Journal articles

2021

- titre
- A velocity-vorticity method for highly viscous 3D flows with application to digital rock physics
- auteur
- Laurène Hume, Philippe Poncet
- article
*Journal of Computational Physics*, Elsevier, 2021, 425, pp.109910. ⟨10.1016/j.jcp.2020.109910⟩- resume
- In this article, we present a numerical iterative method for the solution of internal viscous and incompressible flows in real porous three-dimensional bodies at their pore scale. We use the penalized formulation of the problem involving velocity and vorticity: an operator splitting allows to split apart the diffusion (inherited from Stokes equation) and the penalization phenomena (which takes into account the solid matrix). By means of the numerical analysis of the splitting, we exhibit the penalization coefficient which is actually effective. This method allows to deal only with fast-evaluation operators, that is to say scaling at most as O(n log n) where n is the number of underlying grid points, such as straightforward computations of finite differences schemes or FFT solver. The numerical analysis and implementation solutions are presented, and validated on various digital rock physics geometries acquired by micro-tomography, using numerical and physical diagnostics. To enforce this validation, we also present permeability estimations of several porous samples. The simulation of transport of passive and active scalars is finally investigated in order to perform the practical upscaling to 1D models of transport and diffusion at the Darcy scale.
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2020

- titre
- Improvement of remeshed Lagrangian methods for the simulation of dissolution processes at pore-scale
- auteur
- Jean-Matthieu Etancelin, Peter Moonen, Philippe Poncet
- article
*Advances in Water Resources*, Elsevier, 2020, 146, pp.103780. ⟨10.1016/j.advwatres.2020.103780⟩- resume
- This article shows how to consistently and accurately manage the Lagrangian formulation of chemical reaction equations coupled with the superficial velocity formalism introduced in the late 80s by Quintard and Whitaker. Lagrangian methods prove very helpful in problems in which transport effects are strong or dominant, but they need to be periodically put back in a regular lattice, a process called remeshing. In the context of digital rock physics, we need to ensure positive concentrations and regularity to accurately handle stagnation point neighborhoods. These two conditions lead to the use of kernels resulting in extra-diffusion, which can be prohibitively high when the diffusion coefficient is small. This is the case especially for reactive porous media, and the phenomenon is reinforced in porous rock matrices due to Archie's law. This article shows how to overcome this difficulty in the context of a two-scale porosity model applied in the Darcy-Brinkman-Stokes equations, and how to obtain simultaneous sign preservation, regularity and accurate diffusion, and apply it to dissolution processes at the pore scale of actual rocks.
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- titre
- Simulation of mineral dissolution at the pore scale with evolving fluid-solid interfaces: review of approaches and benchmark problem set
- auteur
- Sergi Molins, Cyprien Soulaine, N. I. Prasianakis, Aida Abbasi, Philippe Poncet, Anthony Ladd, Vitalii Starchenko, Sophie Roman, David Trebotich, Hamdi Tchelepi, Carl I. Steefel
- article
*Computational Geosciences*, Springer Verlag, 2020, ⟨10.1007/s10596-019-09903-x⟩- resume
- This manuscript presents a benchmark problem for the simulation of single-phase flow, reactive transport, and solid geometry evolution at the pore scale. The problem is organized in three parts that focus on specific aspects: flow and reactive transport (part I), dissolution-driven geometry evolution in two dimensions (part II), and an experimental validation of three-dimensional dissolution-driven geometry evolution (part III). Five codes are used to obtain the solution to this benchmark problem, including Chombo-Crunch, OpenFOAM-DBS, a lattice Boltzman code, Vortex, and dissolFoam. These codes cover a good portion of the wide range of approaches typically employed for solving pore-scale problems in the literature, including discretization methods, characterization of the fluid-solid interfaces, and methods to move these interfaces as a result of fluid-solid reactions. A short review of these approaches is given in relation to selected published studies. Results from the simulations performed by the five codes show remarkable agreement both quantitatively-based on upscaled parameters such as surface area, solid volume, and effective reaction rate-and qualitatively-based on comparisons of shape evolution. This outcome is especially notable given the disparity of approaches used by the codes. Therefore, these results establish a strong benchmark for the validation and testing of pore-scale codes developed for the simulation of flow and reactive transport with evolving geometries. They also underscore the significant advances seen in the last decade in tools and approaches for simulating this type of problem.
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2019

- titre
- Analysis of 3D non-linear Stokes problem coupled to transport-diffusion for shear-thinning heterogeneous microscale flows, applications to digital rock physics and mucociliary clearance
- auteur
- David Sanchez, Laurène Hume, Robin Chatelin, Philippe Poncet
- article
*ESAIM: Mathematical Modelling and Numerical Analysis*, EDP Sciences, 2019, 53 (4), pp.1083-1124. ⟨10.1051/m2an/2019013⟩- resume
- The present study provides the analysis of the generalized 3D Stokes problem in a time dependent domain. The fluid viscosity is nonlinear and depends on a transported and diffused quantity. This is a natural model of very low Reynolds number flow, typically at the microscale, involving a heterogeneous shear-thinning miscible medium transported and diffusing in a solvent filling a deformable geometry. This one-way coupling is meaningful when the action produced by the solid on the fluid is dominant or drives the flow. Several mathematical aspects are developed. The penalized version of this problem is introduced, involving the penalization of the solid in a deformable motion but defined in a simple geometry, which is of crucial interest for many numerical methods. All the equations of this partial differential system are analyzed separately, and then the coupled model is shown to be well-posed and to converge toward the solution of the initial problem. In order to illustrate the pertinence of such models, two meaningful micrometer scale real-life problems are presented: the dynamics of a polymer inside the pores of a real rock, in practice a Xanthan biodegradable polymer percolating a Bentheimer sandstone and miscible in water, and the dynamics of the strongly heterogeneous mucus bio-film, covering the human lungs surface, propelled by the vibrating ciliated cells. For these two examples, we show that the mathematical hypothesis are satisfied.
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2018

- titre
- Effective viscosity of a random mixture of fluids
- auteur
- Benoit Noetinger, Laurène Hume, Robin Chatelin, Philippe Poncet
- article
*Physical Review Fluids*, American Physical Society, 2018, 3 (1), pp.014103. ⟨10.1103/PhysRevFluids.3.014103⟩- resume
- We propose an estimation of the effective viscosity of a random mixture of Newtonian fluids that ignores capillary effects. The local viscosity of the mixture is assumed to be a random function of the position. Using perturbation expansions up to the second order, the resulting formula can be recast under the form of a simple power averaging mixing low. Numerical tests are used to assess the validity of the formula and the range of its applicability.
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2017

- titre
- Numerical and experimental investigation of mucociliary clearance breakdown in cystic fibrosis
- auteur
- Robin Chatelin, Dominique Anne-Archard, Marlene Murris-Espin, Marc Thiriet, Philippe Poncet
- article
*Journal of Biomechanics*, Elsevier, 2017, 53, pp.56-63. ⟨10.1016/j.jbiomech.2016.12.026⟩- resume
- The human tracheobronchial tree surface is covered with mucus. A healthy mucus is an heterogeneous material flowing toward the esophagus and a major defense actor against local pathogen proliferation and pollutant deposition. An alteration of mucus or its environment such as in cystic fibrosis dramatically impacts the mucociliary clearance. In the present study, we investigate the mechanical organization and the physics of such mucus in human lungs by means of a joint experimental and numerical work. In particular, we focus on the influence of the shear-thinning mucus mobilized by a ciliated epithelium for mucociliary clearance. The proposed robust numerical method is able to manage variations of more than 5 orders of magnitude in the shear rate and viscosity. It leads to a cartography that allows to discuss major issues on defective mucociliary clearance in cystic fibrosis. Furthermore, the computational rheological analysis based on measurements shows that cystic fibrosis shear-thinning mucus tends to aggregate in regions of lower clearance. Yet, a rarefaction of periciliary fluid has a greater impact than the mucus shear-thinning effects.
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2016

- titre
- A parametric study of mucociliary transport by numerical simulations of 3D non-homogeneous mucus
- auteur
- Robin Chatelin, Philippe Poncet
- article
*Journal of Biomechanics*, Elsevier, 2016, 49 (9), pp.1772-1780. ⟨10.1016/j.jbiomech.2016.04.009⟩- resume
- Mucus 3D computations of non-homogeneous flows Fluid-structure interaction in complex geometry a b s t r a c t Mucociliary clearance is the natural flow of the mucus which covers and protects the lung from the outer world. Pathologies, like cystic fibrosis, highly change the biological parameters of the mucus flow leading to stagnation situations and pathogens proliferation. As the lung exhibits a complex dyadic structure, in-vivo experimental study of mucociliary clearance is almost impossible and numerical simulations can bring important knowledge about this biological flow. This paper brings a detailed study of the biological parameters influence on the mucociliary clearance, in particular for pathological situations such as cystic fibrosis. Using recent suitable numerical methods, a non-homogeneous mucus flow (including non-lin-earities) can be simulated efficiently in 3D, allowing the identification of the meaningful parameters involved in this biological flow. Among these parameters, it is shown that the mucus viscosity, the stiffness transition between pericilliary fluid and mucus, the pericilliary fluid height as well as both cilia length and beating frequency have a great influence on the mucociliary transport.
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- titre
- Analysis of the penalized 3D variable viscosity stokes equations coupled to diffusion and transport
- auteur
- Robin Chatelin, David Sanchez, Philippe Poncet
- article
*ESAIM: Mathematical Modelling and Numerical Analysis*, EDP Sciences, 2016, 50 (2), pp.565-591. ⟨10.1051/m2an/2015056⟩- resume
- The analysis of the penalized Stokes problem, in its variable viscosity formulation, coupled to convection-diffusion equations is presented in this article. It models the interaction between a highly viscous fluid with variable viscosity and immersed moving and deformable obstacles. Indeed, while it is quite common to couple Poisson equations to diffusion-transport equations in plasma physics or fluid dynamics in vorticity formulations, the study of complex fluids requires to consider together the Stokes problem in complex moving geometry and convection-diffusion equations. The main result of this paper shows the existence and the uniqueness of the solution to this equations system with regularity estimates. Then we show that the solution to the penalized problem weakly converges toward the solution to the physical problem. Numerical simulations of fluid mechanics computations in this context are also presented in order to illustrate the practical aspects of such models: lung cells and their surrounding heterogeneous fluid, and porous media flows. Among the main original aspects in the present study, one can highlight the non linear dynamics induced by the coupling, and the tracking of the time-dependence of the domain.
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2015

- titre
- Mucus and Ciliated Cells of Human Lung: Splitting Strategies for Particle Methods and 3D Stokes Flows
- auteur
- R. Chatelin, Philippe Poncet, A. Didier, M. Murris-Espin, Dominique Anne-Archard, M. Thiriet
- article
*Procedia IUTAM*, Elsevier, 2015, 18, pp.114-122. ⟨10.1016/j.piutam.2015.11.012⟩- resume
- This work is introducing a fractional step algorithm for diffusion-transport allowing computation of flows with sharp variations of viscosity. This splitting also allows to perform transport with Lagrangian methods and diffusion with Eulerian methods, using hybrid grid-particle formulation. This splitting algorithm is globally second order. It is applied to computation of mucus mobility in human lungs, where epithelium ciliated cells are beating. A sufficient mobility is required to have healthy configurations. Our goal is to study the dependency of mucus mobility with respect to its viscosity in order to investigate mechanisms involved in pathologies such as cystic fibrosis. Scientific computing involving phenomena where transport effects are dominant, is still a challenge for realistic modeling of complex flows. Among the most famous problems, one can find Euler and Navier-Stokes equations, in velocity-pressure or velocity-vorticity formulations, with various boundary conditions. Particle and vortex methods are a well-known and efficient tool for this kind of problem. In the present paper we describe some aspects on how to use particle methods in the context of highly viscous flows. The application investigated is the propulsion of pulmonary mucus in human lungs. A meaningful model is 3D Stokes flows with a variable viscosity, depending on concentration of proteins (mucins), itself following a diffusion-transport equation. In additions to this coupling, the mucus film covering lung walls is interacting with epithelium ciliated cells, vibrating in the range of 4 to 20 Hz, as shown on figure 1. In the present study we neglect interactions between mucus and air (proven to be a non-dominant interaction), and visco-elastic effects. The aspect we aim at developing in this article is how to split diffusion and transport in this context.
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2014

- titre
- Hybrid grid–particle methods and Penalization: A Sherman–Morrison–Woodbury approach to compute 3D viscous flows using FFT
- auteur
- Robin Chatelin, Philippe Poncet
- article
*Journal of Computational Physics*, Elsevier, 2014, 269, pp.314-328. ⟨10.1016/j.jcp.2014.03.023⟩- resume
- Particle methods are very convenient to compute transport equations in fluid mechanics as their computational cost is linear and they are not limited by convection stability conditions. To achieve large 3D computations the method must be coupled to efficient algorithms for velocity computations, including a good treatment of non-homogeneities and complex moving geometries. The Penalization method enables to consider moving bodies interaction by adding a term in the conservation of momentum equation. This work introduces a new computational algorithm to solve implicitly in the same step the Penalization term and the Laplace operators, since explicit computations are limited by stability issues, especially at low Reynolds number. This computational algorithm is based on the Sherman-Morrison-Woodbury formula coupled to a GMRES iterative method to reduce the computations to a sequence of Poisson problems: this allows to formulate a penalized Poisson equation as a large perturbation of a standard Poisson, by means of algebraic relations. A direct consequence is the possibility to use fast solvers based on Fast Fourier Transforms for this problem with good efficiency from both the computational and the memory consumption point of views, since these solvers are recursive and they do not perform any matrix assembling. The resulting fluid mechanics computations are very fast and they consume a small amount of memory, compared to a reference solver or a linear system resolution. The present applications focus mainly on a coupling between transport equation and 3D Stokes equations, for studying biological organisms motion in a highly viscous flows with variable viscosity.
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2013

- titre
- A Hybrid Grid-Particle Method for Moving Bodies in 3D Stokes Flow with Variable Viscosity
- auteur
- Robin Chatelin, Philippe Poncet
- article
*SIAM Journal on Scientific Computing*, Society for Industrial and Applied Mathematics, 2013, 35 (4), pp.B925-B949. ⟨10.1137/120892921⟩- resume
- This article presents a new approach for the resolution of large three-dimensional Stokes equations with variable viscosity fluids, coupled with transport equations. After building the model, we will write these equations in the context of highly viscous flows and penalization, in order to consider complex geometries moving in a fluid. From a mathematical point of view, the solutions show nonlinear dynamics. Beside the use of standard tools such as finite differences and staggered grids, we have built a new methodology based on large three-dimensional simulations, including operators splitting for an efficient use of fast solvers, multi-index fixed point methods, Lagrangian methods with fast and accurate grid-particle transfers, and the multiresolution description of variables. Among the main original aspects of this method, both accurate incompressibility and variable viscosity are treated in the same fixed point. Hence the computation costs for variable and constant viscosity flows are similar. Several examples are given to validate the order of convergence and conservation rates. Such models are used, among other examples, in biological computing at the cellular scale. The present article eventually describes the ciliated epithelium cells covering a mammal's lungs, beating in a mucus film. This study of human lung diseases explores the efficiency of the mucociliary clearance, a challenging problem in health sciences, especially for the investigation of cystic fibrosis and various chronic obstructive pulmonary diseases.
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2010

- titre
- Efficiency of Multiscale Hybrid Grid-Particle Vortex Methods
- auteur
- Mustapha El Ossmani, Philippe Poncet
- article
*Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal*, Society for Industrial and Applied Mathematics, 2010, 8 (5), pp.1671-1690. ⟨10.1137/090765006⟩- resume
- This article presents a study of computational cost for the vortex method, a La-grangian numerical scheme using particles of fluids for simulation of three-dimensional flows. Its main features are to compute accurately transport effects and to be very robust, that is to say, often without prohibitive stability condition. Special attention is given to hybrid grid-particle vortex methods. They are shown to scale as O(n log n), even when used in a multiscale context. Furthermore, a discussion is provided on the best strategies for simulations in complex geometry. Computational cost is shown to have the same efficiency when performing multiscale simulations of three-dimensional flows in complex geometry.
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- titre
- Mucus dynamics subject to air and wall motion
- auteur
- S. Enault, D. Lombardi, Philippe Poncet, Marc Thiriet
- article
*ESAIM: Proceedings*, EDP Sciences, 2010, 30, pp.125-141- resume
- This study presents a numerical investigation of basic interactions between respiratory mucus motion, air circulation and epithelium ciliated cells vibration. One focuses on identification of meaningful rheological parameters, physiological and numerical simulation dimensioning. These preliminary results are crucial before the study of more general configurations of respiratory mucus motion. The numerical study presented in this work aims at providing a first numerical tool able to simulate the effects of mucus mobility and its ability to carry out pathogens or to deliver aerosol therapy to membrane wall cells. Momentum diffusion is identified as the dominant effect, as expected in this micrometer scale configuration, and its associate momentum diffusion operator is shown to be extremely stiff. Furthermore, epithelium vibration is shown to be much more efficient than air circulation for mucus propulsion.
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2009

- titre
- Analysis of an immersed boundary method for three-dimensional flows in vorticity formulation
- auteur
- Philippe Poncet
- article
*Journal of Computational Physics*, Elsevier, 2009, 228 (19), pp.7268-7288. ⟨10.1016/j.jcp.2009.06.023⟩- resume
- This article presents numerical analysis and practical considerations for three-dimensional flow computation using an implicit immersed boundary method. The Euler equations, or half a step of the Navier-Stokes equations when using fractional step algorithms, are investigated in their vorticity formulation. The context of flow computation around an arbitrarily shaped body is especially investigated. In conventional immersed boundary methods using vorticity, singular vortex are dispatched over the body surface. In the present study, one prefers using sources of potential velocity field, dispatched on the body, whose nature is not vorticity. Such a formulation is compatible to the Euler equations. In practice, these sources of potential flow produce a velocity through this surface, aiming in practice at cancelling a flow-through velocity. This article focuses on the use of the source-to-flow-through linear application, its properties being the key points for fast convergence. Its self-adjointness, or lack thereof, conditioning and preconditioning aspects are investigated. It follows that computing a velocity field with no-flow-through conditions in complex geometry, when using the source-to-flow-through linear application, can be achieved for 4/3 of the computational cost of standard Poisson equation in a Cartesian box. The robustness of immersed boundaries is especially interesting when used together with vortex-in-cell methods, well known for their robustness in time and their ability to compute accurately convective effects. A few examples, based on real-world geometries, illustrate the method capabilities.
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2008

- titre
- Spatially distributed control for optimal drag reduction of the flow past a circular cylinder
- auteur
- Philippe Poncet, Roland Hildebrand, Georges-Henri Cottet, Petros Koumoutsakos
- article
*Journal of Fluid Mechanics*, Cambridge University Press (CUP), 2008, 599, pp.111-120. ⟨10.1017/S0022112008000177⟩- resume
- We report high drag reduction in Direct Numerical Simulations of controled flows past circular cylinders at Reynolds numbers of 300 and 1000. The flow is controlled by the azimuthal component of the tangential velocity of the cylinder surface. Starting from a spanwise uniform velocity profile that leads to high drag reduction, the optimization procedure identifies, for the same energy input, spanwise varying velocity profiles that lead to higher drag reduction. The three dimensional variations of the velocity field, corresponding to the modes A and B of three dimensional wake instabilities, are largely responsible for this drag reduction. The spanwise, wall velocity variations introduce streamwise vortex braids in the wake that are responsible for reducing the drag induced by the primary spanwise vortices shed by the cylinder. The results demonstrate that extending two dimensional controllers to three-dimensional flows is not optimal as three dimensional control strategies can lead efficiently to higher drag reduction.
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2007

- titre
- Analysis of Direct Three-Dimensional Parabolic Panel Methods
- auteur
- Philippe Poncet
- article
*SIAM Journal on Numerical Analysis*, Society for Industrial and Applied Mathematics, 2007, 45 (6), pp.2259-2297. ⟨10.1137/050625849⟩- resume
- Adherence boundary conditions for time dependent partial differential equations, via Chorin algorithm, can be reduced to a parabolic problem with Robin-Fourier boundary conditions in the three-dimensional context. In the spirit of panel methods, one establishes an integral formulation whose key point is the estimation of the potential density, introducing a kind of panel method for tangential kinematic boundary conditions. This paper discusses explicit estimations of this density in the general case of an arbitrarily shaped three-dimensional body, which leads to a fast numerical scheme. An error analysis is also provided, involving body smoothness, the Hölder exponent of the density, and whether the body presents torsion or not.
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2006

- titre
- Finite difference stencils based on particle strength exchange schemes for improvement of vortex methods
- auteur
- Philippe Poncet
- article
*Journal of Turbulence*, Taylor & Francis, 2006, 7, pp.N23. ⟨10.1080/14685240600595586⟩- resume
- The method of Particle Strength Exchange aims at solving the viscous part of the Navier-Stokes equations, when split into diffusive and corrective parts. We present their formulation in the context of vortex methods. The present work follows the classical analysis by Degond and MasGallic whose formulas were obtained by means of integration of continuous functions. It shows that one gets different operators when integration is discrete, and leads to a substantial gain of accuracy. This scheme is then applied to several three-dimensional flows to exhibit convergence rate and impact on conservation laws at moderate Reynolds numbers. A sensitivity analysis is finally provided in order to carry out the good behavior of these schemes in both Eulerian and Lagrangian contexts.
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2005

- titre
- Control of three-dimensional wakes using evolution strategies
- auteur
- Philippe Poncet, Georges-Henri Cottet, Petros Koumoutsakos
- article
*Comptes Rendus Mécanique*, Elsevier Masson, 2005, 333 (1), pp.65-77. ⟨10.1016/j.crme.2004.10.007⟩- resume
- We investigate three-dimensional cylinder wakes of incompressible fully developed flows at Re = 300, resulting from control induced by tangential motions of the cylinder surface. The motion of the cylinder surface, in two dimensions, is optimized using evolution strategies, resulting in significant drag reduction and drastic modification of the wake as compared to the uncontrolled flow. The quasi-optimal velocity profile obtained in 2D is modified by spanwise harmonics and applied to 3D flows. The results indicate important differences in the flow physics induced by two and three dimensional control strategies.
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- titre
- Optimization of Vortex Shedding in 3-D Wakes Using Belt Actuators
- auteur
- Philippe Poncet, Petros Koumoutsakos
- article
*International Journal of Offshore and Polar Engineering (IJOPE)*, The International Society of Offshore and Polar Engineers 2005, 15 (1), pp.007- resume
- This paper discusses the control of cylinder wakes via tangential wall velocity modifications. The wall velocity is piecewise constant (corresponding to belt actuators), and its amplitude is optimized using a clustering real coded genetic algorithm. This type of control significantly affects the vortical structures being shed in the wake, and it is shown that the flow gets significantly modified, resulting in a 3-dimensional body shedding 2-dimensional vortical structures.
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2004

- titre
- Simulation and control of three-dimensional wakes
- auteur
- Georges-Henri Cottet, Philippe Poncet
- article
*Computers and Fluids*, Elsevier, 2004, 33 (5-6), pp.697-713. ⟨10.1016/j.compfluid.2003.01.001⟩- resume
- We illustrate the problem of wake optimization on two examples. In the first example, the destruction of trailing vortices, we show that stochastic optimization is a flexible tool to identify vortex systems enhancing natural instabilities. In the second example, the problem of drag reduction in a cylinder wake, a priori information on the flow dynamics is highly desirable and we give some preliminary results which should be useful to select the most efficient parameters in an optimization strategy based on differential rotation of the cylinder.
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- titre
- Advances in direct numerical simulations of 3D wall-bounded flows by Vortex-in-Cell methods
- auteur
- Georges-Henri Cottet, Philippe Poncet
- article
*Journal of Computational Physics*, Elsevier, 2004, 193 (1), pp.136-158. ⟨10.1016/j.jcp.2003.08.025⟩- resume
- This paper is devoted to the design of Vortex-In-Cell (VIC) methods for the direct numerical simulations of wall-bounded flows. A first method using body-fitted grid is presented in the particular case of a cylinder wake. This method, which has been used in [Phys. Fluids 14(6) (2002) 2021] to investigate the effect on the wake topology of cylinder rotations, is an extension of the VIC method presented in [J. Comput. Phys. 175 (2002) 702] for periodic geometries. Features of the method that are specific to wall-bounded geometries-interpolation operators, field calculations and vorticity flux formulas to enforce no-slip boundary conditions-are described in details. The accuracy of the method in the calculation of the body forces is investigated by comparisons with experiments and benchmark calculations. A second class of methods is in the spirit of the immersed boundary methods. The paper in particular shows that the no-slip conditions are very naturally handled by the vorticity flux formulas, independently of the relative locations of the particles and the body. Numerical experiments on the test-case of a ring impinging on a cylinder suggest that the method is second-order accurate.
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- titre
- Topological aspects of three-dimensional wakes behind rotary oscillating cylinders
- auteur
- Philippe Poncet
- article
*Journal of Fluid Mechanics*, Cambridge University Press (CUP), 2004, 517, pp.27-53. ⟨10.1017/s0022112004000588⟩- resume
- The development of a three-dimensional viscous incompressible flow generated behind an infinitely long circular cylinder, impulsively started into rectilinear motion and rotationally oscillating, is studied computationally. The numerical scheme, an hybrid vortex method, is used to integrate the velocity-vorticity formulation of the Navier-Stokes equations. The Reynolds number considered is Re = 400, which is moderate though beyond the critical values Re 2 190 and Re 2 260 for which the flow becomes spontaneously three-dimensional. The numerical method is explained and its main points are developed. This scheme is then applied to solve some two-dimensional problems, both in order to validate the method and to compute a nominal two-dimensional flow, required to measure the impact of three-dimensionality. The three-dimensional flow past a steady cylinder is also compared to benchmark simulations. Once the flow has become fully three-dimensional, beyond the transient regime and saturation of instabilities, the cylinder begins a rotary oscillation around its axis. Two kinds of rotations are considered: constant amplitude and several frequencies, and constant frequency and various amplitudes. When amplitude and frequency are high enough, the whole flow comes back to its two-dimensional state. This result gives a justification for two-dimensional computations in the literature related to rotating cylinders. For the first super-harmonic frequency of the flow, a parametric study is performed in order to find the impact of the amplitude on the topology of the flow. A bifurcation is clearly identified. Finally, the mechanisms involved in the return to a two-dimensional state are explained: the interaction between transverse instabilities and von K ´ armánarm´armán streets is quantified by means of a correlation analysis.
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2002

- titre
- Vanishing of mode B in the wake behind a rotationally oscillating circular cylinder
- auteur
- Philippe Poncet
- article
*Physics of Fluids*, American Institute of Physics, 2002, 14 (6), pp.2021-2023. ⟨10.1063/1.1479344⟩- resume
- This Brief Communication presents a numerical investigation of the dimensionality of a flow behind a three-dimensional rotationally oscillating circular cylinder. These computations are performed at a Reynolds number Re500, which is at a level where the mode B is the dominant three-dimensional instability. We show that a rotation of high enough amplitude makes the 3D instabilities vanish and brings the flow back to its nominal two-dimensional state.
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- titre
- Particle methods for direct numerical simulations of three-dimensional wakes
- auteur
- Georges-Henri Cottet, Philippe Poncet
- article
*Journal of Turbulence*, Taylor & Francis, 2002, 3, pp.038:1-9. ⟨10.1088/1468-5248/3/1/038⟩- resume
- In this paper we describe recent advances in the development of particle methods for the direct numerical simulations of three-dimensional wakes. Both body-fitted and immersed boundary techniques are considered. The accuracy and numerical cost of the proposed numerical methods are discussed on the benchmark cases of a flow past a cylinder and of a vortex ring impinging on a cylinder.
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1999

- titre
- About Successive Gauss-Seidelisations
- auteur
- Philippe Poncet, François Robert
- article
*Taiwanese Journal of Mathematics, TJM*, Mathematical Society of the Republic of China (Taiwan), 1999, 3, pp.491 - 501. ⟨10.11650/twjm/1500407162⟩- resume
- This note adresses the general problem of the dynamical behavior for successive Gauss-Seidel transformations (shortly called Gauss-Seidelisations) of a given mapping over the n-cube. Complete results are given for n = 2 and n = 3, and then a natural conjecture is proved to be false for greater n. Thus this interesting and simple problem remains still open for n ≥ 4.
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Conference papers

2019

- titre
- Le mucus pulmonaire: impact rhéologique de la mucoviscidose
- auteur
- Philippe Poncet
- article
*Journée MucoSud*, Jun 2019, Bordeaux, France- Accès au bibtex

2018

- titre
- Permeability estimation using vortex-based method and synthetic samples from X-ray micro-CT
- auteur
- Laurène Hume, Philippe Poncet, Peter Moonen
- article
*14me Journées d’étude des milieu poreux (JEMP)*, Oct 2018, Nantes, France- Accès au texte intégral et bibtex

- titre
- Reactive flows at pore scale with hybrid computing
- auteur
- Jean-Matthieu Etancelin, Philippe Poncet
- article
*Fifteenth International Conference Zaragoza-Pau on Mathematics and its Applications*, Sep 2018, Jaca, Spain- Accès au bibtex

- titre
- New trends in Vortex Methods for reactive flows
- auteur
- Philippe Poncet, Laurène Hume, Jean-Matthieu Etancelin
- article
*InterPore 2018*, May 2018, New Orleans, United States- Accès au bibtex

- titre
- Écoulement du mucus pulmonaire: modélisation, simulation et influence de la mucoviscidose
- auteur
- R. Chatelin, Philippe Poncet
- article
*GdR CNRS 3570 MécaBio, Toulouse*, 2018, Toulouse, France- Accès au bibtex

2017

- titre
- Pore scale modelling in porous media: diffusion-transport coupled to 3D Stokes equation at high resolution
- auteur
- Laurène Hume, Philippe Poncet
- article
*V International Conference on Particle-based Methods*, Sep 2017, Hannover, Germany- Accès au bibtex

- titre
- Experimental and numerical cross-validation of flow in real porous media. Part 2: Numerical framework
- auteur
- Laurène Hume, Fabrice Guerton, Peter Moonen, Philippe Poncet
- article
*3rd International Conference on Tomography of Materials and Structures*, Jun 2017, Lund, Sweden- resume
- The main objective of this contribution is the cross-validation of a novel approach to fluid flow simulation inside porous media. The originality of the model stands both in the use of rather raw 3D volumes from X-ray tomography and the model's computational efficiency on standard calculation means. The performance of the model is assessed in different geometries and configurations, based on reference data obtained via a dedicated experimental setup.
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- titre
- Experimental and numerical cross-validation of flow in real porous media. Part 1: Experimental framework
- auteur
- Fabrice Guerton, Laurène Hume, Philippe Poncet, Peter Moonen
- article
*3rd International Conference on Tomography of Materials and Structures (ICTMS 2017)*, Jun 2017, Lund, Sweden- resume
- The main objective of this contribution is the cross-validation of a novel approach to fluid flow simulation inside porous media. The originality of the model stands both in the use of rather raw 3D volumes from X-ray tomography and the model's computational efficiency on standard calculation means. The performance of the model is assessed in different geometries and configurations, based on reference data obtained via a dedicated experimental setup.
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- titre
- Pore-scale modelling in porous media: diffusion-transport coupled to 3D Stokes equation at high resolution
- auteur
- Laurène Hume, Philippe Poncet
- article
*Vth International Conference on Particle-Based Methods*, 2017, Hanovre, Germany- Accès au bibtex

- titre
- Modélisation de l’écoulement d’un fluide miscible rhéofluidifiant à l’échelle micrométrique et applications
- auteur
- Laurène Hume, D. Sanchez, R. Chatelin, Philippe Poncet
- article
*Congrès SMAI 2017*, 2017, La Tramblade, France- Accès au bibtex

2016

- titre
- Phénomènes de transport complexe à l'échelle des pores et modélisation de la rugosité
- auteur
- Laurène Hume, Philippe Poncet
- article
*13èmes Journéess d'études des Milieux Poreux 2016*, Oct 2016, Anglet, France- resume
- Phénomènes de transport complexe à l'échelle des pores et modélisation de la rugosité
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- titre
- Numerical modeling of coupled two-phase multicomponent flow with reactive geochemical transport in porous media
- auteur
- Etienne Ahusborde, Brahim Amaziane, Mustapha El Ossmani, Philippe Poncet
- article
*13èmes Journéess d'études des Milieux Poreux 2016*, Oct 2016, Anglet, France- resume
- Numerical modeling of coupled two-phase multicomponent flow with reactive geochemical transport in porous media
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- titre
- Benchmark for coupled two-phase multicomponent flow with reactive geochemical transport in porous media
- auteur
- Etienne Ahusborde, Brahim Amaziane, Mustapha El Ossmani, Philippe Poncet
- article
*Subsurface Environmental Simulation Benchmarking Workshop V*, Oct 2016, La Corogne, Spain- Accès au bibtex

- titre
- 3D computations of mucocilliary clearance
- auteur
- Robin Chatelin, Dominique Anne-Archard, Philippe Poncet
- article
*European Congress on Computational Methods in Applied Sciences and Engineering - ECCOMAS 2016*, Jun 2016, Hersonissos, Greece- resume
- In this talk, an overview of the results obtained in the ANR project BioFiReaDy will be presented. This interdisciplinary project aims at understanding mucociliary clearance: the natural flow of mucus in the lung, which protects bronchus from inhaled agents such as dust, pathogens or pollution particles... 3D simulations are used, as well as experimental rheology to characterize the non-Newtonian rheological behaviour of this complex biological fluid. Using suitable numerical algorithms specially developed to compute this 3D fluid-structure interaction (FSI) problem (mucus is propelled by beating cilia), it is now possible to study the biological parameters influence on the mucus flow. These algorithms are based on (i) a novative projection method which integrates the resolution of rheological shear-thinning effects, (ii) a penalization techniques coupled to a Shermann--Morrison--Woodbury approach to compute the FSI, (iii) the use of fast solvers (FFT or multigrid) and (iv) a lagrangian discretization of the convection effects. These numerical methods are particularly suitable for 3D computations since the global computational cost is linear with respect to the number of grid points, which is very interesting for a parametric study. Several results will be presented, in particular for pathological situations such as cystic fibrosis.
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- titre
- Complex transport at pore scale and roughness modeling
- auteur
- Laurène Hume, Philippe Poncet
- article
*XIII JEMP – French Interpore Chapter, Anglet, France*, 2016, Anglet, France- Accès au bibtex

- titre
- Design and realization of electroactive polymer actuators for transparent and flexible haptic feedback interfaces
- auteur
- Philippe Poncet, F. Casset, A. Latour, S. dos Santos, R. Gwoziecki, S. Fanget
- article
*2016 17th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EuroSimE)*, 2016, Montpellier, France- Accès au bibtex

2015

- titre
- A sequential numerical simulator for two-phase multicomponent flow with reactive transport in porous media
- auteur
- Etienne Ahusborde, Brahim Amaziane, Mustapha El Ossmani, Philippe Poncet
- article
*MoMaS Multiphase Seminar Days - Journées MoMaS Multiphasiques*, Oct 2015, Nice, France- Accès au bibtex

- titre
- Release of fine particles from Birch pollen grains following impaction
- auteur
- N. Visez, G. Chassard, N. Azarkan, O. Naas, H. Sénéchal, J.P. Sutra, Philippe Poncet, M. Choël
- article
*Mediterranean Palynology 2015 Symposium*, Sep 2015, Rome, Italy- Accès au bibtex

- titre
- Méthodes hybrides grille–particules pour la simulation 3D d'écoulements complexes non newtoniens
- auteur
- Robin Chatelin, Philippe Poncet
- article
*12e colloque national en calcul des structures*, CSMA, May 2015, Giens, France- resume
- Quelques méthodes numériques récemment développées dans le cadre du projet ANR BioFiReaDy sont regroupées dans cette communication. L'objectif est de simuler en 3D des écoulements de fluides complexes interagissant avec des obstacles immergés. L'algorithme repose sur une discrétisation hybride grille-particules, des algorithmes à pas fractionnaires et l'utilisation de solveurs rapide afin de réduire les coûts de calcul. Cela ouvre la porte à des simulations 3D de grande dimension, tout en permettant d'étudier l'influence de nombreux jeux de paramètres sur l'écoulement.
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- titre
- An integrated two-phase ow with reactive transport model in a parallel reservoir simulator
- auteur
- Etienne Ahusborde, Brahim Amaziane, Mustapha El Ossmani, Philippe Poncet
- article
*Proceedings of the 6th MAMERN Conference, Pau, France*, 2015, pau, France- Accès au bibtex

- titre
- Méthodes particulaires pour le transport en milieux poreux à l’échelle des pores
- auteur
- Philippe Poncet
- article
*Journée CEA-GAMNI sur la mécanique des fluides numérique, IHP*, 2015, Paris, France- Accès au bibtex

- titre
- Numerical simulation of flows with reactive transport in porous media: towards a validation for two-phase flow
- auteur
- Etienne Ahusborde, Brahim Amaziane, Mustapha El Ossmani, Philippe Poncet
- article
*Reactive Transport Modeling in the Geological Sciences Workshop, Institut Henri Poincaré, Paris, France*, 2015, Paris, France- Accès au bibtex

- titre
- Particle methods for transport in porous media at the pore scale
- auteur
- Etienne Ahusborde, Brahim Amaziane, Philippe Poncet
- article
*Complex Fluid Flow in Porous Media Conference (ComFlowPore), Bordeaux, France*, 2015, Bordeaux, France- Accès au bibtex

2014

- titre
- Passive and active flow control using vortex methods
- auteur
- Georges-Henri Cottet, Roland Hildebrand, Petros Koumoutsakos, Chloé Mimeau, Iraj Mortazavi, Philippe Poncet
- article
*6th International Conference on Vortex Flows and Vortex Models*, Nov 2014, Nagoya, Japan- resume
- This paper deals with the application of remeshed particle methods for the simulation and control of wakes behind obstacles. We consider passive as well as active control strategies. Active control consists of jets at the boundary and we show how enhancing natural 3D instabilities of the flow can reduce the drag. Passive control strategies consist of porous coatings on the obstacle and we investigate optimal locations of the porous coating to regularize the wake downstream and reduce the drag.
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- titre
- Réactions croisées entre pollen et graine de Brassica napus : implication des allergènes non hydrosolubles
- auteur
- O. Rivera Martos, Pierre Sutra, H. Sénéchal, S. D’andréa, Philippe Poncet
- article
*9ème Congrès Francophone d'Allergologie*, Apr 2014, Paris, France. pp.237, ⟨10.1016/j.reval.2014.02.040⟩- Accès au bibtex

- titre
- Particle methods for transport at the pore scale
- auteur
- Philippe Poncet, R. Chatelin
- article
*International Conference on Numerical and Mathematical Modeling of Flow and Transport in Porous Media, Dubrovnik, Croatia*, 2014, Dubrovnik, Croatia- Accès au bibtex

- titre
- Approche Sherman-Morrison-Woodbury pour la résolution rapide des termes de pénalisation. Application l’écoulement du mucus pulmonaire
- auteur
- R. Chatelin, Philippe Poncet
- article
*CANUM 2014, Carry-le-Rouet, France*, 2014, Carry-le-Rouet, France- Accès au bibtex

- titre
- Apprentissage machine pour l'étude de la mobilité du mucus dans le poumon
- auteur
- F. Dardalhon, Philippe Poncet
- article
*CANUM 2014, Carry-le-Rouet, France*, 2014, Carry-le-Rouet, France- Accès au bibtex

- titre
- Méthodes particulaires pour les écoulements en milieux poreux à l’échelle des pores
- auteur
- Philippe Poncet
- article
*12èmes Journées d’Etude des Milieux Poreux (JEMP), Toulouse*, 2014, Toulouse, France- Accès au bibtex

- titre
- Symposium on Penalization and Applications
- auteur
- Gilles Carbou, P. Chatelain, Philippe Poncet, David Sanchez
- article
*Thirteenth International Conference Zaragoza-Pau on Mathematics and its Applications, Jaca, Spain*, 2014, Jaca, Spain- Accès au bibtex

2012

- titre
- PARTICLE METHODS FOR 3D BIOLOGICAL FLOWS WITH VARIABLE DENSITY AND VISCOSITY
- auteur
- Robin Chatelin, Philippe Poncet
- article
*European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012)*, Sep 2012, Vienna, Austria- resume
- This work is investigating a mechanical dysfunction of human lungs: Mucociliary clearance. The goal is to understand how diseases like cystic fibrosis impact mucus motion (by altering mucus viscosity, cilia vibrations, respiration cycle...) and to identify stagnation situations where pathogens proliferate. Strategies for computation of mucus flow around an epithelium ciliated cell are presented. Governing equations are proposed to take into account this complex geometry in motion in the surrounding highly viscous fluid. The characteristic Reynolds number of the flow is very small and mucus is modelled by Stokes problem with both variable viscosity and density. Epithelium displacement is taken into account using a penal-ization method. As density and viscosity follow a convection equation, these assumptions lead to an elliptic/hyperbolic coupling and non-linear equations. Numerical implementations are discussed to get a fast, robust and accurate algorithm in three dimension based on fast elliptic solvers and Lagrangian methods.
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2010

- titre
- COMPUTATIONAL ASPECTS OF MUCUS PROPULSION BY CILIATED EPITHELIUM
- auteur
- Robin Chatelin, Philippe Poncet, Mayya Tokman
- article
*2nd European Conference on Microfluidics*, Dec 2010, Toulouse, France- resume
- This work is focused on the investigation of the mechanical dysfunction and airway clearance efficiency (or lack thereof) of the respiratory system by a quantitative analysis of mucus motion. We perform three-dimensional numerical simulations of the mucus and the ciliated cells present at lung walls (epithelium). In this preliminary work we consider a simplified configuration where the fluid is assumed to be Newtonian, the air/mucus interface is flat, and density and viscosity are constant. We formulate equations describing the simplified configuration, which are also relevant in the general context, and solve this system numerically to obtain mucus motion over a full cilia cycle. The stability condition of the explicit time integrators for the set of equations is prohibitively restrictive on the size of the integration time step. In order to begin development of effective numerical strategies for this problem we construct a one-dimensional model equation that contains the numerical challenges of a more general system. We study performance of several implicit and exponential time integrators using the model equation and use the results of the comparison to outline promising strategies for building efficient time integrators for a more complex system.
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2009

- titre
- New trends in Multi-Scale Simulation using Hybrid Grid-Particle Vortex Methods
- auteur
- Mustapha El Ossmani, Philippe Poncet
- article
*3rd International Conference on Approximation Methods and Numerical Modelling in Environment and Natural Resources*, Jun 2009, Pau, France- resume
- The purpose of this talk is to present a multi-scale Vortex-in-Cell method, for direct numerical simulation of three dimensional fluid flows. Vortex methods are Lagrangian. They are among the most challenging particle method, and have been dramatically improved during the last decade. This talk will present the main ideas of these methods, implemented in the open source package COMMA, funded by ANR and based on high performance packages FISHPACK and AGRIF. Among numerical methods aiming at solving the equations of fluid dynamics, the particle methods are renowned for their ability to compute accurately transport and convective effects. Most of the current challenges focus on the Navier-Stokes equations and their generalizations : multi-fluids, reactive flows, combustion, complex boundary conditions. In this short communication , one focuses on the basic three-dimensional Navier-Stokes equations, for an incompress-ible fluid, given by ∂u ∂t + u · u − ν∆u = f ρ − p ρ (1) where u is the divergence-free velocity field satisfying the no-slip condition u = 0 on Γ, p the pressure, ρ the density, ν the kinematic viscosity, and f the external force. For an incompressible fluid of constant density and viscosity, taking the curl of equation (1) and introducing the vorticity as ω = curlu, leads to ∂ω ∂t + u · ω − ω · u − ν∆ω = 0 (2) with kinematic no-slip boundary conditions u = 0, for external forces deriving from a potential (ie is the gradient of a scalar function).
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2005

- titre
- Bimodal control of three-dimensional wakes
- auteur
- Philippe Poncet
- article
*ERCOFTAC Workshop DLES-6*, 2005, Poitiers, France- resume
- This paper investigates control strategies for drag reduction of three-dimensional wake generated by a circular cylinder at Reynolds number Re = 300, such flows presenting mode B instabilities whose main feature is streamwise finger-shaped eddies. The control is performed thanks to a field of tangential velocities on the cylinder. One first focuses on two-dimensional velocity fiels (spanwise invariant), using both a clustering genetic algorithm (see [7]) and Newton algorithm in Fourier space with five Fourier modes. Besically the same field comes out, whatever the control technique used. A square-root regression of the drag reduction versus amplitude of the control leads to the formulation of an efficiency criterion. One then considers a class of spanwise harmonic perturbation of this quasi-optimal profile, leading to a two paramater optimization problem, involving amplitude and wavelength of the perturbation. A cartography of the efficiency with respect to these two parameters is finally obtained, showing regions of interest.
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2003

- titre
- Open-loop control of three-dimensional wakes
- auteur
- Philippe Poncet, Georges-Henri Cottet
- article
*2nd MIT Conference on Computational Fluid and Solid Mechanics*, 2003, Cambridge, United States- resume
- We illustrate the problem of wake optimisation on the generic case of the three-dimensional cylinder. After a description of the numerical method, several optimisation strategies are presented, aiming at identifying vortex systems providing drag reduction or special topology of the flow. On the one hand, strategies as simple as cylinder rotation or bidimensional stationary profiles of velocity are considered. On the other hand, an outlook of full 3D tangential controls is presented, which may be mixed or not with the simple strategies above.
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2000

- titre
- A few Multiresolution Schemes for the Black-Scholes equation
- auteur
- Philippe Poncet, Remi Abgrall
- article
*CEMRACS 2000*, Jul 2000, Marseille, France- resume
- In this note, we apply mutliresolution technics to a discretisation of the linear Black Schole equation , which is a convection diffusion equation. In particular, we describe a few numerical schemes in order to perform one step of decimation and reconstruction. This suggests a new discretisation method of this equation, which should easily extend to more than one dimension, and lead to very efficient numerical schemes.
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Book sections

2017

- titre
- Modeling Cystic Fibrosis and Mucociliary Clearance
- auteur
- Dominique Anne-Archard, Robin Chatelin, Marlène Murris-Espin, David Sanchez, Marc Thiriet, Alain Didier, Philippe Poncet
- article
- Sid M. Becker.
*Modeling of microscale transport in biological processes*, Academic Press, pp.113-154, 2017, 978-0-12-804595-4. ⟨10.1016/B978-0-12-804595-4.00005-5⟩ - resume
- This chapter considers the investigation of airway clearance efficiency and dysfunction using rheological measurements, modeling and simulations of mucus flows. The work is mainly dedicated to the respiratory epithelium dysfunction subsequently to the consumption of cilia-inhibiting drugs (e.g. nicotine) or viral and/or bacterial infections, and cystic fibrosis, although many respiratory diseases are associated with an altered mucus transport.
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Habilitation à diriger des recherches

2007

- titre
- Méthodes lagrangiennes pour le calcul et le contrôle des écoulements tridimensionnels
- auteur
- Philippe Poncet
- article
- Mathématiques [math]. Université Paul Sabatier - Toulouse (France), 2007
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Theses

2001

- titre
- Méthodes particulaires pour la simulation des sillages tridimensionnels
- auteur
- Philippe Poncet
- article
- Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2001. Français
- resume
- Ce travail est consacré au développement des méthodes particulaires pour la résolution des équations de Navier-Stokes incompressibles en dimension 3. L'évaluation des formules de Biot-Savart ayant un coût de calcul prohibitif en dimension trois, on utilise un couplage grille-particules. On applique alors cette technique à la simulation et au contrôle de sillages produits par un cylindre. La première partie est consacrée à la méthode numérique proprement dite. On commence par présenter le modèle lagrangien et la méthode utilisée pour calculer le champ de vitesse, qui est la clef de voûte du schéma. On décrit ensuite, au chapitre 2 comment sont calculées les couches limites. Enfin, on présente au chapitre 3 l'algorithme à pas fractionnaire utilisé, ainsi que les méthodes de transfert entre jeux de particules et grilles sous-jacentes, et le calcul de la diffusion. Le code est alors validé par des simulations d'anneaux tourbillonnaires qui se propulsent sur un obstacle cylindrique, pour des nombres de Reynolds modérés (entre 400 et 2000).La seconde partie utilise la méthode numérique décrite précédemment, en l'appliquant dans un premier temps à la simulation des sillages turbulents qui se développent derrière un cylindre circulaire (chapitre 4), puis au contrôle de ces écoulements au chapitre 4.Il est connu que les solutions bidimensionnelles sont instables pour des nombres de Reynolds suffisamment élevés. Les instabilités tridimensionnelles sont identifiée grâce à leur profil spectral. Elles ont un effet important sur les forces de traînée et sur la fréquence propre de l'écoulement.Le chapitre 5, relatif au contrôle, se propose de mettre en évidence plusieurs phénomènes. On considère un contrôle en boucle ouverte, réalisé par une rotation à pulsation et amplitude constante. On étudie des rotations basse et haute fréquences. Le coefficient de traînée est alors diminué de 43% par le contrôle à haute fréquence, pour un nombre de Reynolds de 550. De plus, on montre que l'écoulement turbulent revient à un état bidimensionnel si l'amplitude de rotation est suffisamment élevée.
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