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

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 53
Author(s):  
Stefania Cherubini ◽  
Francesco Picella ◽  
Jean-Christophe Robinet
Keyword(s):  
Channel Flow ◽  
Degrees Of Freedom ◽  
Slip Length ◽  
Superhydrophobic Surfaces ◽  
Transition To Turbulence ◽  
Growth Mechanisms ◽  
Boundary Slip ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Optimal Energy

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.

scholarly journals SUPERHYDROPHOBIC SURFACES FOR DRAG REDUCTION

2014 ◽  
Author(s):  
Alessandro Bottaro
Keyword(s):  
Drag Reduction ◽  
Slip Length ◽  
Superhydrophobic Surfaces ◽  
Transition To Turbulence ◽  
Effective Coupling ◽  
Instability Waves ◽  
Low Surface Energy ◽  
Apparent Slip ◽  
Energy Material ◽  
Relevant Length Scale

Properties of superhydrophobic materials are examined in light of their possible use for drag reduction in naval applications. To achieve superhydrophobicity a low-surface-energy material must be structured so as to minimize the liquid-solid interactions. The crucial aspect is that of maintaining a layer of gas in between the (rough) wall and the liquid, and this can be achieved by hierarchical micro- and nano-structuring of the solid surface, to ensure a sufficiently large apparent slip of the fluid at the wall, thus reducing skin friction. The behavior of the liquid is quantified by a slip length; recent results have shown that this length can be as large as 400 μm. As far as transition to turbulence is concerned, we show that superhydrophobic surfaces are effective (i.e. they delay the onset of travelling instability waves) only in channels with characteristic dimensions of a few millimeters. Conversely, when the fluid flow has already attained a turbulent state, the gain in term of drag reduction can be very significant also in macroscopic configurations. This occurs because the relevant length scale of the boundary layer is now the thickness of the viscous sub-layer, which can be of magnitude comparable to the slip length, so that an effective coupling emerges. Finally, some procedures to produce superhydrophobic surfaces are examined, in light of the possible application of such innovative coatings on the hull of ships.

Linear and nonlinear evolution of a localized disturbance in polymeric channel flow

2014 ◽  
Vol 760 ◽  
pp. 278-303 ◽  
Author(s):  
Akshat Agarwal ◽  
Luca Brandt ◽  
Tamer A. Zaki
Keyword(s):  
Channel Flow ◽  
Large Scale ◽  
Reynolds Shear Stress ◽  
Polymer Concentration ◽  
Streamwise Velocity ◽  
Transition To Turbulence ◽  
Velocity Perturbation ◽  
Nonlinear Growth ◽  
Energy Growth ◽  
Newtonian Flows

AbstractThe evolution of an initially localized disturbance in polymeric channel flow is investigated, with the FENE-P model used to characterize the viscoelastic behaviour of the flow. In the linear growth regime, the flow response is stabilized by viscoelasticity, and the maximum attainable disturbance-energy amplification is reduced with increasing polymer concentration. The reduction in the energy growth rate is attributed to the polymer work, which plays a dual role. First, a spanwise polymer-work term develops, and is explained by the tilting action of the wall-normal vorticity on the mean streamwise conformation tensor. This resistive term weakens the spanwise velocity perturbation thus reducing the energy of the localized disturbance. The second action of the polymer is analogous, with a wall-normal polymer work term that weakens the vertical velocity perturbation. Its indirect effect on energy growth is substantial since it reduces the production of Reynolds shear stress and in turn of the streamwise velocity perturbation, or streaks. During the early stages of nonlinear growth, the dominant effect of the polymer is to suppress the large-scale streaky structures which are strongly amplified in Newtonian flows. As a result, the process of transition to turbulence is prolonged and, after transition, a drag-reduced turbulent state is attained.

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

2019 ◽  
Vol 881 ◽  
pp. 462-497 ◽  
Author(s):  
Francesco Picella ◽  
J.-Ch. Robinet ◽  
S. Cherubini
Keyword(s):  
Channel Flow ◽  
Initial Conditions ◽  
Slip Length ◽  
Low Noise ◽  
Superhydrophobic Surfaces ◽  
Partial Slip ◽  
Turbulent Transition ◽  
Onset Of Turbulence ◽  
Fully Developed Turbulent Flow ◽  
Laminar Turbulent Transition

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.

scholarly journals Localization of flow structures using -norm optimization

2013 ◽  
Vol 729 ◽  
pp. 672-701 ◽  
Author(s):  
D. P. G. Foures ◽  
C. P. Caulfield ◽  
P. J. Schmid
Keyword(s):  
Channel Flow ◽  
Energy Production ◽  
Worst Case Analysis ◽  
Two Dimensional ◽  
Time Intervals ◽  
Worst Case ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Perturbation Problem ◽  
Production Zone

AbstractStability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density ${E}_{1} (t)= (1/ {V}_{\Omega } )\int \nolimits _{\Omega } e(\boldsymbol{x}, t)\hspace{0.167em} \mathrm{d} \Omega $ (where $e(\boldsymbol{x}, t)= \vert \boldsymbol{u}{\vert }^{2} / 2$) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain $\Omega $ of volume ${V}_{\Omega } $. This paper explores the use of $p$-norms in determining optimal perturbations for ‘energy’ growth over prescribed time intervals of length $T$. For $p= 1$ the traditional energy-based stability analysis is recovered, while for large $p\gg 1$, localization of the optimal perturbations is observed which identifies confined regions, or ‘hotspots’, in the domain where significant energy growth can be expected. In addition, the $p$-norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the $\infty $-norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the $\infty $-norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed $p$-norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth.

RESONANCE OF NONMODAL PERTURBATIONS IN THE BATCHELOR VORTEX

2010 ◽  
Vol 24 (13) ◽  
pp. 1449-1452
Author(s):  
ZHI-WEI GUO ◽  
DE-JUN SUN
Keyword(s):  
Resonant Mode ◽  
Resonance Phenomenon ◽  
Mode Switch ◽  
Energy Growth ◽  
Optimal Energy ◽  
Maximum Value ◽  
Resonant Character ◽  
Swirl Parameter

The resonance phenomenon for nonmodal perturbation of Batchelor vortex is studied. For azimuthal wavenumber n = - 1, two resonant peaks appear and the left one is always dominant. For n = 1, the resonant character becomes very complicated. There is a resonant mode switch from right peak to left peak as swirl parameter q increases from 2 to infinity. The resonant wavenumber k is the largest when q approaches to infinity for n = - 1 while it is the smallest for n = 1. The maximum value of the optimal energy growth for n = 1 is at q approaches to infinity, whereas it decreases monotonically as q increases for n = - 1. The resonance for n = - 1 is the more important one.

Turbulent flow over superhydrophobic surfaces with streamwise grooves

2014 ◽  
Vol 747 ◽  
pp. 186-217 ◽  
Author(s):  
S. Türk ◽  
G. Daschiel ◽  
A. Stroh ◽  
Y. Hasegawa ◽  
B. Frohnapfel
Keyword(s):  
Boundary Conditions ◽  
Turbulent Flow ◽  
Drag Reduction ◽  
Secondary Flow ◽  
Slip Length ◽  
Laminar Flows ◽  
Superhydrophobic Surfaces ◽  
Mean Velocity ◽  
Effective Slip Length ◽  
Effective Slip

AbstractWe investigate the effects of superhydrophobic surfaces (SHS) carrying streamwise grooves on the flow dynamics and the resultant drag reduction in a fully developed turbulent channel flow. The SHS is modelled as a flat boundary with alternating no-slip and free-slip conditions, and a series of direct numerical simulations is performed with systematically changing the spanwise periodicity of the streamwise grooves. In all computations, a constant pressure gradient condition is employed, so that the drag reduction effect is manifested by an increase of the bulk mean velocity. To capture the flow properties that are induced by the non-homogeneous boundary conditions the instantaneous turbulent flow is decomposed into the spatial-mean, coherent and random components. It is observed that the alternating no-slip and free-slip boundary conditions lead to the generation of Prandtl’s second kind of secondary flow characterized by coherent streamwise vortices. A mathematical relationship between the bulk mean velocity and different dynamical contributions, i.e. the effective slip length and additional turbulent losses over slip surfaces, reveals that the increase of the bulk mean velocity is mainly governed by the effective slip length. For a small spanwise periodicity of the streamwise grooves, the effective slip length in a turbulent flow agrees well with the analytical solution for laminar flows. Once the spanwise width of the free-slip area becomes larger than approximately 20 wall units, however, the effective slip length is significantly reduced from the laminar value due to the mixing caused by the underlying turbulence and secondary flow. Based on these results, we develop a simple model that allows estimating the gain due to a SHS in turbulent flows at practically high Reynolds numbers.

Examination of the Slip Boundary Condition by µ-PIV and Lattice Boltzmann Simulations

2002 ◽  
Author(s):  
Derek C. Tretheway ◽  
Luoding Zhu ◽  
Linda Petzold ◽  
Carl D. Meinhart
Keyword(s):  
Boundary Condition ◽  
Shear Rate ◽  
Channel Flow ◽  
Lattice Boltzmann ◽  
Slip Velocity ◽  
Slip Length ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
Lattice Boltzmann Simulations ◽  
Bounce Back

This work examines the slip boundary condition by Lattice Boltzmann simulations, addresses the validity of the Navier’s hypothesis that the slip velocity is proportional to the shear rate and compares the Lattice Boltzmann simulations to the experimental results of Tretheway and Meinhart (Phys. of Fluids, 14, L9–L12). The numerical simulation models the boundary condition as the probability, P, of a particle to bounce-back relative to the probability of specular reflection, 1−P. For channel flow, the numerically calculated velocity profiles are consistent with the experimental profiles for both the no-slip and slip cases. No-slip is obtained for a probability of 100% bounce-back, while a probability of 0.03 is required to generate a slip length and slip velocity consistent with the experimental results of Tretheway and Meinhart for a hydrophobic surface. The simulations indicate that for microchannel flow the slip length is nearly constant along the channel walls, while the slip velocity varies with wall position as a results of variations in shear rate. Thus, the resulting velocity profile in a channel flow is more complex than a simple combination of the no-slip solution and slip velocity as is the case for flow between two infinite parallel plates.

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