinear coupling of the finite amplitude disturbances is crucial to sustain such streamwise inclination, as well as to generate dislocations within the flow structures, and local inflectional velocity distributions. The analysis provides a picture of the transition process characterized by a sequence of structures appearing successively in the flow, namely, $ \mrm{\Lambda} $ vortices, hairpin vortices and streamwise streaks. Finally, a disturbance regeneration cycle is conceived, initiated by the fast nonlinear amplification of the minimal seed, providing a possible scenario for the continuous regeneration of the same fundamental flow structures at smaller space and time scales. [ABSTRACT FROM AUTHOR]

The three-dimensional global optimal dynamics of a flat-plate boundary layer is studied by means of an adjoint-based optimization in a spatial domain of long — but finite — streamwise dimension. The localized optimal initial perturbation is characterized by a pair of streamwise-modulated counter-rotating vortices, tilted upstream, yielding at the optimal time elongated streaks of alternating sign in the streamwise direction. This indicates that perturbations with non-zero streamwise wavenumber have a role in the transient dynamics of a boundary layer. A scaling law is provided, describing the variation of the streamwise modulation of the optimal initial perturbation with respect to the streamwise domain length and to the Reynolds number. For spanwise-extended domains, a near-optimal three-dimensional perturbation is extracted during the optimization process; it is localized also in the spanwise direction, resulting in a wave packet of elongated disturbances modulated in the spanwise and streamwise directions. The nonlinear evolution of the optimal and near-optimal perturbations is investigated by means of direct numerical simulations. Both perturbations are found to induce transition at lower levels of the initial energy than local optimal and suboptimal perturbations. Moreover, it is observed that transition occurs in a well-defined region of the convected wave packet, close to its centre, via a mechanism including at the same time oscillations of the streaks of both quasi-sinuous and quasi-varicose nature. Hairpin vortices are observed before transition; they have an active role in the breakdown of the streaks and result in a turbulent spot which spreads out in the boundary layer. [ABSTRACT FROM AUTHOR]

The effects of non-normality and nonlinearity of the two-dimensional Navier–Stokes differential operator on the dynamics of a large laminar separation bubble over a flat plate have been studied in both subcritical and slightly supercritical conditions. The global eigenvalue analysis and direct numerical simulations have been employed in order to investigate the linear and nonlinear stability of the flow. The steady-state solutions of the Navier–Stokes equations at supercritical and slightly subcritical Reynolds numbers have been computed by means of a continuation procedure. Topological flow changes on the base flow have been found to occur close to transition, supporting the hypothesis of some authors that unsteadiness of separated flows could be due to structural changes within the bubble. The global eigenvalue analysis and numerical simulations initialized with small amplitude perturbations have shown that the non-normality of convective modes allows the bubble to act as a strong amplifier of small disturbances. For subcritical conditions, nonlinear effects have been found to induce saturation of such an amplification, originating a wave-packet cycle similar to the one established in supercritical conditions, but which is eventually damped. A transient amplification of finite amplitude perturbations has been observed even in the attached region due to the high sensitivity of the flow to external forcing, as assessed by a linear sensitivity analysis. For supercritical conditions, the non-normality of the modes has been found to generate low-frequency oscillations (flapping) at large times. The dependence of such frequencies on the Reynolds number has been investigated and a scaling law based on a physical interpretation of the phenomenon has been provided, which is able to explain the onset of a secondary flapping frequency close to transition. [ABSTRACT FROM AUTHOR]

Abstract Coherent structures in turbulent shear flows take the form of packets of hairpin vortices reaching the outer region of the boundary layer along with streaks of different size, going from the near-wall to the outer region. The latter can be explained by the linear transient growth of the perturbations of the mean turbulent profile. Whereas, the former are recently found to be optimally-growing only in the presence of nonlinear effects, as ascertained for a turbulent channel flow at a low friction Reynolds number. The present work aims at investigating whether large-scale streaks can be optimally-growing in a nonlinear framework for a turbulent channel flow. Changing the friction Reynolds number from 180 to 590, the nonlinear optimal perturbation tends towards more robust large-scale streaks and vortical structures of smaller size. These streaks are generated by a coherent large-scale lift-up mechanism, acting as a source term in the energy balance, inducing a positive turbulent kinetic energy production at the outer scale. This indicates that the outer energy production peak arising between the two considered Reynolds numbers can be associated with the growth of optimal large-scale streaks, which represent a robust feature of turbulent channel flows. [ABSTRACT FROM AUTHOR]