In this contribution we propose a general presentation of Eulerian multi-fluid modeling and numeri... more In this contribution we propose a general presentation of Eulerian multi-fluid modeling and numerical methods for the simulation of polydisperse evaporating sprays. By spray, we denote a cloud of spherical liquid droplets of various sizes ranging from submicronic scales up to several hundred microns which interact with the carrier gaseous phase and among themselves. We deal with sprays for which the physics of such a two-phase flow is governed at the “kinetic” level, also called mesoscopic level, by a Williams-Boltzmann spray equation, where the elementary phenomena such as evaporation, heating, coalescence and secondary break-up can be described properly. Our ob jective is to provide a hierarchy of models of Eulerian type with two main criteria : 1- to take into account accurately the polydispersion of the spray, that is the large size spectrum, as well as size-conditioned dynamics, evaporation and heating, 2- to keep a rigourous link with the Williams-Boltzmann spray equation at the...
In this paper, we tackle the numerical simulation of reaction-diffusion equations modeling multis... more In this paper, we tackle the numerical simulation of reaction-diffusion equations modeling multiscale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts which are spatially very localized. In a series of previous studies, the numerical analysis of operator splitting techniques has been conducted and such an approach has shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, even if a firm theoretical background is available, an optimal strategy for high performance numerical simulation is still needed. In this paper, we introduce a new strategy for reaction-diffusion systems based on time operator splitting in the context of very localized and very stiff reaction fronts. It provides an optimal combination of adaptive spatial multiresolution, implicit resolution of reaction and explicit resolution of diffusion. The optimality is reached in terms of the choice of the operator splitting time step which, in the framework of self-similar reaction waves, allows a very good combination of the various dedicated solvers used in the proposed strategy. The computational efficiency is then evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for spiral waves and scroll waves as an illustration.
This contribution deals with the modeling of collisional multicomponent magnetized plasmas in the... more This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model to investigate the influence on shock solutions of a nonconservative product present in the electron energy equation. Then, we derive jump conditions based on traveling wave solutions and propose an original numerical treatment in order to avoid non-physical shocks for the solution that remains valid in the case of coarse-resolution simulations. A key element for the numerical scheme proposed is the presence of diffusion in the electron variables, consistent with the physically-sound scaling used in the model developed by Graille et al. following a multiscale Chapman-Enskog expansion method [M3AS, 19 (2009) 527-599]. The numerical strategy is assessed in the framework of a solar physics test case. The computational method is able to capture the traveling wave solutions in both the highly-and coarsely-resolved cases.
This study employs DNS of two-phase flows to enhance primary atomization understanding and modeli... more This study employs DNS of two-phase flows to enhance primary atomization understanding and modeling to be used in numerical simulation in RANS or LES framework. In particular, the work has been aimed at improving the information on the liquid-gas interface evolution for modeling approaches, such as the Eulerian-Lagrangian Spray Atomization (ELSA) framework. Even though this approach has been already successfully employed to describe the complete liquid atomization process from the primary region to the dilute spray, improvements are still expected on the derivation of the drop size distribution (DSD). The main aim of the present work is the introduction of a new framework to achieve a continuous description of the DSD formation during the atomization process. The attention is here focused on the extraction from DNS data of the behavior of geometrical variable of the liquid-gas interface, such as the mean (H) and Gauss (G) surface curvatures. The use of a Surface Curvature Distribution is also proposed and studied. A Rayleigh-Plateau instability along a column of liquid and a droplet collision case are first of all considered to analyze and to verify the capabilities of the code to correctly predicting the curvature distributions. A statistical analysis
This paper exposes a novel exploratory formalism, the end goal of which is the numerical simulati... more This paper exposes a novel exploratory formalism, the end goal of which is the numerical simulation of the dynamics of a cloud of particles weakly or strongly coupled with a turbulent fluid. Given the large panel of expertise of the list of authors, the content of this paper scans a wide range of connex notions, from the physics of turbulence to the rigorous definition of stochastic processes. Our approach is to develop reduced-order models for the dynamics of both carrying and carried phases which remain consistant within this formalism, and to set up a numerical process to validate these models. The novelties of this paper lie in the gathering of a large panel of mathematical and physical definitions and results within a common framework and an agreed vocabulary (sections 1 and 2), and in some preliminary results and achievements within this context, section 3. While the first three sections have been simplified to the context of a gas field providing that the disperse phase only ...
We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reac- tio... more We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reac- tion waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reaction fronts, spatially very lo- calized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. Based on recent theoretical studies of numerical analysis, such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by restric- tive diffusive step stability limits, but only by the physics of the phenomenon. We thus aim at solving accurately complete models including all time and space scales of the phenomenon, considering large simulation domains with co...
In this paper, we tackle the issue of the accurate simulation of evaporating and reactive polydis... more In this paper, we tackle the issue of the accurate simulation of evaporating and reactive polydisperse sprays strongly coupled to unsteady gaseous flows. In solid propulsion, aluminum particles are included in the propellant to improve the global performances but the distributed combustion of these droplets in the chamber is suspected to be a driving mechanism of hydrodynamic and acoustic instabilities. The faithful prediction of two-phase interactions is a determining step for future solid rocket motor optimization. When looking at saving computational ressources as required for industrial applications, performing reliable simulations of two-phase flow instabilities appears as a challenge for both modeling and scientific computing. The size polydispersity, which conditions the droplet dynamics, is a key parameter that has to be accounted for. For moderately dense sprays, a kinetic approach based on a statistical point of view is particularly appropriate. The spray is described by a number density function and its evolution follows a Williams-Boltzmann transport equation. To solve it, we use Eulerian Multi-Fluid methods, based on a continuous discretization of the size phase space into sections, which offer an accurate treatment of the polydispersion. The objective of this paper is threefold: first to derive a new Two Size Moment Multi-Fluid model that is able to tackle evaporating polydisperse sprays at low cost while accurately describing the main driving mechanisms, second to develop a dedicated evaporation scheme to treat simultaneously mass, moment and energy exchanges with the gas and between the sections. Finally, to design a time splitting operator strategy respecting both reactive two-phase flow physics and cost/accuracy ratio required for industrial computations. Using a research code, we provide 0D validations of the new scheme before assessing the splitting technique's ability on a reference two-phase flow acoustic case. Implemented in the industrial-oriented CEDRE code, all developments allow to simulate realistic solid rocket motor configurations featuring the first polydisperse reactive computations with a fully Eulerian method.
Operator splitting techniques were originally introduced with the main objective of saving comput... more Operator splitting techniques were originally introduced with the main objective of saving computational costs. A multi-physics problem is thus split in subproblems of different nature with a significant reduction of the algorithmic complexity and computational requirements of the numerical solvers. Nevertheless, splitting errors are introduced in the numerical approximations due to the separate evolution of the split subproblems and can compromise a reliable reproduction of the coupled dynamics. In this chapter we present a numerical technique to estimate such splitting errors on the fly and dynamically adapt the splitting time steps according to a user-defined accuracy tolerance. The method applies to the numerical solution of time-dependent stiff PDEs, illustrated here by propagating laminar flames investigated in combustion applications.
We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapt... more We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts. The method is an extension of the multiresolution finite volume scheme used to solve hyperbolic and parabolic time-dependent PDEs. Such an approach guarantees a numerical solution of the Poisson equation within a user-defined accuracy tolerance. Most adaptive meshing approaches in the literature solve elliptic PDEs level-wise and hence at uniform resolution throughout the set of adapted grids. Here we introduce a numerical procedure to represent the elliptic operators on the adapted grid, strongly coupling inter grid relations that guarantee the conservation and accuracy properties of multiresolution finite volume schemes. The discrete Poisson equation is solved at once over the entire computational domain as a completely separate process. The accuracy and numerical performance of the method is assessed in the context of streamer discharge simulations.
We recall here the essential results of moment quadrature in a one-dimensional configuration. The... more We recall here the essential results of moment quadrature in a one-dimensional configuration. The complete developments can be found in (Chalons et al. 2011b). 2.1. Quadrature Consider the solution f = f (t, x, v) of the free transport kinetic equation ∂ t
The purpose of the present contribution is to introduce a new high-order moment formalism for par... more The purpose of the present contribution is to introduce a new high-order moment formalism for particle/droplet trajectory crossing (PTC) in the framework of large-eddy simulation (LES) of gas-particle flows. Thus far, the ability to treat PTC has been examined by several investigators for direct numerical simulations (DNS) using quadraturebased moment methods based on a sum of Dirac delta functions (Yuan & Fox (2010), Kah et al. (2010)). However, for LES, such methods require too many moments in order to capture both the effect of subgrid-scale turbulence on the disperse phase as well as PTC due to large-scale eddies in a Eulerian mesoscopic framework. The challenge is thus twofold: first, to propose a new generation of quadrature with less singular behavior as well as associated proper mathematical properties and related algorithms, and second to limit the number of moments used for applicability in multi-dimensional configurations without losing accuracy in the representation of spatial fluxes.
In this contribution we propose a general presentation of Eulerian multi-fluid modeling and numeri... more In this contribution we propose a general presentation of Eulerian multi-fluid modeling and numerical methods for the simulation of polydisperse evaporating sprays. By spray, we denote a cloud of spherical liquid droplets of various sizes ranging from submicronic scales up to several hundred microns which interact with the carrier gaseous phase and among themselves. We deal with sprays for which the physics of such a two-phase flow is governed at the “kinetic” level, also called mesoscopic level, by a Williams-Boltzmann spray equation, where the elementary phenomena such as evaporation, heating, coalescence and secondary break-up can be described properly. Our ob jective is to provide a hierarchy of models of Eulerian type with two main criteria : 1- to take into account accurately the polydispersion of the spray, that is the large size spectrum, as well as size-conditioned dynamics, evaporation and heating, 2- to keep a rigourous link with the Williams-Boltzmann spray equation at the...
In this paper, we tackle the numerical simulation of reaction-diffusion equations modeling multis... more In this paper, we tackle the numerical simulation of reaction-diffusion equations modeling multiscale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts which are spatially very localized. In a series of previous studies, the numerical analysis of operator splitting techniques has been conducted and such an approach has shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, even if a firm theoretical background is available, an optimal strategy for high performance numerical simulation is still needed. In this paper, we introduce a new strategy for reaction-diffusion systems based on time operator splitting in the context of very localized and very stiff reaction fronts. It provides an optimal combination of adaptive spatial multiresolution, implicit resolution of reaction and explicit resolution of diffusion. The optimality is reached in terms of the choice of the operator splitting time step which, in the framework of self-similar reaction waves, allows a very good combination of the various dedicated solvers used in the proposed strategy. The computational efficiency is then evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for spiral waves and scroll waves as an illustration.
This contribution deals with the modeling of collisional multicomponent magnetized plasmas in the... more This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model to investigate the influence on shock solutions of a nonconservative product present in the electron energy equation. Then, we derive jump conditions based on traveling wave solutions and propose an original numerical treatment in order to avoid non-physical shocks for the solution that remains valid in the case of coarse-resolution simulations. A key element for the numerical scheme proposed is the presence of diffusion in the electron variables, consistent with the physically-sound scaling used in the model developed by Graille et al. following a multiscale Chapman-Enskog expansion method [M3AS, 19 (2009) 527-599]. The numerical strategy is assessed in the framework of a solar physics test case. The computational method is able to capture the traveling wave solutions in both the highly-and coarsely-resolved cases.
This study employs DNS of two-phase flows to enhance primary atomization understanding and modeli... more This study employs DNS of two-phase flows to enhance primary atomization understanding and modeling to be used in numerical simulation in RANS or LES framework. In particular, the work has been aimed at improving the information on the liquid-gas interface evolution for modeling approaches, such as the Eulerian-Lagrangian Spray Atomization (ELSA) framework. Even though this approach has been already successfully employed to describe the complete liquid atomization process from the primary region to the dilute spray, improvements are still expected on the derivation of the drop size distribution (DSD). The main aim of the present work is the introduction of a new framework to achieve a continuous description of the DSD formation during the atomization process. The attention is here focused on the extraction from DNS data of the behavior of geometrical variable of the liquid-gas interface, such as the mean (H) and Gauss (G) surface curvatures. The use of a Surface Curvature Distribution is also proposed and studied. A Rayleigh-Plateau instability along a column of liquid and a droplet collision case are first of all considered to analyze and to verify the capabilities of the code to correctly predicting the curvature distributions. A statistical analysis
This paper exposes a novel exploratory formalism, the end goal of which is the numerical simulati... more This paper exposes a novel exploratory formalism, the end goal of which is the numerical simulation of the dynamics of a cloud of particles weakly or strongly coupled with a turbulent fluid. Given the large panel of expertise of the list of authors, the content of this paper scans a wide range of connex notions, from the physics of turbulence to the rigorous definition of stochastic processes. Our approach is to develop reduced-order models for the dynamics of both carrying and carried phases which remain consistant within this formalism, and to set up a numerical process to validate these models. The novelties of this paper lie in the gathering of a large panel of mathematical and physical definitions and results within a common framework and an agreed vocabulary (sections 1 and 2), and in some preliminary results and achievements within this context, section 3. While the first three sections have been simplified to the context of a gas field providing that the disperse phase only ...
We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reac- tio... more We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reac- tion waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reaction fronts, spatially very lo- calized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. Based on recent theoretical studies of numerical analysis, such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by restric- tive diffusive step stability limits, but only by the physics of the phenomenon. We thus aim at solving accurately complete models including all time and space scales of the phenomenon, considering large simulation domains with co...
In this paper, we tackle the issue of the accurate simulation of evaporating and reactive polydis... more In this paper, we tackle the issue of the accurate simulation of evaporating and reactive polydisperse sprays strongly coupled to unsteady gaseous flows. In solid propulsion, aluminum particles are included in the propellant to improve the global performances but the distributed combustion of these droplets in the chamber is suspected to be a driving mechanism of hydrodynamic and acoustic instabilities. The faithful prediction of two-phase interactions is a determining step for future solid rocket motor optimization. When looking at saving computational ressources as required for industrial applications, performing reliable simulations of two-phase flow instabilities appears as a challenge for both modeling and scientific computing. The size polydispersity, which conditions the droplet dynamics, is a key parameter that has to be accounted for. For moderately dense sprays, a kinetic approach based on a statistical point of view is particularly appropriate. The spray is described by a number density function and its evolution follows a Williams-Boltzmann transport equation. To solve it, we use Eulerian Multi-Fluid methods, based on a continuous discretization of the size phase space into sections, which offer an accurate treatment of the polydispersion. The objective of this paper is threefold: first to derive a new Two Size Moment Multi-Fluid model that is able to tackle evaporating polydisperse sprays at low cost while accurately describing the main driving mechanisms, second to develop a dedicated evaporation scheme to treat simultaneously mass, moment and energy exchanges with the gas and between the sections. Finally, to design a time splitting operator strategy respecting both reactive two-phase flow physics and cost/accuracy ratio required for industrial computations. Using a research code, we provide 0D validations of the new scheme before assessing the splitting technique's ability on a reference two-phase flow acoustic case. Implemented in the industrial-oriented CEDRE code, all developments allow to simulate realistic solid rocket motor configurations featuring the first polydisperse reactive computations with a fully Eulerian method.
Operator splitting techniques were originally introduced with the main objective of saving comput... more Operator splitting techniques were originally introduced with the main objective of saving computational costs. A multi-physics problem is thus split in subproblems of different nature with a significant reduction of the algorithmic complexity and computational requirements of the numerical solvers. Nevertheless, splitting errors are introduced in the numerical approximations due to the separate evolution of the split subproblems and can compromise a reliable reproduction of the coupled dynamics. In this chapter we present a numerical technique to estimate such splitting errors on the fly and dynamically adapt the splitting time steps according to a user-defined accuracy tolerance. The method applies to the numerical solution of time-dependent stiff PDEs, illustrated here by propagating laminar flames investigated in combustion applications.
We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapt... more We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts. The method is an extension of the multiresolution finite volume scheme used to solve hyperbolic and parabolic time-dependent PDEs. Such an approach guarantees a numerical solution of the Poisson equation within a user-defined accuracy tolerance. Most adaptive meshing approaches in the literature solve elliptic PDEs level-wise and hence at uniform resolution throughout the set of adapted grids. Here we introduce a numerical procedure to represent the elliptic operators on the adapted grid, strongly coupling inter grid relations that guarantee the conservation and accuracy properties of multiresolution finite volume schemes. The discrete Poisson equation is solved at once over the entire computational domain as a completely separate process. The accuracy and numerical performance of the method is assessed in the context of streamer discharge simulations.
We recall here the essential results of moment quadrature in a one-dimensional configuration. The... more We recall here the essential results of moment quadrature in a one-dimensional configuration. The complete developments can be found in (Chalons et al. 2011b). 2.1. Quadrature Consider the solution f = f (t, x, v) of the free transport kinetic equation ∂ t
The purpose of the present contribution is to introduce a new high-order moment formalism for par... more The purpose of the present contribution is to introduce a new high-order moment formalism for particle/droplet trajectory crossing (PTC) in the framework of large-eddy simulation (LES) of gas-particle flows. Thus far, the ability to treat PTC has been examined by several investigators for direct numerical simulations (DNS) using quadraturebased moment methods based on a sum of Dirac delta functions (Yuan & Fox (2010), Kah et al. (2010)). However, for LES, such methods require too many moments in order to capture both the effect of subgrid-scale turbulence on the disperse phase as well as PTC due to large-scale eddies in a Eulerian mesoscopic framework. The challenge is thus twofold: first, to propose a new generation of quadrature with less singular behavior as well as associated proper mathematical properties and related algorithms, and second to limit the number of moments used for applicability in multi-dimensional configurations without losing accuracy in the representation of spatial fluxes.
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