Beyond pressureless gas dynamics: Quadrature-based velocity moment models

Following the seminal work of F. Bouchut on zero pressure gas dynamics which has been extensively used for gas particle-flows, the present contribution investigates quadrature-based velocity moments models for kinetic equations in the framework of the infinite Knudsen number limit, that is, for dilute clouds of small particles where the collision or coalescence probability asymptotically approaches zero. Such models define a hierarchy based on the number of moments and associated quadrature nodes, the first level of which leads to pressureless gas dynamics. We focus in particular on the four moment model where the flux closure is provided by a two-node quadrature in the velocity phase space and provide the right framework for studying both smooth and singular solutions. The link with both the kinetic underlying equation as well as with zero pressure gas dynamics is provided and we define the notion of measure solutions as well as the mathematical structure of the resulting system of four PDEs. We exhibit a family of entropies and entropy fluxes and define the notion of entropic solution. We study the Riemann problem and provide a series of entropic solutions in particular cases. This leads to a rigorous link with the possibility of the system of macroscopic PDEs to allow particle trajectory crossing (PTC) in the framework of smooth solutions. Generalized $\delta$-choc solutions resulting from Riemann problem are also investigated. Finally, using a kinetic scheme proposed in the literature without mathematical background in several areas, we validate such a numerical approach in the framework of both smooth and singular solutions.Comment: Submitted to Communication in Mathematical Science

Sensor validation of a Structural Health Monitoring Process for Aircraft Nacelle

This paper details the implementation process of an embedded structural health monitoring (SHM) system enabling condition-based maintenance of aircraft nacelles. One critical issue before being able to make use of such system is to ensure the effective bonding of the chosen actuators and sensors with their host structure, especially as the latter will be exposed to harsh environments and wide operational variability. In this work, we are concerned with the composite components of the nacelle and we use piezoelectric elements as both sensors and actuators. We propose an integrated approach that allows to validate a combination “Substrate—Glue—Piezoelectric” (SGP) and thus provides criteria to choose and size these assemblies. This validation scheme is based on the observation of the variations of the static capacity of the piezoelectric element after enduring various temperature and stress conditions when bonded to its host structure. Based on those SGP combinations, an active SHM strategy interrogating the structure by means of elastic wave propagation is currently being developed and preliminary results on samples representative of the nacelle are presented and discussed.Projet AIRCELLE (EPICE/CORALIE

Thermoacoustic instability - a dynamical system and time domain analysis

This study focuses on the Rijke tube problem, which includes features relevant to the modeling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source, and nonlinearities. This thermo-acoustic system features a complex dynamical behavior. In order to synthesize accurate time-series, we tackle this problem from a numerical point-of-view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness, in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by means of the projection method proposed by Jarlebring (2008), which alleviates the linearization with respect to the retarded time. The results are then compared to the analytical solution of the undamped system, and to Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We illustrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case, predicted by using Galerkin expansion, show spurious oscillations resulting from the Gibbs phenomenon. By comparing the modes of the linear to that of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching. Finally, time-series in the fully nonlinear regime, where a limit cycle is established, are analyzed and dominant modes are extracted. The analysis of the saturated limit cycles shows the presence of higher frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not captured when the coupling between different frequencies is not accounted for.Comment: Submitted to Journal of Fluid Mechanic

Task-based adaptive multiresolution for time-space multi-scale reaction-diffusion systems on multi-core architectures

A new solver featuring time-space adaptation and error control has been recently introduced to tackle the numerical solution of stiff reaction-diffusion systems. Based on operator splitting, finite volume adaptive multiresolution and high order time integrators with specific stability properties for each operator, this strategy yields high computational efficiency for large multidimensional computations on standard architectures such as powerful workstations. However, the data structure of the original implementation, based on trees of pointers, provides limited opportunities for efficiency enhancements, while posing serious challenges in terms of parallel programming and load balancing. The present contribution proposes a new implementation of the whole set of numerical methods including Radau5 and ROCK4, relying on a fully different data structure together with the use of a specific library, TBB, for shared-memory, task-based parallelism with work-stealing. The performance of our implementation is assessed in a series of test-cases of increasing difficulty in two and three dimensions on multi-core and many-core architectures, demonstrating high scalability

High order time integration and mesh adaptation with error control for incompressible Navier-Stokes and scalar transport resolution on dual grids

International audienceRelying on a building block developed by the authors in order to resolve the incompressible Navier-Stokes equation with high order implicit time stepping and dynamic mesh adaptation based on multiresolution analysis with collocated variables, the present contribution investigates the ability to extend such a strategy for scalar transport at relatively large Schmidt numbers using a finer level of refinement compared to the resolution of the hydrody-namic variables, while preserving space adaptation with error control. This building block is a key part of a strategy to construct a low-Mach number code based on a splitting strategy for combustion applications, where several spatial scales are into play. The computational efficiency and accuracy of the proposed strategy is assessed on a well-chosen three-vortex simulation

On the Direct Numerical Simulation of moderate-Stokes-number turbulent particulate flows using Algebraic-Closure-Based and Kinetic-Based Moment Methods

In turbulent particulate flows, the occurrence of particle trajectory crossings (PTC) is the main constraint on classical monokinetic Eulerian methods. To handle such PTC, accounting for high-order moments of the particle velocity distribution is mandatory. In the simplest case, second-order moments are needed. To retrieve these moments, two solutions are proposed in the literature: the Algebraic-Closure-Based Moment Method (ACBMM) and the Kinetic-Based Moment Method (KBMM). The ACBMM provides constitutive relations for the random-uncorrelated-motion (RUM) particle kinetic stress tensor as algebraic closures based on physical arguments (Simonin et al. 2002; Kaufmann et al. 2008; Masi 2010; Masi & Simonin 2012). These closures rely on the internal energy, namely the RUM particle kinetic energy, which is obtained using an additional transport equation. Alternatively, it is possible to directly solve for the second-order moment by providing a closure for the third-order correlation. The KBMM proposes a kinetic description, that is, the number density function (NDF) is reconstructed based on the resolved moments and on a presumed shape. In the present work, an isotropic Gaussian and the anisotropic Gaussian closure of Vié et al. (2012) are used. The goal of the present study is to provide a first comparison between ACBMM and KBMM, using the same robust numerical methods, in order to highlight differences and common points. The test case is a 2D turbulent flow with a mean shear

High-order adaptive multi-domain time integration scheme for microscale lithium-ion batteries simulations

We investigate the modeling and simulation of ionic transport and charge conservation in lithium-ion batteries (LIBs) at the microscale. It is a multiphysics problem that involves a wide range of time scales. The associated computational challenges motivate the investigation of numerical techniques that can decouple the time integration of the governing equations in the liquid electrolyte and the solid phase (active materials and current collectors). First, it is shown that semi-discretization in space of the non-dimensionalized governing equations leads to a system of index-1 semi-explicit differential algebraic equations (DAEs). Then, a new generation of strategies for multi-domain integration is presented, enabling high-order adaptive coupling of both domains in time. A simple 1D LIB half-cell code is implemented as a demonstrator of the new strategy for the simulation of different modes of cell operation. The integration of the decoupled subsystems is performed with high-order accurate implicit nonlinear solvers. The accuracy of the space discretization is assessed by comparing the numerical results to the analytical solutions. Then, temporal convergence studies demonstrate the accuracy of the new multi-domain coupling approach. Finally, the accuracy and computational efficiency of the adaptive coupling strategy are discussed in the light of the conditioning of the decoupled subproblems compared to the one of the fully-coupled problem. This new approach will constitute a key ingredient for the full scale 3D LIB high-fidelity simulations based on actual electrode microstructures

Radiation induces turbulence in particle-laden fluids.

When a transparent fluid laden with solid particles is subject to radiative heating, non-uniformities in particle distribution result in local fluid temperature fluctuations. Under the influence of gravity, buoyancy induces vortical fluid motion which can lead to strong preferential concentration, enhancing the local heating and more non- uniformities in particle distribution. By employing direct numerical simulations this study shows that the described feedback loop can create and sustain turbulence. The velocity and length scale of the resulting turbulence is not known a priori, and is set by balance between viscous forces and buoyancy effects. When the particle response time is comparable to a viscous time scale, introduced in our analysis, the system exhibits intense fluctuations of turbulent kinetic energy and strong preferential concentration of particle

Adaptive time splitting method for multi-scale evolutionary partial differential equations

Accepted to publication in Confluentes Mathematici. Dedication : Cet article est dédié à la mémoire de Michelle Schatzman. Spécialiste des méthodes de décomposition d'opérateur, sa grande clairvoyance scientifique lui a permis d'orienter plusieurs chercheurs débutants sur ce sujet à un moment où il pouvait sembler achevé. Michelle aimait dire qu'il n'y a pas de frontière entre les branches des mathématiques et que seule une grande culture permet de naviguer dans cette forêt et d'y trouver les bonnes techniques pour résoudre un problème. Ce travail est un hommage; à la croisée des mathématiques et de leurs applications effectives, il tente d'illustrer cette assertion. Michelle, ton dynamisme, ton humour et ton plaisir à parler mathématiques nous manquent.International audienceThis paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady problems. The strategy considers a second order Strang method and another lower order embedded splitting scheme that takes into account potential loss of order due to the stiffness featured by time-space multi-scale phenomena. The scheme is then built upon a precise numerical analysis of the method and a complementary numerical procedure, conceived to overcome classical restrictions of adaptive time stepping schemes based on lower order embedded methods, whenever asymptotic estimates fail to predict the dynamics of the problem. The performance of the method in terms of control of integration errors is evaluated by numerical simulations of stiff propagating waves coming from nonlinear chemical dynamics models as well as highly multi-scale nanosecond repetitively pulsed gas discharges, which allow to illustrate the method capabilities to consistently describe a broad spectrum of time scales and different physical scenarios for consecutive discharge/post-discharge phases