A hyperbolic two-fluid model for compressible flows with arbitrary material-density ratios
A hyperbolic two-fluid model for gas–particle flow derived using the Boltzmann–Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy...
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Published in | Journal of fluid mechanics Vol. 903 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
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Cambridge, UK
Cambridge University Press
25.11.2020
Cambridge University Press (CUP) |
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Abstract | A hyperbolic two-fluid model for gas–particle flow derived using the Boltzmann–Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy balances for the particle phase, augmented to include the portion of the particle wake moving with the particle velocity. The resulting compressible two-fluid model contains seven balance equations (mass, momentum and energy for each phase, plus added mass) and employs a stiffened-gas model for the equation of state for the fluid. Using Sturm's theorem, the model is shown to be globally hyperbolic for arbitrary ratios of the material densities $Z = \rho _f / \rho _p$ (where $\rho _f$ and $\rho _p$ are the fluid and particle material densities, respectively). An eight-equation extension to include the pseudo-turbulent kinetic energy (PTKE) in the fluid phase is also proposed; however, PTKE has no effect on hyperbolicity. In addition to the added mass, the key physics needed to ensure hyperbolicity for arbitrary $Z$ is a fluid-mediated contribution to the particle-phase pressure tensor that is taken to be proportional to the volume fraction of the added mass. A numerical solver for hyperbolic equations is developed for the one-dimensional model, and numerical examples are employed to illustrate the behaviour of solutions to Riemann problems for different material-density ratios. The relation between the proposed two-fluid model and prior work on effective-field models is discussed, as well as possible extensions to include viscous stresses and the formulation of the model in the limit of an incompressible continuous phase. |
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AbstractList | A hyperbolic two-fluid model for gas–particle flow derived using the Boltzmann–Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy balances for the particle phase, augmented to include the portion of the particle wake moving with the particle velocity. The resulting compressible two-fluid model contains seven balance equations (mass, momentum and energy for each phase, plus added mass) and employs a stiffened-gas model for the equation of state for the fluid. Using Sturm's theorem, the model is shown to be globally hyperbolic for arbitrary ratios of the material densities $Z = \rho _f / \rho _p$ (where $\rho _f$ and $\rho _p$ are the fluid and particle material densities, respectively). An eight-equation extension to include the pseudo-turbulent kinetic energy (PTKE) in the fluid phase is also proposed; however, PTKE has no effect on hyperbolicity. In addition to the added mass, the key physics needed to ensure hyperbolicity for arbitrary $Z$ is a fluid-mediated contribution to the particle-phase pressure tensor that is taken to be proportional to the volume fraction of the added mass. A numerical solver for hyperbolic equations is developed for the one-dimensional model, and numerical examples are employed to illustrate the behaviour of solutions to Riemann problems for different material-density ratios. The relation between the proposed two-fluid model and prior work on effective-field models is discussed, as well as possible extensions to include viscous stresses and the formulation of the model in the limit of an incompressible continuous phase. Abstract A hyperbolic two-fluid model for gas–particle flow derived using the Boltzmann–Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy balances for the particle phase, augmented to include the portion of the particle wake moving with the particle velocity. The resulting compressible two-fluid model contains seven balance equations (mass, momentum and energy for each phase, plus added mass) and employs a stiffened-gas model for the equation of state for the fluid. Using Sturm's theorem, the model is shown to be globally hyperbolic for arbitrary ratios of the material densities $Z = \rho _f / \rho _p$ (where $\rho _f$ and $\rho _p$ are the fluid and particle material densities, respectively). An eight-equation extension to include the pseudo-turbulent kinetic energy (PTKE) in the fluid phase is also proposed; however, PTKE has no effect on hyperbolicity. In addition to the added mass, the key physics needed to ensure hyperbolicity for arbitrary $Z$ is a fluid-mediated contribution to the particle-phase pressure tensor that is taken to be proportional to the volume fraction of the added mass. A numerical solver for hyperbolic equations is developed for the one-dimensional model, and numerical examples are employed to illustrate the behaviour of solutions to Riemann problems for different material-density ratios. The relation between the proposed two-fluid model and prior work on effective-field models is discussed, as well as possible extensions to include viscous stresses and the formulation of the model in the limit of an incompressible continuous phase. A hyperbolic two-fluid model for gas-particle flow derived using the Boltzmann-Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy balances for the particle phase, augmented to include the portion of the particle wake moving with the particle velocity. The resulting compressible two-fluid model contains seven balance equations (mass, momentum, and energy for each phase, plus added mass) and employs a stiffened-gas model for the equation of state for the fluid. Using Sturm's theorem, the model is shown to be globally hyperbolic for arbitrary ratios of the material densities Z = ρ f /ρ p. An eight-equation extension to include the pseudo-turbulent kinetic energy (PTKE) in the fluid phase is also proposed; however, PTKE has no effect on hyperbolicity. In addition to the added mass, the key physics needed to ensure hyperbolicity for arbitrary Z is a fluid-mediated contribution to the particle-phase pressure tensor that is taken to be proportional to the volume fraction of the added mass. A numerical solver for hyperbolic equations is developed for the 1-D model, and numerical examples are employed to illustrate the behaviour of solutions to Riemann problems for different material-density ratios. The relation between the proposed two-fluid model and prior work on effective-field models is discussed, as well as possible extensions to include viscous stresses and the formulation of the model in the limit of an incompressible continuous phase. |
ArticleNumber | A5 |
Author | Vié, Aymeric Fox, Rodney O. Laurent, Frédérique |
Author_xml | – sequence: 1 givenname: Rodney O. orcidid: 0000-0003-1944-1861 surname: Fox fullname: Fox, Rodney O. email: rofox@iastate.edu organization: 1Department of Chemical and Biological Engineering, Iowa State University, 618 Bissell Road, Ames, IA50011-1098, USA – sequence: 2 givenname: Frédérique surname: Laurent fullname: Laurent, Frédérique organization: 3Fédération de Mathématiques de CentraleSupélec, CNRS, 3, rue Joliot-Curie, 91192Gif-sur-Yvette CEDEX, France – sequence: 3 givenname: Aymeric surname: Vié fullname: Vié, Aymeric organization: 3Fédération de Mathématiques de CentraleSupélec, CNRS, 3, rue Joliot-Curie, 91192Gif-sur-Yvette CEDEX, France |
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CitedBy_id | crossref_primary_10_1016_j_ijmultiphaseflow_2023_104591 crossref_primary_10_1103_PhysRevFluids_6_104306 crossref_primary_10_1146_annurev_fluid_121021_015818 crossref_primary_10_1115_1_4064660 crossref_primary_10_1016_j_ces_2024_119909 crossref_primary_10_1016_j_ijmultiphaseflow_2023_104456 crossref_primary_10_1016_j_ijmultiphaseflow_2023_104698 crossref_primary_10_1016_j_ijmultiphaseflow_2022_104008 crossref_primary_10_1016_j_ijmultiphaseflow_2023_104715 crossref_primary_10_1016_j_ijmultiphaseflow_2024_104902 crossref_primary_10_1016_j_ijmultiphaseflow_2021_103563 crossref_primary_10_1063_5_0123445 crossref_primary_10_1063_5_0147347 |
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Keywords | particle/fluid flow multiphase flow disperse multiphase flow compressible flow hyperbolic two-fluid model material-density ratio |
Language | English |
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Snippet | A hyperbolic two-fluid model for gas–particle flow derived using the Boltzmann–Enskog kinetic theory is generalized to include added mass. In place of the... Abstract A hyperbolic two-fluid model for gas–particle flow derived using the Boltzmann–Enskog kinetic theory is generalized to include added mass. In place of... A hyperbolic two-fluid model for gas-particle flow derived using the Boltzmann-Enskog kinetic theory is generalized to include added mass. In place of the... |
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SubjectTerms | Compressible flow Computational fluid dynamics Density Energy balance Equations of state Fluid flow Incompressible flow JFM Papers Kinetic energy Kinetic theory Mass Mathematical models Mathematics Momentum Numerical Analysis One dimensional models Orbital velocity Physics Ratios Reynolds number Tensors Two fluid models Velocity |
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Title | A hyperbolic two-fluid model for compressible flows with arbitrary material-density ratios |
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