Variational Nonlinear Optimization in Fluid Dynamics: The Case of a Channel Flow with Superhydrophobic Walls

Variational optimization has been recently applied to nonlinear systems with many degrees of freedom such as shear flows undergoing transition to turbulence. This technique has unveiled powerful energy growth mechanisms able to produce typical coherent structures currently observed in transition and turbulence. However, it is still not clear the extent to which these nonlinear optimal energy growth mechanisms are robust with respect to external disturbances or wall imperfections. Within this framework, this work aims at investigating how nano-roughnesses such as those of superhydrophobic surfaces affect optimal energy growth mechanisms relying on nonlinearity. Nonlinear optimizations have been carried out in a channel flow with no-slip and slippery boundaries, mimicking the presence of superhydrophobic surfaces. For increasing slip length, the energy threshold for obtaining hairpin-like nonlinear optimal perturbations slightly rises, scaling approximately with Re−2.36 no matter the slip length. The corresponding energy gain increases with Re with a slope that reduces with the slip length, being almost halved for the largest slip and Reynolds number considered. This suggests a strong effect of boundary slip on the energy growth of these perturbations. While energy is considerably decreased, the shape of the optimal perturbation barely changes, indicating the robustness of optimal perturbations with respect to wall slip

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

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

On the influence of the modelling of superhydrophobic surfaces on laminar–turbulent transition

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

A synthetic forcing to trigger laminar-turbulent transition in parallel wall bounded flows via receptivity

Research on laminar-turbulent transition of wall-bounded parallel flows has usually focused on controlled scenarios where transition is triggered by perturbations having simple shapes and spectra. These disturbances strongly differ from the environmental noise usually present in experimental setups or industrial applications, where uncontrolled transition is usually observed. In this paper a new method is proposed to trigger uncontrolled transition to turbulence in wall-bounded parallel flows exploiting the receptivity of the flow to a volume forcing. Using some concepts provided by linear stability and sensitivity analysis, such as the resolvent, we propose a method for constructing a volume forcing capable of inducing stochastic velocity perturbations with a prescribed energy level, eventually leading to laminar-turbulent transition as a response of the system to external noise. The method has been tested in a channel flow configuration, using direct numerical simulations of the fully nonlinear Navier-Stokes equations in the presence of the volume forcing constructed on the basis of optimal forcing functions. Subcritical transition to turbulence induced by the prescribed forcing has been investigated and compared to other transition scenarios, where deterministic perturbations are imposed for obtaining a turbulent flow. Finally, the fully developed turbulent flows induced by the proposed method has been analysed, showing that low-order statistics and energy balance equations are practically unaffected by the continuous synthetic forcing

Successive bifurcations in a fully three-dimensional open cavity flow

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

Delay transition to turbulence by mimicking Lotus leaves

​ Nombreuses stratégies de contrôle ont été récemment proposées par la communauté scientifique afin depouvoir réduire la traînée dans les écoulements pariétaux. Entre autres, les Surfaces Superhydrophobes (SHS) ontmontré leurs capacités de pouvoir réduire considérablement le frottement pariétal d’un écoulement liquide grâce à laprésence de microbulles de gaz piégées dans les nano-rugosités de la surface. Dans des conditions géométrique etthermodynamique données pour lesquelles la transition de mouillage est évitée (condition pour laquelle normalementla taille des rugosités qui caractérise la SHS est de plusieurs ordres de grandeur plus petite que l'échellecaractéristique de l'écoulement principal), on peut atteindre ce qu’on appelle ‘l'effet Lotus’, pour lequel l'écoulementglisse à la paroi, avec une vitesse différente de zéro.. Dans ce cadre, nous nous sommes proposés d’étudier, à l’aidede simulations numériques l’influence des SHS sur la transition laminaire-turbulent dans un écoulement de canal.Pour cela, nous avons réalisé une série de simulations numériques directes (DNS), allant de l'état laminaire au casturbulent pleinement développé, en traitant la plupart de scénarios de transition connu en littérature. Des analyses destabilité locale et globale ont aussi été réalisées afin de déterminer l’influence de ces surfaces sur la première phasedu processus de transition. Bien que la procédure de déclenchement de la transition contrôlée (type K, H, C,...) soitbien décrite dans la littérature, cela n’est pas le cas pour les transitions naturelles. À cette fin, une nouvelle méthode aété développée pour déclencher puis étudier la transition naturelle dans des écoulements de type canal. Cette méthodeest basée sur des mécanismes de réceptivité de l'écoulement (resolvent global) permettant de construire un forçagevolumique spécifique. Plusieurs approches pour modéliser les SHS ont été utilisées, de complexités croissantes, touten tenant en compte des caractéristiques physiques de ces surfaces. Dans un premier temps, une condition deglissement homogène a été utilisée et son influence analysée. Chaque rugosité a été ensuite discrétisée spatialement,d’abord avec une alternance de condition limite sur une surface plate, ensuite en tenant compte de la dynamique del’interface gaz-liquide par une méthode Lagrangienne-Eulerienne Arbitraire (ALE). Nous avons montré que les SHSpermettent d’efficacement retarder les transitions contrôlées mais qu’en revanche elles ont peu d’influence sur lestransitions naturelles (développant des stries de vitesse). En effet, ce comportement dérive de l'équilibre entre deuxeffets contradictoires. D’un côté, le glissement pariétal nuit au développement des structures cohérentes de typehairpin ​ , en altérant le processus de ​ vortex stretching-tilting ​ . D’autre part, le mouvement de l’interface gaz-liquideinteragit avec les structures cohérentes de l'écoulement, en produisant des vitesses normales à la paroi favorisantdavantage le processus de ​ sweep-ejection et entraînant le développement de structures en forme d’arche. Nous avonsmontré que les interfaces gaz-liquide statiques retardent la transition de façon analogue à une condition aux limiteshomogène (si l’hétérogénéité pariétale est petite). En revanche la prise en compte de leur dynamique limite le retardde la transition, montrant l’importance du modèle de SHS dans les écoulements transitionnels.Many passive control strategies have been recently proposed for reducing drag in wall-bounded shearflows. Among them, underwater SuperHydrophobic Surfaces (SHS) have proven to be capable of dramaticallyreducing the skin friction of a liquid flowing on top of them, due to the presence of gas bubbles trapped within thesurface nano-sculptures. In specific geometrical and thermodynamical conditions for which wetting transition isavoided (in particular, when the roughness elements characterizing the SHS are several orders of magnitude smallerthan the overlying flow), the so-called ’Lotus effect’ is achieved, for which the flow appears to slip on the surfacewith a non zero velocity. In this framework, we propose to study, by means of numerical simulations, the influence ofSHS on laminar-turbulent transition in a channel flow. To do so we have performed a series of direct numericalsimulations (DNS), from the laminar to the fully turbulent state, covering the majority of transition scenarios knownin the literature, as well as local and global stability analysis so to determine the influence of SHS onto the initialstages of the process. While the conditions for observing controlled K-type transition in a temporal channel flow arewell defined, this is not the case for uncontrolled ones. To this end, a novel theoretical numerical framework has beendeveloped so to enable the observation of natural transition in wall-bounded flows. This method, similarly to theFree-Stream-Turbulence framework available for the boundary layer flow, is capable of triggering uncontrolledtransition t​ hrough flow receptivity to a purpose-built forcing. Different surface modellings for the superhydrophobicsurfaces are tested. First, homogeneous slip conditions are used. Then, the spatial heterogeneity of the SHS has beenconsidered by modelling it as a flat surface with alternating slip no-slip boundary conditions. Finally, the dynamics ofeach microscopic liquid-gas free-surface has been taken into account by means of a fully coupled fluid-structuresolver, using an Arbitrary Lagrangian Eulerian formulation. We show that while SHS are ineffective in controllingtransition in noisy environment​ , they can strongly delay transition to turbulence for the K-type scenario​ . Thisbehaviour results from the balance of two opposing effects. On one hand slippery surfaces inhibit the development ofcharacteristic hairpin vortices by altering the vortex stretching-tilting process. On the other hand, the movement ofthe gas-liquid free-surfaces interacts with the overlying coherent structures, producing wall-normal velocities thatenhance the sweep-ejection process, leading to a rapid formation of hairpin-like head vortices. Thus, whenconsidering flat interfaces transition time is strongly increased, while taking into account the interface dynamicsinduces smaller changes with respect to the no-slip case, indicating the need for an appropriate modelling of SHS fortransition delay purposes

Retarder la transition vers la turbulence en imitant les feuilles de lotus

Many passive control strategies have been recently proposed for reducing drag in wall-bounded shearflows. Among them, underwater SuperHydrophobic Surfaces (SHS) have proven to be capable of dramaticallyreducing the skin friction of a liquid flowing on top of them, due to the presence of gas bubbles trapped within thesurface nano-sculptures. In specific geometrical and thermodynamical conditions for which wetting transition isavoided (in particular, when the roughness elements characterizing the SHS are several orders of magnitude smallerthan the overlying flow), the so-called ’Lotus effect’ is achieved, for which the flow appears to slip on the surfacewith a non zero velocity. In this framework, we propose to study, by means of numerical simulations, the influence ofSHS on laminar-turbulent transition in a channel flow. To do so we have performed a series of direct numericalsimulations (DNS), from the laminar to the fully turbulent state, covering the majority of transition scenarios knownin the literature, as well as local and global stability analysis so to determine the influence of SHS onto the initialstages of the process. While the conditions for observing controlled K-type transition in a temporal channel flow arewell defined, this is not the case for uncontrolled ones. To this end, a novel theoretical numerical framework has beendeveloped so to enable the observation of natural transition in wall-bounded flows. This method, similarly to theFree-Stream-Turbulence framework available for the boundary layer flow, is capable of triggering uncontrolledtransition t​ hrough flow receptivity to a purpose-built forcing. Different surface modellings for the superhydrophobicsurfaces are tested. First, homogeneous slip conditions are used. Then, the spatial heterogeneity of the SHS has beenconsidered by modelling it as a flat surface with alternating slip no-slip boundary conditions. Finally, the dynamics ofeach microscopic liquid-gas free-surface has been taken into account by means of a fully coupled fluid-structuresolver, using an Arbitrary Lagrangian Eulerian formulation. We show that while SHS are ineffective in controllingtransition in noisy environment​ , they can strongly delay transition to turbulence for the K-type scenario​ . Thisbehaviour results from the balance of two opposing effects. On one hand slippery surfaces inhibit the development ofcharacteristic hairpin vortices by altering the vortex stretching-tilting process. On the other hand, the movement ofthe gas-liquid free-surfaces interacts with the overlying coherent structures, producing wall-normal velocities thatenhance the sweep-ejection process, leading to a rapid formation of hairpin-like head vortices. Thus, whenconsidering flat interfaces transition time is strongly increased, while taking into account the interface dynamicsinduces smaller changes with respect to the no-slip case, indicating the need for an appropriate modelling of SHS fortransition delay purposes.​ Nombreuses stratégies de contrôle ont été récemment proposées par la communauté scientifique afin depouvoir réduire la traînée dans les écoulements pariétaux. Entre autres, les Surfaces Superhydrophobes (SHS) ontmontré leurs capacités de pouvoir réduire considérablement le frottement pariétal d’un écoulement liquide grâce à laprésence de microbulles de gaz piégées dans les nano-rugosités de la surface. Dans des conditions géométrique etthermodynamique données pour lesquelles la transition de mouillage est évitée (condition pour laquelle normalementla taille des rugosités qui caractérise la SHS est de plusieurs ordres de grandeur plus petite que l'échellecaractéristique de l'écoulement principal), on peut atteindre ce qu’on appelle ‘l'effet Lotus’, pour lequel l'écoulementglisse à la paroi, avec une vitesse différente de zéro.. Dans ce cadre, nous nous sommes proposés d’étudier, à l’aidede simulations numériques l’influence des SHS sur la transition laminaire-turbulent dans un écoulement de canal.Pour cela, nous avons réalisé une série de simulations numériques directes (DNS), allant de l'état laminaire au casturbulent pleinement développé, en traitant la plupart de scénarios de transition connu en littérature. Des analyses destabilité locale et globale ont aussi été réalisées afin de déterminer l’influence de ces surfaces sur la première phasedu processus de transition. Bien que la procédure de déclenchement de la transition contrôlée (type K, H, C,...) soitbien décrite dans la littérature, cela n’est pas le cas pour les transitions naturelles. À cette fin, une nouvelle méthode aété développée pour déclencher puis étudier la transition naturelle dans des écoulements de type canal. Cette méthodeest basée sur des mécanismes de réceptivité de l'écoulement (resolvent global) permettant de construire un forçagevolumique spécifique. Plusieurs approches pour modéliser les SHS ont été utilisées, de complexités croissantes, touten tenant en compte des caractéristiques physiques de ces surfaces. Dans un premier temps, une condition deglissement homogène a été utilisée et son influence analysée. Chaque rugosité a été ensuite discrétisée spatialement,d’abord avec une alternance de condition limite sur une surface plate, ensuite en tenant compte de la dynamique del’interface gaz-liquide par une méthode Lagrangienne-Eulerienne Arbitraire (ALE). Nous avons montré que les SHSpermettent d’efficacement retarder les transitions contrôlées mais qu’en revanche elles ont peu d’influence sur lestransitions naturelles (développant des stries de vitesse). En effet, ce comportement dérive de l'équilibre entre deuxeffets contradictoires. D’un côté, le glissement pariétal nuit au développement des structures cohérentes de typehairpin ​ , en altérant le processus de ​ vortex stretching-tilting ​ . D’autre part, le mouvement de l’interface gaz-liquideinteragit avec les structures cohérentes de l'écoulement, en produisant des vitesses normales à la paroi favorisantdavantage le processus de ​ sweep-ejection et entraînant le développement de structures en forme d’arche. Nous avonsmontré que les interfaces gaz-liquide statiques retardent la transition de façon analogue à une condition aux limiteshomogène (si l’hétérogénéité pariétale est petite). En revanche la prise en compte de leur dynamique limite le retardde la transition, montrant l’importance du modèle de SHS dans les écoulements transitionnels

Confined self-propulsion of an isotropic active colloid

To spontaneously break their intrinsic symmetry and self-propel at the micron scale, isotropic active colloidal particles and droplets exploit the non-linear convective transport of chemical solutes emitted/consumed at their surface by the surface-driven fluid flows generated by these solutes. Significant progress was recently made to understand the onset of self-propulsion and non-linear dynamics. Yet, most models ignore a fundamental experimental feature, namely the spatial confinement of the colloid, and its effect on propulsion. In this work, the self-propulsion of an isotropic colloid inside a capillary tube is investigated numerically. A flexible computational framework is proposed based on a finite-volume approach on adaptative octree-grids and embedded boundary methods. This method is able to account for complex geometric confinement, the nonlinear coupling of chemical transport and flow fields, and the precise resolution of the surface boundary conditions, that drive the system's dynamics. Somewhat counter-intuitively, spatial confinement promotes the colloid's spontaneous motion by reducing the minimum advection-to-diffusion ratio or Péclet number, Pe, required to self-propel; furthermore, self-propulsion velocities are significantly modified as the colloid-to-capillary size ratio κ is increased, reaching a maximum at fixed Pe for an optimal confinement 0 < κ < 1. These properties stem from a fundamental change in the dominant chemical transport mechanism with respect to the unbounded problem : with diffusion now restricted in most directions by the confining walls, the excess solute is predominantly convected away downstream from the colloid, enhancing front-back concentration contrasts. These results are confirmed quantitatively using conservation arguments and lubrication analysis of the tightly-confined limit, κ → 1

Variational Nonlinear Optimization in Fluid Dynamics: The Case of a Channel Flow with Superhydrophobic Walls

Variational optimization has been recently applied to nonlinear systems with many degrees of freedom such as shear flows undergoing transition to turbulence. This technique has unveiled powerful energy growth mechanisms able to produce typical coherent structures currently observed in transition and turbulence. However, it is still not clear the extent to which these nonlinear optimal energy growth mechanisms are robust with respect to external disturbances or wall imperfections. Within this framework, this work aims at investigating how nano-roughnesses such as those of superhydrophobic surfaces affect optimal energy growth mechanisms relying on nonlinearity. Nonlinear optimizations have been carried out in a channel flow with no-slip and slippery boundaries, mimicking the presence of superhydrophobic surfaces. For increasing slip length, the energy threshold for obtaining hairpin-like nonlinear optimal perturbations slightly rises, scaling approximately with Re&minus;2.36 no matter the slip length. The corresponding energy gain increases with Re with a slope that reduces with the slip length, being almost halved for the largest slip and Reynolds number considered. This suggests a strong effect of boundary slip on the energy growth of these perturbations. While energy is considerably decreased, the shape of the optimal perturbation barely changes, indicating the robustness of optimal perturbations with respect to wall slip