Jean-Christophe Nave

A gradient-augmented level set method with an optimally local, coherent advection scheme

The level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the extra information carried by the derivatives. The method is of comparable quality to WENO schemes, but with optimally local stencils (performing updates in time by using information from only a single adjacent grid cell). In addition, structures smaller than the grid size can be located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.

Exploration of in-fiber nanostructures from capillary instability

A new class of multi-material fiber that incorporates micrometer-thickness concentric-cylindrical sheets of glass into polymer matrix has emerged. The ultimate lower limit of feature size and recent observation of interesting instability phenomenon in fiber system motivate us to examine fluid instabilities during the complicated thermal drawing fabrication processing. In this paper, from the perspective of a single instability mechanism, classical Plateau-Rayleigh instabilities in the form of radial fluctuation, we explore the stability of various microstructures (such as shells and filaments) in our composite fibers. The attained uniform structures are consistent with theoretical analysis. Furthermore, a viscous materials map is established from calculations and agrees well with various identified materials. These results not only shed insights into other forms of fluid instabilities, but also provide guidance to achieve more diverse nanostructures (such as filaments, wires, and particles) in the microstructured fibers.

A Correction Function Method for Poisson problems with interface jump conditions

Abstract In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard “black box” solvers, without compromising accuracy. The basic idea of the new approach is similar to the Ghost Fluid Method (GFM). The GFM relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the right-hand-side (RHS) of the equations, producing a problem with the same linear system as if there were no jumps, only with a different RHS. However, achieving high accuracy is very hard (if not impossible) with the “standard” approaches used to compute the GFM correction terms. In this paper we generalize the GFM correction terms to a correction function, defined on a band around the interface. This function is then shown to be characterized as the solution to a PDE, with appropriate boundary conditions. This PDE can, in principle, be solved to any desired order of accuracy. As an example, we apply this new method to devise a 4th order accurate scheme for the constant coefficients Poisson equation with discontinuities in 2D. This scheme is based on (i) the standard 9-point stencil discretization of the Poisson equation, (ii) a representation of the correction function in terms of bicubics, and (iii) a solution of the correction function PDE by a least squares minimization. Several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries, its capability to capture sharp discontinuities, and its high convergence rate.

Aerodynamic Ground Effect in Fruitfly Sized Insect Takeoff

Aerodynamic ground effect in flapping-wing insect flight is of importance to comparative morphologies and of interest to the micro-air-vehicle (MAV) community. Recent studies, however, show apparently contradictory results of either some significant extra lift or power savings, or zero ground effect. Here we present a numerical study of fruitfly sized insect takeoff with a specific focus on the significance of leg thrust and wing kinematics. Flapping-wing takeoff is studied using numerical modelling and high performance computing. The aerodynamic forces are calculated using a three-dimensional Navier–Stokes solver based on a pseudo-spectral method with volume penalization. It is coupled with a flight dynamics solver that accounts for the body weight, inertia and the leg thrust, while only having two degrees of freedom: the vertical and the longitudinal horizontal displacement. The natural voluntary takeoff of a fruitfly is considered as reference. The parameters of the model are then varied to explore possible effects of interaction between the flapping-wing model and the ground plane. These modified takeoffs include cases with decreased leg thrust parameter, and/or with periodic wing kinematics, constant body pitch angle. The results show that the ground effect during natural voluntary takeoff is negligible. In the modified takeoffs, when the rate of climb is slow, the difference in the aerodynamic forces due to the interaction with the ground is up to 6%. Surprisingly, depending on the kinematics, the difference is either positive or negative, in contrast to the intuition based on the helicopter theory, which suggests positive excess lift. This effect is attributed to unsteady wing-wake interactions. A similar effect is found during hovering.

Linear stability analysis of capillary instabilities for concentric cylindrical shells

Center for Materials Science and Engineering at MIT (National Science Foundation (U.S.) (MRSEC program award DMR-0819762) )

A Sharp-Interface Active Penalty Method for the Incompressible Navier---Stokes Equations

The volume penalty method provides a simple, efficient approach for solving the incompressible Navier–Stokes equations in domains with boundaries or in the presence of moving objects. Despite the simplicity, the method is typically limited to first order spatial accuracy. We demonstrate that one may achieve high order accuracy by introducing an active penalty term. One key difference from other works is that we use a sharp, unregularized mask function. We discuss how to construct the active penalty term, and provide numerical examples, in dimensions one and two. We demonstrate second and third order convergence for the heat equation, and second order convergence for the Navier–Stokes equations. In addition, we show that modifying the penalty term does not significantly alter the time step restriction from that of the conventional penalty method.

Reply to the comment by Mike R. James et al. on "it takes three to tango: 2. Bubble dynamics in basaltic volcanoes and ramifications for modeling normal Strombolian activity"

[1] The current paradigm for normal eruptions at Stromboli volcano and Strombolian‐type activity more generally posits that each eruption represents the burst of a large pocket of gas, commonly referred to as slug, at the free surface of themagma column. This slug model has been investigated and refined primarily through analog fluid dynamical experiments at the laboratory scale. There is no doubt that these studies have advanced our understanding of Strombolian eruptions considerably. However, given the very fundamental status of the slug model for our current thinking about Strombolian‐type eruptions, it is paramount to carefully assess all underlying assumptions of the model. One important uncertainty lies in the scaling behavior of the observed slug dynamics in the laboratory. Scale invariance requires all nondimensional numbers to be identical, but it is generally not feasible to match all of the nondimensional numbers characterizing the volcanic conduit exactly in a laboratory experiment. Numerical computations offer a means of directly comparing slug dynamics at drastically different scales [Suckale et al., 2010b]. We find that the very large slugs in volcanic conduits are more prone to dynamic instabilities and breakup than the comparatively small slugs in laboratory settings. This finding in itself does of course not ‘disprove’ the slug model: it merely points to potentially important differences in slug stability at volcanic scales as opposed to laboratory scales. [2] Being careful and critical of the errors and interpretations of numerical simulations is important; similarly, the appropriateness of laboratory analogs must also be examined. James et al. [2011] raise a valid point asking whether our simulations are capable of reproducing stable slug rise in water. Figure 1 shows a computation performed for the experimental parameters provided by James et al. [2011]. In agreement with the analog experiments, we do indeed find that this slug rises stably in water, i.e., the slug ascends buoyantly within the conduit without experiencing ‘catastrophic’ breakup as defined by Suckale et al. [2010b]. Thus the experiments by James et al. [2011] do not appear to contradict our computations. We also reproduce computationally stable slug flow at finite Reynolds number (Re) shown in the experiment specified by Jaupart and Vergniolle [1989] [Suckale et al., 2010b]. It is worth noting that in both cases, the set of nondimensional numbers characterizing the experiment are not identical to those thought to be representative of the volcanic conduit and thus potential differences in the fluid dynamical behavior are not unexpected. [3] That we observe stable slugs in water but not in magma supports an argument raised by James et al. [2011], namely that Re is an insufficient parameter for comparing slug dynamics at different spatial scales. We welcome their discussion of the drawbacks of using a single Re to describe slugs. Focusing on Re to characterize our computations was a purely pragmatic decision: First, Re is commonly used for the purpose of comparing fluid dynamical computations at different scales. Second, we were able to use estimates for Re characterizing Strombolian slugs based on observational data measured at Stromboli [Vergniolle and Brandeis, 1996]. Third, we observed a correlation between Re and breakup within the regime we investigated, which is not unexpected since Re is an indirect measure of the size of a gas bubble or slug. That being said, a more comprehensive set of nondimensional numbers is undoubtedly needed to evaluate scaling behavior of slugs for different fluid dynamical regimes, including nondimensional criteria that capture the wavelength of potential interface instabilities in relation to the curvature of the interface and the time scale associated with instability growth to the time scale associated with advection of the instability along the interface. Our study is a first step toward gaining a better understanding of slug stability. [4] From a theoretical point of view, very large and dynamic gas volumes are not expected to be indefinitely stable. The light gas slug moves underneath a column of very heavy magma and this unstable density stratification is prone to the formation of Rayleigh‐Taylor instabilities at the essentially flat upper surface of the slug. Grace et al. [1978] developed a semiempirical rationalization of this process that affords rough estimates of maximum stable sizes as listed in Table 1 of Suckale et al. [2010b]. The appeal of this model lies primarily in its simplicity, but this very simplicity also Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada.

Sufficient Conditions for Global Minimality of Metastable States in a Class of Non-convex Functionals: A Simple Approach Via Quadratic Lower Bounds

We consider mass-constrained minimizers for a class of non-convex energy functionals involving a double-well potential. Based upon global quadratic lower bounds to the energy, we introduce a simple strategy to find sufficient conditions on a given critical point (metastable state) to be a global minimizer. We show that this strategy works well for the one exact and known metastable state: the constant state. In doing so, we numerically derive an almost optimal lower bound for both the order–disorder transition curve of the Ohta–Kawasaki energy and the liquid–solid interface of the phase-field crystal energy. We discuss how this strategy extends to non-constant computed metastable states, and the resulting symmetry issues that one must overcome. We give a preliminary analysis of these symmetry issues by addressing the global optimality of a computed lamellar structure for the Ohta–Kawasaki energy in one (1D) and two (2D) space dimensions. We also consider global optimality of a non-constant state for a spatially in-homogenous perturbation of the 2D Ohta–Kawasaki energy. Finally we use one of our simple quadratic lower bounds to rigorously prove that for certain values of the Ohta–Kawasaki parameter and aspect ratio of an asymmetric torus, any global minimizer

High order solution of Poisson problems with piecewise constant coefficients and interface jumps

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