ABSTRACT Relying on two recent contributions by Massot et al. [SIAM J. Appl. Math. 70 (2010), 320... more ABSTRACT Relying on two recent contributions by Massot et al. [SIAM J. Appl. Math. 70 (2010), 3203--3234] and Kah et al. [J. Comput. Phys. 231 (2012), 394--422], where a Eulerian Multi-Size Moment (EMSM) model for the simulation of polydisperse evaporating sprays has been introduced, we investigate the potential of such an approach for the robust and accurate simulation of the injection of a liquid disperse phase into a gas for automotive engine applications. The original model used a high order moment method in droplet size to resolve polydispersity, with built-in realizability preserving numerical algorithm of high order in space and time, but only dealt with one-way coupling and was restricted to fixed meshes. Extending the approach to internal combustion engine and fuel injection requires solving two major steps forward, while preserving the properties of robustness, accuracy and realizability: 1- the extension of the method and numerical strategy to two-way coupling with stable integration of potential stiff source terms, 2- the introduction of a moving geometry and meshes. We therefore present a detailed account on how we have solved these two issues, provide a series of verification of the proposed algorithm, showing its potential in simplified configurations. The method is then implemented in the IFP-C3D unstructured solver for reactive compressible flows in engines and validated through comparisons with a structured fixed mesh solver; it finally proves its potential on a free spray jet injection where it is compared to a Lagrangian approach and its accuracy and robustness are assessed, thus making it a good candidate for realistic injection applications.
In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is di... more In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gas-droplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N, N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N-dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the Eulerian Multi Size Moment (EMSM) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.
The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous fl... more The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous flows with large scale vortical structures is both a crucial issue for industrial applications and a challenge for modeling and scientific computing. The difficulties encountered by the usual Lagrangian approaches make the use of Eulerian models attractive, aiming at a lower cost and an easier coupling with the carrier gaseous phase. Among these models, the multi-fluid model allows the detailed description of polydispersity and size/velocity correlations for droplets of various sizes. The purpose of the present study is two-fold. First, we extend the multi-fluid model in order to cope with droplet trajectory crossings by using the quadrature method of moments in velocity phase space conditioned by size. We identify the numerical difficulties and provide dedicated numerical schemes in order to preserve the velocity moment space. Second, we conduct a comparison study and demonstrate the capability of such an approach to capture the dynamics of an evaporating polydisperse spray in a two-dimensional free jet configuration. We evaluate the accuracy and computational cost of Eulerian models and related discretization schemes versus Lagrangian solvers. It shows that, even for finite Stokes number, the standard Eulerian multi-fluid model is accurate at reasonable cost. To cite this article: S. de Chaisemartin et al., C. R. Mecanique 337 (2009).
The secondary bag break-up mechanism is described and modeled at a mesoscopic scale using populat... more The secondary bag break-up mechanism is described and modeled at a mesoscopic scale using population balance equations as a first step towards industrial simulations of turbulent atomization. A stochastic process accounting for temporality and experimental asymptotics of turbulence-liquid interactions is built with a self similar kernel whereas larger droplets are only subject to a mean-flow-based discrete and instantaneous breakup kernel. For turbulence to influence only the small droplets, they are collected into a second population balance equation and renamed ligaments in relation with their transient shape during the process. The evolution of velocity distributions is introduced in the modeling as a driving parameter of the break-up process. Computations are made for William's full spray equation on simplified cases. Finally the model is reduced to a system of conservation equations in space and time with closed source terms thanks to the Eulerian Multi-Fluid approach, in order to be applicable for more realistic configurations.
This work investigates the issue of describing polydispersivity in an Eulerian framework of a dis... more This work investigates the issue of describing polydispersivity in an Eulerian framework of a disperse spray with potentially mesh movement. In this perspective, the multi-fluid model and and associated numerical schemes provide robust tools (de Chaisemartin (2009)). ...
The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous fl... more The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous flows with large-scale vortical structures is both a crucial issue for industrial applications and a challenge for modeling and scientific computing. The difficulties encountered by the usual Lagrangian approaches make the use of Eulerian models attractive, aiming at a lower cost and an easier coupling with the carrier gaseous phase. Among these models, the multi-fluid model allows for a detailed description of the ...
In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, o... more In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, of polydisperse multiphase systems, constituted of dispersed particles or droplets, a general class of systems which include aerosols. Their modeling starts at a mesoscopic scale with an equation on the number density function NDF of particles/droplets which satisfies a population balance equation. (PBE, also called Williams equation in the spray community). In order to limit the computational cost, moment methods provide a system of conservation equation with an eventual closure problem which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments. However, a drift velocity, that is, the rate of change due to continuous phenomena of the internal coordinate such as the size of the particles, has sometimes to be taken into account; it can be either positive like molecular growth, or negative such as for evaporation of droplets in aerosols or oxidation of soots. When negative, it leads to the disappearance of droplets/particles thus creating a negative flux at zero size. Its closure requires an evaluation of the reconstructed NDF at zero size from the knowledge of a given finite set of moments. The nature of this information, pointwise in internal coordinate, and its influence on moment dynamics results in a difficulty from both a modeling and a numerical point of view. We obtain in the present contribution a comprehensive solution to this important issue. Since we introduce some new tools in order to resolve the flux evaluation, we also introduce a new Eulerian type of description which will combine both the flexibility of Eulerian models for which the size phase space is discretized into "sections" (i.e. size intervals) and the efficiency of Direct Quadrature Method of Moments (DQMOM). It yields a precise and stable description of moment dynamics with a minimal number of variables which should lead to a low computational cost in multi-dimensional configurations.
Dilute liquid sprays can be modeled at the mesoscale using a kinetic equation, namely the William... more Dilute liquid sprays can be modeled at the mesoscale using a kinetic equation, namely the Williams–Boltzmann equation, containing terms for spatial transport, evaporation and fluid drag. The most common method for simulating the Williams–Boltzmann equation uses Lagrangian particle tracking wherein a finite ensemble of numerical “parcels” provides a statistical estimate of the joint surface area, velocity number density function (NDF). An alternative approach is to discretize the NDF into droplet size intervals, called sections, and to neglect velocity fluctuations conditioned on droplet size, resulting in an Eulerian multi-fluid model. In comparison to Lagrangian particle tracking, multi-fluid models contain no statistical error (due to the finite number of parcels) but they cannot reproduce the particle trajectory crossings observed in Lagrangian simulations of non-collisional kinetic equations. Here, in order to overcome this limitation, a quadrature-based moment method is used to describe the velocity moments. When coupled with the sectional description of droplet sizes, the resulting Eulerian multi-fluid, multi-velocity model is shown to capture accurately both particle trajectory crossings and the size-dependent dynamics of evaporation and fluid drag. Model validation is carried out using direct comparisons between the Lagrangian and Eulerian models for an unsteady free-jet configuration with mono- and polydisperse droplets with and without evaporation. Comparisons between the Eulerian and Lagrangian instantaneous number density and gas-phase fuel mass fraction fields show excellent agreement, suggesting that the multi-fluid, multi-velocity model is well suited for describing spray combustion.
Following the seminal work of F. Bouchut on zero pressure gas dynamics which has been extensively... more 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.
In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is di... more In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gasdroplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N , N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N -dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the EMSM (Eulerian Multi Size Moment) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.
ABSTRACT Relying on two recent contributions by Massot et al. [SIAM J. Appl. Math. 70 (2010), 320... more ABSTRACT Relying on two recent contributions by Massot et al. [SIAM J. Appl. Math. 70 (2010), 3203--3234] and Kah et al. [J. Comput. Phys. 231 (2012), 394--422], where a Eulerian Multi-Size Moment (EMSM) model for the simulation of polydisperse evaporating sprays has been introduced, we investigate the potential of such an approach for the robust and accurate simulation of the injection of a liquid disperse phase into a gas for automotive engine applications. The original model used a high order moment method in droplet size to resolve polydispersity, with built-in realizability preserving numerical algorithm of high order in space and time, but only dealt with one-way coupling and was restricted to fixed meshes. Extending the approach to internal combustion engine and fuel injection requires solving two major steps forward, while preserving the properties of robustness, accuracy and realizability: 1- the extension of the method and numerical strategy to two-way coupling with stable integration of potential stiff source terms, 2- the introduction of a moving geometry and meshes. We therefore present a detailed account on how we have solved these two issues, provide a series of verification of the proposed algorithm, showing its potential in simplified configurations. The method is then implemented in the IFP-C3D unstructured solver for reactive compressible flows in engines and validated through comparisons with a structured fixed mesh solver; it finally proves its potential on a free spray jet injection where it is compared to a Lagrangian approach and its accuracy and robustness are assessed, thus making it a good candidate for realistic injection applications.
In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is di... more In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gas-droplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N, N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N-dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the Eulerian Multi Size Moment (EMSM) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.
The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous fl... more The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous flows with large scale vortical structures is both a crucial issue for industrial applications and a challenge for modeling and scientific computing. The difficulties encountered by the usual Lagrangian approaches make the use of Eulerian models attractive, aiming at a lower cost and an easier coupling with the carrier gaseous phase. Among these models, the multi-fluid model allows the detailed description of polydispersity and size/velocity correlations for droplets of various sizes. The purpose of the present study is two-fold. First, we extend the multi-fluid model in order to cope with droplet trajectory crossings by using the quadrature method of moments in velocity phase space conditioned by size. We identify the numerical difficulties and provide dedicated numerical schemes in order to preserve the velocity moment space. Second, we conduct a comparison study and demonstrate the capability of such an approach to capture the dynamics of an evaporating polydisperse spray in a two-dimensional free jet configuration. We evaluate the accuracy and computational cost of Eulerian models and related discretization schemes versus Lagrangian solvers. It shows that, even for finite Stokes number, the standard Eulerian multi-fluid model is accurate at reasonable cost. To cite this article: S. de Chaisemartin et al., C. R. Mecanique 337 (2009).
The secondary bag break-up mechanism is described and modeled at a mesoscopic scale using populat... more The secondary bag break-up mechanism is described and modeled at a mesoscopic scale using population balance equations as a first step towards industrial simulations of turbulent atomization. A stochastic process accounting for temporality and experimental asymptotics of turbulence-liquid interactions is built with a self similar kernel whereas larger droplets are only subject to a mean-flow-based discrete and instantaneous breakup kernel. For turbulence to influence only the small droplets, they are collected into a second population balance equation and renamed ligaments in relation with their transient shape during the process. The evolution of velocity distributions is introduced in the modeling as a driving parameter of the break-up process. Computations are made for William's full spray equation on simplified cases. Finally the model is reduced to a system of conservation equations in space and time with closed source terms thanks to the Eulerian Multi-Fluid approach, in order to be applicable for more realistic configurations.
This work investigates the issue of describing polydispersivity in an Eulerian framework of a dis... more This work investigates the issue of describing polydispersivity in an Eulerian framework of a disperse spray with potentially mesh movement. In this perspective, the multi-fluid model and and associated numerical schemes provide robust tools (de Chaisemartin (2009)). ...
The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous fl... more The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous flows with large-scale vortical structures is both a crucial issue for industrial applications and a challenge for modeling and scientific computing. The difficulties encountered by the usual Lagrangian approaches make the use of Eulerian models attractive, aiming at a lower cost and an easier coupling with the carrier gaseous phase. Among these models, the multi-fluid model allows for a detailed description of the ...
In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, o... more In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, of polydisperse multiphase systems, constituted of dispersed particles or droplets, a general class of systems which include aerosols. Their modeling starts at a mesoscopic scale with an equation on the number density function NDF of particles/droplets which satisfies a population balance equation. (PBE, also called Williams equation in the spray community). In order to limit the computational cost, moment methods provide a system of conservation equation with an eventual closure problem which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments. However, a drift velocity, that is, the rate of change due to continuous phenomena of the internal coordinate such as the size of the particles, has sometimes to be taken into account; it can be either positive like molecular growth, or negative such as for evaporation of droplets in aerosols or oxidation of soots. When negative, it leads to the disappearance of droplets/particles thus creating a negative flux at zero size. Its closure requires an evaluation of the reconstructed NDF at zero size from the knowledge of a given finite set of moments. The nature of this information, pointwise in internal coordinate, and its influence on moment dynamics results in a difficulty from both a modeling and a numerical point of view. We obtain in the present contribution a comprehensive solution to this important issue. Since we introduce some new tools in order to resolve the flux evaluation, we also introduce a new Eulerian type of description which will combine both the flexibility of Eulerian models for which the size phase space is discretized into "sections" (i.e. size intervals) and the efficiency of Direct Quadrature Method of Moments (DQMOM). It yields a precise and stable description of moment dynamics with a minimal number of variables which should lead to a low computational cost in multi-dimensional configurations.
Dilute liquid sprays can be modeled at the mesoscale using a kinetic equation, namely the William... more Dilute liquid sprays can be modeled at the mesoscale using a kinetic equation, namely the Williams–Boltzmann equation, containing terms for spatial transport, evaporation and fluid drag. The most common method for simulating the Williams–Boltzmann equation uses Lagrangian particle tracking wherein a finite ensemble of numerical “parcels” provides a statistical estimate of the joint surface area, velocity number density function (NDF). An alternative approach is to discretize the NDF into droplet size intervals, called sections, and to neglect velocity fluctuations conditioned on droplet size, resulting in an Eulerian multi-fluid model. In comparison to Lagrangian particle tracking, multi-fluid models contain no statistical error (due to the finite number of parcels) but they cannot reproduce the particle trajectory crossings observed in Lagrangian simulations of non-collisional kinetic equations. Here, in order to overcome this limitation, a quadrature-based moment method is used to describe the velocity moments. When coupled with the sectional description of droplet sizes, the resulting Eulerian multi-fluid, multi-velocity model is shown to capture accurately both particle trajectory crossings and the size-dependent dynamics of evaporation and fluid drag. Model validation is carried out using direct comparisons between the Lagrangian and Eulerian models for an unsteady free-jet configuration with mono- and polydisperse droplets with and without evaporation. Comparisons between the Eulerian and Lagrangian instantaneous number density and gas-phase fuel mass fraction fields show excellent agreement, suggesting that the multi-fluid, multi-velocity model is well suited for describing spray combustion.
Following the seminal work of F. Bouchut on zero pressure gas dynamics which has been extensively... more 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.
In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is di... more In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gasdroplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N , N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N -dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the EMSM (Eulerian Multi Size Moment) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.
Uploads
Papers by Damien Kah