Paolo Luzzatto-Fegiz

Structure and stability of the finite-area von Kármán street

By using a recently developed numerical method, we explore in detail the possible inviscid equilibrium flows for a Karman street comprising uniform, large-area vortices. In order to determine stability, we make use of an energy-based stability argument (originally proposed by Lord Kelvin), whose previous implementation had been unsuccessful in determining stability for the Karman street [P. G. Saffman and J. C. Schatzman, “Stability of a vortex street of finite vortices,” J. Fluid Mech. 117, 171–186 (1982)10.1017/S0022112082001578]. We discuss in detail the issues affecting this interpretation of Kelvins ideas, and show that this energy-based argument cannot detect subharmonic instabilities. To find superharmonic instabilities, we employ a recently introduced approach, which constitutes a reliable implementation of Kelvins stability ideas [P. Luzzatto-Fegiz and C. H. K. Williamson, “Stability of conservative flows and new steady fluid solutions from bifurcation diagrams exploiting a variational argument...

Resonant instability in two-dimensional vortex arrays

We examine conditions for the development of an oscillatory instability in two-dimensional vortex arrays. By building on the theory of Krein signatures for Hamiltonian systems, and considering constraints owing to impulse conservation, we show that a resonant instability (developing through coalescence of two eigenvalues) cannot occur for one or two vortices. We illustrate this deduction by examining available linear stability results for one or two vortices. Our work indicates that a resonant instability may, however, occur for three or more vortices. For these more complex flows, we propose a simple model, based on an elliptical vortex representation, to detect the onset of an oscillatory instability. We provide an example in support of our theory by examining three co-rotating vortices, for which we also perform a linear stability analysis. The stability boundary in our model is in good agreement with the full stability calculation. In addition, we show that eigenmodes associated with an overall rotation or an overall displacement of the vortices always have eigenvalues equal to zero and ±iΩ, respectively, where Ω is the angular velocity of the array. These results, for overall rotation and displacement modes, can also be used to immediately check the accuracy of a detailed stability calculation.

Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces

Significance Whereas superhydrophobic surfaces (SHSs) have long promised large drag reductions, experiments have provided inconsistent results, with many textures yielding little or no benefit. Given the vast potential impact of SHSs on energy utilization, finding an explanation and mitigating strategies is crucially important. A recent hypothesis suggests surfactant-induced Marangoni stresses may be to blame. However, paradoxically, adding surfactants has a barely measurable effect, casting doubt on this hypothesis. By performing surfactant-laden simulations and unsteady experiments we demonstrate the impact of surfactants and how extremely low concentrations, unavoidable in practice, can increase drag up to complete immobilization of the air–liquid interface. Our approach can be used to test other SHS textures for sensitivity to surfactant-induced stresses. Superhydrophobic surfaces (SHSs) have the potential to achieve large drag reduction for internal and external flow applications. However, experiments have shown inconsistent results, with many studies reporting significantly reduced performance. Recently, it has been proposed that surfactants, ubiquitous in flow applications, could be responsible by creating adverse Marangoni stresses. However, testing this hypothesis is challenging. Careful experiments with purified water already show large interfacial stresses and, paradoxically, adding surfactants yields barely measurable drag increases. To test the surfactant hypothesis while controlling surfactant concentrations with precision higher than can be achieved experimentally, we perform simulations inclusive of surfactant kinetics. These reveal that surfactant-induced stresses are significant at extremely low concentrations, potentially yielding a no-slip boundary condition on the air–water interface (the “plastron”) for surfactant concentrations below typical environmental values. These stresses decrease as the stream-wise distance between plastron stagnation points increases. We perform microchannel experiments with SHSs consisting of stream-wise parallel gratings, which confirm this numerical prediction, while showing near-plastron velocities significantly slower than standard surfactant-free predictions. In addition, we introduce an unsteady test of surfactant effects. When we rapidly remove the driving pressure following a loading phase, a backflow develops at the plastron, which can only be explained by surfactant gradients formed in the loading phase. This demonstrates the significance of surfactants in deteriorating drag reduction and thus the importance of including surfactant stresses in SHS models. Our time-dependent protocol can assess the impact of surfactants in SHS testing and guide future mitigating designs.

Bifurcation structure and stability in models of opposite-signed vortex pairs

We employ a recently developed numerical method to examine in detail the properties of opposite-signed, translating vortex pairs. We first consider a uniform-vortex approximation; for this flow, previous studies have found essential differences between rotating and translating configurations, and have encountered numerical difficulties at the boundary between the two types of equilibria. Recently, Luzzatto-Fegiz and Williamson (2012 J. Fluid Mech. 706 323–50) used an imperfect velocity-impulse (IVI) diagram to show that the rotating pairs have a translating counterpart, arising from a bifurcation of the classical translating configurations. In this paper, we expand this IVI diagram to find two new branches of steady vortices, including antisymmetric pairs, as well as vortices without any symmetry. We next consider more realistic models for flows at moderate Reynolds number Re, by computing solution families based on a discretized Chaplygin–Lamb dipole. We find that, as the accuracy of the discretization improves, the bifurcated branches shrink rapidly, while the unstable portion of the basic solution family becomes smaller. These results indicate that the bifurcation structure of moderate-Re flows can be very different from that of solutions that use a single patch per vortex.

Video: Soap opera in the maze: Geometry matters in Marangoni flows

Fernando Temprano-Coleto,1,* François J. Peaudecerf,2 Julien R. Landel,3 Frédéric Gibou,1,4 and Paolo Luzzatto-Fegiz1 1Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, California 93106, United States 2Department of Civil, Environmental, and Geomatic Engineering, ETH Zürich, 8093 Zürich, Switzerland 3School of Mathematics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom 4Department of Computer Science, University of California Santa Barbara, Santa Barbara, California 93106, United States

Stability of steady vortices and new equilibrium flows from “Imperfect-Velocity-Impulse” diagrams

In 1875, Lord Kelvin proposed an energy-based argument for determining the stability of steady inviscid flows [1]. While the key underpinnings of the method are well established, its practical use has been the subject of extensive debate. In this work, we draw on ideas from dynamical systems and imperfection theory to construct a methodology that represents a rigorous implementation of Kelvin’s argument. Besides yielding stability properties, which are found to be in precise agreement with the results of linear analysis, our approach also implicitly yields new bifurcated solutions branches, as we shall describe below.