It has been recently shown that the presence of macrotextures on superhydrophobic materials can markedly modify the dynamics of water impacting them, and in particular significantly reduce the contact time of bouncing drops, compared with what is observed on a flat surface. This finding constitutes a significant step in the maximization of water repellency, since it enables to minimize even further the contact between solid and liquid. It also opens a new axis of research on the design of super-structures to induce specific functions such as anti-freezing, liquid fragmentation and/or recomposition, guiding, trapping and so on. Here we show that the contact time of drops bouncing on a repellent macrotexture takes discrete values when varying the impact speed. This allows us to propose a quantitative analysis of the reduction of contact time and thus to understand how and why macrotextures can control the dynamical properties of bouncing drops.

Abstract

International audienceData assimilation can be used to combine experimental and numerical realizations of the same flow to produce hybrid flow fields. These have the advantages of less noise contamination and higher resolution while simultaneously reproducing the main physical features of the measured flow. This study investigates data assimilation of the mean flow around an idealized airfoil (Re = 13,500) obtained from time-averaged two-dimensional particle image velocimetry (PIV) data. The experimental data, which constitute a low-dimensional representation of the full flow field due to resolution and field-of-view limitations, are incorporated into a simulation governed by the two-dimensional, incompressible Reynolds-averaged Navier–Stokes (RANS) equations with an unknown momentum forcing. This forcing, which corresponds to the divergence of the Reynolds stress tensor, is calculated from a direct-adjoint optimization procedure to match the experimental and numerical mean velocity fields. The simulation is projected onto the low-dimensional subspace of the experiment to calculate the discrepancy and a smoothing procedure is used to recover adjoint solutions on the higher dimensional subspace of the simulation. The study quantifies how well data assimilation can reconstruct the mean flow and the minimum experimental measurements needed by altering the resolution and domain size of the time-averaged PIV

We seek to quantify non-normality of the most amplified resolvent modes and predict their features based on the characteristics of the base or mean velocity profile. A 2-by-2 model linear Navier-Stokes (LNS) operator illustrates how non-normality from mean shear distributes perturbation energy in different velocity components of the forcing and response modes. The inverse of their inner product, which is unity for a purely normal mechanism, is proposed as a measure to quantify non-normality. When the operator is normal but not self-adjoint, a phase difference between the forcing and response modes is indicated. In flows where there is downstream spatial dependence of the base/mean, mean flow advection separates the spatial support of forcing and response modes which impacts the inner product and leads to non-normal amplification. Success of mean flow (linear) stability analysis for a particular frequency depends on the normality of amplification. If the amplification is normal, the resolvent operator can be rewritten in its dyadic representation to reveal that the adjoint and forward stability modes are proportional to the forcing and response resolvent modes at that frequency. If the amplification is non-normal, then resolvent analysis is required to understand the origin of observed flow structures. Eigenspectra and pseudospectra are used to characterize these phenomena. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The first deals mainly with normal mechanisms hence the success of both classical and mean stability analysis with respect to predicting the critical Reynolds number and global frequency of the saturated flow. Quantification of non-normality using the inverse inner product of the leading forcing and response modes agrees well with the product of the resolvent norm and distance between the imaginary axis and least stable eigenvalue. An approximate wavemaker computed from the resolvent modes scales with the length of the mean recirculation bubble. In the case of turbulent channel flow, structures result from both normal and non-normal mechanisms, suggesting that both are necessary elements to sustain turbulence. Mean shear is exploited most efficiently by stationary disturbances while bounds on the pseudospectra illustrate how non-normality is responsible for the most amplified disturbances at spatial wavenumbers and temporal frequencies corresponding to well-known turbulent structures. Some implications for flow control are discussed. * ssymon@caltech.edu arXiv:1712.05473v1 [physics.flu-dyn]

We study the evolution of velocity fluctuations due to an isolated spatio-temporal impulse using the linearized Navier-Stokes equations. The impulse is introduced as an external body force in incompressible channel flow at Re τ = 10000. Velocity fluctuations are defined about the turbulent mean velocity profile. A turbulent eddy viscosity is added to the equations to fix the mean velocity as an exact solution, which also serves to model the dissipative effects of the background turbulence on large-scale fluctuations. An impulsive body force produces flowfields that evolve into coherent structures containing long streamwise velocity streaks that are flanked by quasi-streamwise vortices; some of these impulses produce hairpin vortices. As these vortex-streak structures evolve, they grow in size to be nominally self-similar geometrically with an aspect ratio (streamwise to wallnormal) of approximately 10, while their kinetic energy density decays monotonically. The topology of the vortex-streak structures is not sensitive to the location of the impulse, but is dependent on the direction of the impulsive body force. All of these vortex-streak structures are attached to the wall, and their Reynolds stresses collapse when scaled by distance from the wall, consistent with Townsend's attached eddy hypothesis. †

The flows around a NACA 0018 airfoil at a chord-based Reynolds number of Re = 10250 and angles of attack of α = 0 • and α = 10 • are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity profiles are data-assimilated so that they are solutions of the incompressible Reynolds-averaged Navier-Stokes equations forced by Reynolds stress terms which are derived from experimental data. Spectral proper orthogonal decompositions of the velocity fluctuations and nonlinear forcing suggest different modelling approaches should be taken based on the angle of attack under consideration. For the α = 0 • case, the cross-spectral density tensors of both the velocity fluctuations and nonlinear forcing are low-rank at the shedding frequency and its higher harmonics. In the α = 10 • case, low-rank behaviour is observed for the velocity fluctuations in two bands of frequencies. Resolvent analysis of the data-assimilated means identifies lowrank behaviour only in the vicinity of the shedding frequency for α = 0 • and none of its harmonics. The resolvent operator for the α = 10 • case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by spectral proper orthogonal decomposition. It is also shown that the second linear mechanism, corresponding to the Kelvin-Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as a mean flow for resolvent analysis. This is due to the fact that experimental data are missing near the leading edge of the airfoil. The α = 0 • case is classified as an oscillator where the flow is organized around an intrinsic instability mechanism while the α = 10 • case behaves like an amplifier whose forcing is unstructured. For both cases, resolvent modes resemble those from spectral proper orthogonal decomposition when the operator is low-rank. To model the higher harmonics where this is not the case, we add parasitic resolvent modes, as opposed to classical resolvent modes which are the most amplified, by approximating the nonlinear forcing from limited triadic interactions of known resolvent modes. The amplifier case is modelled without parasitic modes at frequencies where the resolvent is low-rank. The two cases suggest that resolvent-based modelling can be achieved for more complex flows with limited experimental measurements and the nonlinear forcing need not be approximated unless the flow behaves like an oscillator.

We use the feedback loop of McKeon & Sharma (2010), where the nonlinear term in the Navier-Stokes equations is treated as an intrinsic forcing of the linear resolvent operator, to educe the structure of fluctuations in the range of scales (wavenumbers) where linear mechanisms are not active. In this region, the absence of dominant linear mechanisms is reflected in the lack of low-rank characteristics of the resolvent and in the disagreement between the structure of resolvent modes and actual flow features. To demonstrate the procedure, we choose low Reynolds number cylinder flow and the Couette equilibrium solution EQ1, which are representative of very low-rank flows dominated by one linear mechanism. The former is evolving in time, allowing us to compare resolvent modes with Dynamic Mode Decomposition (DMD) modes at the first and second harmonics of the shedding frequency. There is a match between the modes at the first harmonic but not at the second harmonic where there is no separation of the resolvent operator's singular values. We compute the self-interaction of the resolvent mode at the shedding frequency and illustrate its similarity to the nonlinear forcing of the second harmonic. When it is run through the resolvent operator, the 'forced' resolvent mode shows better agreement with the DMD mode. A similar phenomenon is observed for the fundamental streamwise wavenumber of the EQ1 solution and its second harmonic. The importance of parasitic modes, labeled as such since they are driven by the amplified frequencies, is their contribution to the nonlinear forcing of the main amplification mechanisms as shown for the shedding mode, which has subtle discrepancies with its DMD counterpart.

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