In this paper it is shown that a modal detuned instability of periodic near-wall streaks originates a large-scale structure in the bulk of the turbulent channel flow. The effect of incoherent turbulent fluctuations is included in the linear operator by means of an eddy viscosity. The base flow is an array of periodic two-dimensional streaks, extracted from numerical simulations in small domains, superposed to the turbulent mean profile. The stability problem for a large number of periodic units is efficiently solved using the block-circulant matrix method proposed by Schmid et al. (Phys. Rev. Fluids, vol. 2, 2017, 113902). For friction Reynolds numbers equal or higher than $590$, it is shown that an unstable branch is present in the eigenspectra. The most unstable eigenmodes display large-scale modulations whose characteristic wavelengths are compatible with the large-scale end of the premultiplied velocity fluctuation spectra reported in previous computational studies. The wall-normal location of the large-wavelength near-wall peak in the spanwise spectrum of the eigenmode exhibits a power-law dependence on the friction Reynolds number, similarly to that found in experiments of pipes and boundary layers. Lastly, the shape of the eigenmode in the streamwise-wall-normal plane is reminiscent of the superstructures reported in the recent experiments of Deshpande et al. (J. Fluid Mech., vol. 969, 2023, A10). Therefore, there is evidence that such large-wavelength instabilities generate large-scale motions in wall-bounded turbulent flows.

Superhydrophobic surfaces dramatically reduce skin friction of overlying liquid flows. These surfaces are complex and numerical simulations usually rely on models to reduce this complexity. One of the simplest consists of finding an equivalent boundary condition through a homogenisation procedure, which in the case of channel flow over oriented riblets, leads to the presence of a small spanwise component in the homogenised base flow velocity. This work aims at investigating the influence of such a three-dimensionality of the base flow on stability and transition in a channel with walls covered by oriented riblets. Linear stability for this base flow is investigated: a new instability region, linked to cross-flow effects, is observed. Tollmien–Schlichting waves are also retrieved but the most unstable are three-dimensional. Transient growth is also affected as oblique streaks with non-zero streamwise wavenumber become the most amplified perturbations. When transition is induced by Tollmien–Schlichting waves, after an initial exponential growth regime, streaky structures with large spanwise wavenumber rapidly arise. Modal mechanisms appear to play a leading role in the development of these structures and a secondary stability analysis is performed to retrieve successfully some of their characteristics. The second scenario, initiated with cross-flow vortices, displays a strong influence of nonlinearities. The flow develops into large quasi-spanwise-invariant structures before breaking down to turbulence. Secondary stability on the saturated cross-flow vortices sheds light on this stage of transition. In both cases, cross-flow effects dominate the flow dynamics, suggesting the need to consider the anisotropicity of the wall condition when modelling superhydrophobic surfaces.

This work aims at studying the mechanisms behind the occurrence of extreme dissipation events in a channel flow, identifying nonlinear optimal perturbations as potential precursors of these events. Nonlinear optimal perturbations with respect to a generic turbulent instantaneous snapshot are computed for the first time using a direct-adjoint algorithm in the channel flow at $Re_{\tau }\approx 180$. The resulting initial perturbation displays the upstream tilting characteristic of Orr's mechanism and is positioned along the interfaces between two opposite-sign velocity streaks of the pre-existing turbulent field. Such a perturbation induces a sudden breakdown of the pre-existing structures and a heavier tail in the dissipation probability density function distribution. Different mechanisms are at play during this process: the high shear present at the interface between coherent low- and high-momentum regions is exploited to break down the larger structures and drive energy to small scales. This energy cascade is fed by an enhanced lift-up effect that produces intense streaks near the wall. It is found that the optimal perturbation grows exponentially during the first phase of its evolution reflecting the existence of a secondary modal instability of the streaks. To corroborate the results, the conditional spatiotemporal proper orthogonal decomposition (POD) analysis of Hack & Schimdt (J. Fluid Mech., vol. 907, 2021, A9) is performed both in the perturbed and in the unperturbed flow, showing a clear agreement between the two cases and with the reference study. Thus, the optimal perturbation at initial time can be considered as a precursor of extreme events.

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 $Re \lesssim 1200$, the minimal seed evolves in time, creating an isolated oblique band, whereas for $Re\gtrsim 1200$, 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.

Triggering is the process by which a linearly stable thermoacoustic system can reach self-sustained oscillations. This nonlinear phenomenon is activated only for sufficiently large amplitudes of perturbations to the equilibrium state. In this work, using a nonlinear variational optimisation method coupled with energy bisection, we compute the minimal thresholds for triggering in the Rijke tube. In particular, extending previous works, we take into account the effect of the time delay by optimising not only the perturbations at initial time, but also the velocity at the hot-wire position in the time-delay interval. We found that, for sufficiently large time delays, the nonlinearity linked to the delayed flow velocity bears a strong potential for energy growth, leading to transient amplifications of the energy reaching ${O}(10^{2})$, two orders of magnitude larger than those reported in previous studies. Notably, the gain increases with the time delay, but decreases with the initial energy of the perturbation, thus reaching very high values close to the triggering threshold of the system. The minimal energy for triggering self-sustained oscillations achieves energy values as low as ${O}(10^{-4})$, two orders of magnitude smaller than previous estimates. This indicates that, for thermoacoustic systems characterised by a non-negligible time delay, taking into account the effect of the time-delayed variables, as well as the system nonlinearity, is crucial for correctly evaluating the triggering energy thresholds.

Recently, many authors have investigated the origin and growth of turbulent bands in shear flows, highlighting the role of streaks and their inflectional instability in the process of band generation and sustainment. Recalling that streaks are created by an optimal transient growth mechanism, and motivated by the observation of a strong increase of the disturbance kinetic energy corresponding to the creation of turbulent bands, we use linear and nonlinear energy optimisations in a tilted domain to unveil the main mechanisms allowing the creation of a turbulent band in a channel flow. Linear transient growth analysis shows an optimal growth for wavenumbers having an angle of approximately $35^\circ$, close to the peak values of the premultiplied energy spectra of direct numerical simulations. This linear optimal perturbation generates oblique streaks, which, for a sufficiently large initial energy, induce turbulence in the whole domain, due to the lack of spatial localisation. However, spatially localised perturbations obtained by adding nonlinear effects to the optimisation or by artificially confining the linear optimal to a localised region in the transverse direction are characterised by a large-scale flow and lead to the generation of a localised turbulent band. These results suggest that two main elements are needed for inducing turbulent bands in a tilted domain: (i) a linear energy growth mechanism, such as the lift-up, for generating large-amplitude flow structures, which produce inflection points; (ii) spatial localisation, linked to the presence or generation of large-scale vortices. We show that these elements alone generate isolated turbulent bands also in large non-tilted domains.

Superhydrophobic surfaces dramatically reduce the skin friction of overlying liquid flows, providing a lubricating layer of gas bubbles trapped within their surface nano-sculptures. Under wetting-stable conditions, different models can be used to numerically simulate their effect on the overlying flow, ranging from spatially homogeneous slip conditions at the wall, to spatially heterogeneous slip–no-slip conditions taking into account or not the displacement of the gas–water interfaces. These models provide similar results in both laminar and turbulent regimes, but their effect on transitional flows has not been investigated yet. In this work we study, by means of numerical simulations and global stability analyses, the influence of the modelling of superhydrophobic surfaces on laminar–turbulent transition in a channel flow. For the K-type scenario, a strong transition delay is found using spatially homogeneous or heterogeneous slippery boundaries with flat, rigid liquid–gas interfaces. Whereas, when the interface dynamics is taken into account, the time to transition is reduced, approaching that of a no-slip channel flow. It is found that the interface deformation promotes ejection events creating hairpin heads that are prone to breakdown, reducing the transition delay effect with respect to flat slippery surfaces. Thus, in the case of modal transition, the interface dynamics must be taken into account for accurately estimating transition delay. Contrariwise, non-modal transition triggered by a broadband forcing is unaffected by the presence of these surfaces, no matter the surface modelling. Thus, superhydrophobic surfaces may or not influence transition to turbulence depending on the interface dynamics and on the considered transition process.

Superhydrophobic surfaces are capable of trapping gas pockets within the micro-roughnesses on their surfaces when submerged in a liquid, with the overall effect of lubricating the flow on top of them. These bio-inspired surfaces have proven to be capable of dramatically reducing skin friction of the overlying flow in both laminar and turbulent regimes. However, their effect in transitional conditions, in which the flow evolution strongly depends on the initial conditions, has still not been deeply investigated. In this work the influence of superhydrophobic surfaces on several scenarios of laminar–turbulent transition in channel flow is studied by means of direct numerical simulations. A single phase incompressible flow has been considered and the effect of the micro-structured superhydrophobic surfaces has been modelled imposing a slip condition with given slip length at both walls. The evolution from laminar, to transitional, to fully developed turbulent flow has been followed starting from several different initial conditions. When modal disturbances issued from linear stability analyses are used for perturbing the laminar flow, as in supercritical conditions or in the classical K-type transition scenario, superhydrophobic surfaces are able to delay or even avoid the onset of turbulence, leading to a considerable drag reduction. Whereas, when transition is triggered by non-modal mechanisms, as in the optimal or uncontrolled transition scenarios, which are currently observed in noisy environments, these surfaces are totally ineffective for controlling transition. Superhydrophobic surfaces can thus be considered effective for delaying transition only in low-noise environments, where transition is triggered mostly by modal mechanisms.

Transitional turbulence in shear flows is supported by a network of unstable exact invariant solutions of the Navier–Stokes equations. The network is interconnected by heteroclinic connections along which the turbulent trajectories evolve between invariant solutions. While many invariant solutions in the form of equilibria, travelling waves and periodic orbits have been identified, computing heteroclinic connections remains a challenge. We propose a variational method for computing orbits dynamically connecting small neighbourhoods around equilibrium solutions. Using local information on the dynamics linearized around these equilibria, we demonstrate that we can choose neighbourhoods such that the connecting orbits shadow heteroclinic connections. The proposed method allows one to approximate heteroclinic connections originating from states with multi-dimensional unstable manifold and thereby provides access to heteroclinic connections that cannot easily be identified using alternative shooting methods. For plane Couette flow, we demonstrate the method by recomputing three known connections and identifying six additional previously unknown orbits.

The transition to unsteadiness of a three-dimensional open cavity flow is investigated using the joint application of direct numerical simulations and fully three-dimensional linear stability analyses, providing a clear understanding of the first two bifurcations occurring in the flow. The first bifurcation is characterized by the emergence of Taylor–Görtler-like vortices resulting from a centrifugal instability of the primary vortex core. Further increasing the Reynolds number eventually triggers self-sustained periodic oscillations of the flow in the vicinity of the spanwise end walls of the cavity. This secondary instability causes the emergence of a new set of Taylor–Görtler vortices experiencing a spanwise drift directed toward the spanwise end walls of the cavity. While a two-dimensional stability analysis would fail to capture this secondary instability due to the neglect of the lateral walls, it is the first time to our knowledge that this drifting of the vortices can be entirely characterized by a three-dimensional linear stability analysis of the flow. Good agreements with experimental observations and measurements strongly support our claim that the initial stages of the transition to turbulence of three-dimensional open cavity flows are solely governed by modal instabilities.

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.

The gradient of the fluid stresses exerted on curved boundaries, conventionally computed in terms of directional derivatives of a tensor, is here analyzed by using the notion of intrinsic derivative which represents the geometrically appropriate tool for measuring tensor variations projected on curved surfaces. Relevant differences in the two approaches are found by using the classical Stokes analytical solution for the slow motion of a fluid over a fixed sphere and a numerically generated three dimensional dynamical scenario. Implications for theoretical fluid dynamics and for applied sciences are finally discussed.

Transition to turbulence in shear flows is often subcritical, thus the dynamics of the flow strongly depends on the shape and amplitude of the perturbation of the laminar state. In the state space, initial perturbations which directly relaminarize are separated from those that go through a chaotic trajectory by a hypersurface having a very small number of unstable dimensions, known as the edge of chaos. Even for the simple case of plane Couette flow in a small domain, the edge of chaos is characterized by a fractal, folded structure. Thus, the problem of determining the threshold energy to trigger subcritical transition consists in finding the states on this complex hypersurface with minimal distance (in the energy norm) from the laminar state. In this work we have investigated the minimal-energy regions of the edge of chaos, by developing a minimization method looking for the minimal-energy perturbations capable of approaching the edge state (within a prescribed tolerance) in a finite target time $T$. For sufficiently small target times, the value of the minimal energy has been found to vary with $T$ following a power law, whose best fit is given by $E_{min}\propto T^{-1.75}$. For large values of $T$, the minimal energy achieves a constant value which corresponds to the energy of the minimal seed, namely the perturbation of minimal energy asymptotically approaching the edge state (Rabin et al., J. Fluid Mech., vol. 738, 2012, R1). For $T\geqslant 40$, all of the symmetries of the edge state are broken and the minimal perturbation appears to be localized in space with a basic structure composed of scattered patches of streamwise velocity with inclined streamwise vortices on their flanks. Finally, we have found that minimal perturbations originate in a small low-energy zone of the state space and follow very fast similar trajectories towards the edge state. Such trajectories are very different from those of linear optimal disturbances, which need much higher initial amplitudes to approach the edge state. The time evolution of these minimal perturbations represents the most efficient path to subcritical transition for Couette flow.

The present work provides an optimal control strategy, based on the nonlinear Navier–Stokes equations, aimed at hampering the rapid growth of unsteady finite-amplitude perturbations in a Blasius boundary-layer flow. A variational procedure is used to find the blowing and suction control law at the wall providing the maximum damping of the energy of a given perturbation at a given target time, with the final aim of leading the flow back to the laminar state. Two optimally growing finite-amplitude initial perturbations capable of leading very rapidly to transition have been used to initialize the flow. The nonlinear control procedure has been found able to drive such perturbations back to the laminar state, provided that the target time of the minimization and the region in which the blowing and suction is applied have been suitably chosen. On the other hand, an equivalent control procedure based on the linearized Navier–Stokes equations has been found much less effective, being not able to lead the flow to the laminar state when finite-amplitude disturbances are considered. Regions of strong sensitivity to blowing and suction have been also identified for the given initial perturbations: when the control is actuated in such regions, laminarization is also observed for a shorter extent of the actuation region. The nonlinear optimal blowing and suction law consists of alternating wall-normal velocity perturbations, which appear to modify the core flow structures by means of two distinct mechanisms: (i) a wall-normal velocity compensation at small times; (ii) a rotation-counterbalancing effect al larger times. Similar control laws have been observed for different target times, values of the cost parameter, and streamwise extents of the blowing and suction zone, meaning that these two mechanisms are robust features of the optimal control strategy, provided that the nonlinear effects are taken into account.

The “analogue gravity formalism”, an interdisciplinary theoretical scheme developed in the past for studying several non relativistic classical and quantum systems through effective relativistic curved space-times, is here applied to largely de-formable elastic bodies described by the nonlinear theory of solid mechanics. Assuming the simplest nonlinear constitutive relation for the elastic material given by a Kirchhoff-St Venant strain-energy density function, it is possible to write for the perturbations an effective space-time metric if the deformation is purely longitudinal and depends on one spatial coordinate only. Theoretical and numerical studies of the corresponding dynamics are performed in selected cases and physical implications of the results obtained are finally discussed.

This work provides a global optimization analysis, looking for perturbations inducing the largest energy growth at a finite time in a boundary-layer flow in the presence of smooth three-dimensional roughness elements. Amplification mechanisms are described which can bypass the asymptotical growth of Tollmien–Schlichting waves. Smooth axisymmetric roughness elements of different height have been studied, at different Reynolds numbers. The results show that even very small roughness elements, inducing only a weak deformation of the base flow, can localize the optimal disturbance characterizing the Blasius boundary-layer flow. Moreover, for large enough bump heights and Reynolds numbers, a strong amplification mechanism has been recovered, inducing an increase of several orders of magnitude of the energy gain with respect to the Blasius case. In particular, the highest value of the energy gain is obtained for an initial varicose perturbation, differently to what found for a streaky parallel flow. Optimal varicose perturbations grow very rapidly by transporting the strong wall-normal shear of the base flow, which is localized in the wake of the bump. Such optimal disturbances are found to lead to transition for initial energies and amplitudes considerably smaller than sinuous optimal ones, inducing hairpin vortices downstream of the roughness element.

The 18O(p, α)15N reaction rate has been extracted by means of the Trojan-Horse method. For the first time the contribution of the 20-keV peak has been directly evaluated, giving a value about 35% larger than previously estimated. The present approach has allowed to improve the accuracy of a factor 8.5, as it is based on the measured strength instead of educated guesses or spectroscopic measurements. The contribution of the 90-keV resonance has been determined as well, which turned out to be of negligible importance to astrophysics.

This paper provides the analysis of bursting and transition to turbulence in a Couette flow, based on the growth of nonlinear optimal disturbances. We use a global variational procedure to identify such optimal disturbances, defined as those initial perturbations yielding the largest energy growth at a given target time, for given Reynolds number and initial energy. The nonlinear optimal disturbances are found to be characterized by a basic structure, composed of inclined streamwise vortices along localized regions of low and high momentum. This basic structure closely recalls that found in boundary-layer flow (Cherubini et al., J. Fluid Mech., vol. 689, 2011, pp. 221–253), indicating that this structure may be considered the most ‘energetic’ one at short target times. However, small differences in the shape of these optimal perturbations, due to different levels of the initial energy or target time assigned in the optimization process, may produce remarkable differences in their evolution towards turbulence. In particular, direct numerical simulations have shown that optimal disturbances obtained for large initial energies and target times induce bursting events, whereas for lower values of these parameters the flow is directly attracted towards the turbulent state. For this reason, the optimal disturbances have been classified into two classes, the highly dissipative and the short-path perturbations. Both classes lead the flow to turbulence, skipping the phases of streak formation and secondary instability which are typical of the classical transition scenario for shear flows. The dynamics of this transition scenario exploits three main features of the nonlinear optimal disturbances: (i) the large initial value of the streamwise velocity component; (ii) the streamwise dependence of the disturbance; (iii) the presence of initial inclined streamwise vortices. The short-path perturbations are found to spend a considerable amount of time in the vicinity of the edge state (Schneider et al., Phys. Rev. E, vol. 78, 2008, 037301), whereas the highly dissipative optimal disturbances pass closer to the edge, but they are rapidly repelled away from it, leading the flow to high values of the dissipation rate. After this dissipation peak, the trajectories do not lead towards the turbulent attractor, but they spend some time in the vicinity of an unstable periodic orbit (UPO). This behaviour led us to conjecture that bursting events can be obtained not only as homoclinic orbits approaching the UPO, as recently found by van Veen & Kawahara (Phys. Rev. Lett., vol. 107, 2011, p. 114501), but also as heteroclinic orbits between the equilibrium solution on the edge and the UPO.

Cancer spread is a dynamical process occurring not only in time but also in space which, for solid tumors at least, can be modeled quantitatively by reaction and diffusion equations with a bistable behavior: tumor cell colonization happens in a portion of tissue and propagates, but in some cases the process is stopped. Such a cancer proliferation/extintion dynamics is obtained in many mathematical models as a limit of complicated interacting biological fields. In this article we present a very basic model of cancer proliferation adopting the bistable equation for a single tumor cell dynamics. The reaction-diffusion theory is numerically and analytically studied and then extended in order to take into account dispersal effects in cancer progression in analogy with ecological models based on the porous medium equation. Possible implications of this approach for explanation and prediction of tumor development on the lines of existing studies on brain cancer progression are discussed. The potential role of continuum models in connecting the two predominant interpretative theories about cancer, once formalized in appropriate mathematical terms, is discussed.

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, 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.