A compressible two-phase model for dispersed particle flows with application from dense to dilute regimes

McGrath, Thomas P.; Clair, Jeffrey G. St.; Balachandar, S.

doi:10.1063/1.4948301

Cited by 24 publications

(8 citation statements)

References 24 publications

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“…Relative phase velocity may be a necessary component in the particle jet formation, and therefore, it is not surprising that the radial and circumferential particle distributions are not well reproduced. Examples of continuum models more suitable for the present problem that include relative phase velocities are models based on the Baer-Nunziato equations [3] and variations thereof, e.g., [19]. Investigations on the use of these models are postponed to future studies.…”

Section: Base Flowmentioning

confidence: 99%

“…The model equations are formulated to account for the volume and density of the dispersed phase. There are numerous issues with such simplified dispersed flow models, such as non-hyperbolic equation sets, as discussed in Lhuillier et al 19 , Theofanous and Chang 41 . The second approach circumvents these problems by utilizing particle-resolved simulations.…”

Section: Introductionmentioning

confidence: 99%

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Numerical simulation of particle jet formation induced by shock wave acceleration in a Hele-Shaw cell

2017

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First of all I would like to thank my main supervisor, Prof. Bjørn Anders Pettersson Reif for his guidance and support throughout my Ph.D. studies. His insight and intuition for flow physics is exceptional, and we have had countless interesting discussions over the last years. He always manages to keep focus on the important questions, and this ability has shaped this thesis. I would also like to thank my other supervisors Magnus Vartdal and Marianne Gjestvold Omang. There is never a problem too difficult for Magnus, which is always inspiring for everyone around him. His involvement in the details in the studies within this thesis has been invaluable. Marianne has also provided valuable advice throughout these years. My period at the Norwegian Defence Estates Agency was very enjoyable and taught me a lot. During these years I have been fortunate to be allowed to work with the fluid dynamics group at the Norwegian Defence Research Establishment. Special thanks to Espen Åkervik, who I have shared an office with and can always ask for advice. I would also like to thank

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Section: Base Flowmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical simulation of particle jet formation induced by shock wave acceleration in a Hele-Shaw cell

2017

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“…Related approaches have been used by many other authors (see, e.g. Buyevich & Shchelchkova 1978; Nigmatulin 1979; Jackson 1997; Lhuillier & Theofanous 2010; McGrath II, St. Clair & Balachandar 2016; Saurel, Chinnayya & Carmouze 2017).…”

Section: Introductionmentioning

confidence: 99%

A kinetic-based hyperbolic two-fluid model for binary hard-sphere mixtures

Fox

2019

J. Fluid Mech.

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Starting from coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures is derived with separate mean velocities and energies for each phase. In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, using an elastic hard-sphere collision model, and the Archimedes (buoyancy) force due to spatial gradients of the pressure in each phase, as well as other forces involving spatial gradients (e.g. lift). In the derivation, the particles in a given phase have identical mass and volume, and have no internal degrees of freedom (i.e. the particles are adiabatic). The ‘hard-sphere-fluid’ phase is obtained in the limit where the particle diameter in one phase tends to zero with fixed phase density so that the number of fluid particles tends to infinity. The moment system resulting from the two BE kinetic equations is closed at second order by invoking the anisotropic Gaussian closure. The resulting two-fluid model for a binary, hard-sphere mixture therefore consists (for each phase $\unicode[STIX]{x1D6FC}=1,2$) of transport equations for the mass $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}$, mean momentum $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}$ (where $\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}$ is the velocity) and a symmetric, second-order, kinetic energy tensor $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D640}_{\unicode[STIX]{x1D6FC}}=\frac{1}{2}\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}\otimes \boldsymbol{u}_{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}})$. The trace of the fluctuating energy tensor $\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}}$ is $\text{tr}(\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}})=3\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FC}}$ where $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FC}}$ is the phase temperature (or granular temperature). Thus, $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}E_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\text{tr}(\unicode[STIX]{x1D640}_{\unicode[STIX]{x1D6FC}})$ is the total kinetic energy, the sum over $\unicode[STIX]{x1D6FC}$ of which is the total kinetic energy of the system, a conserved quantity. From the analysis, it is found that the BE finite-size correction leads to exact phase pressure (or stress) tensors that depend on the mean-slip velocity $\boldsymbol{u}_{12}=\boldsymbol{u}_{1}-\boldsymbol{u}_{2}$, as well as the phase temperatures for both phases. These pressure tensors also appear in the momentum-exchange terms in the mean momentum equations that produce the Archimedes force, as well as drag contributions due to fluid compressibility and a lift force due to mean fluid-velocity gradients. The closed BE energy flux tensors show a similar dependence on the mean-slip velocity. The characteristic polynomial of the flux matrix from the one-dimensional model is computed symbolically and depends on five parameters: the particle volume fractions $\unicode[STIX]{x1D711}_{1}$, $\unicode[STIX]{x1D711}_{2}$, the phase density ratio ${\mathcal{Z}}=\unicode[STIX]{x1D70C}_{f}/\unicode[STIX]{x1D70C}_{p}$, the phase temperature ratio $\unicode[STIX]{x1D6E9}_{r}=\unicode[STIX]{x1D6E9}_{2}/\unicode[STIX]{x1D6E9}_{1}$ and the mean-slip Mach number $Ma_{s}=\boldsymbol{u}_{12}/\sqrt{5\unicode[STIX]{x1D6E9}_{1}/3}$. By applying Sturm’s Theorem to the characteristic polynomial, it is demonstrated that the model is hyperbolic over a wide range of these parameters, in particular, for the physically most relevant values.

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“…Besides combustion, two-phase mixture models have been used to describe the hydrodynamics of physical systems composed of separated-phase continua in a variety of contexts. Examples include bubbly liquid flows [4,5] and solid-particle loaded fluid flows [6]. In this paper, we study a fundamental problem associated with these models, adopting the continuum-mixture theory point of view.…”

Section: Introductionmentioning

confidence: 99%

Shock structure for the seven-equation, two-phase continuum-mixture model

Bdzil

Kapila

Hennessey

2021

Combustion Theory and Modelling

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Continuum models of two-phase flow have been employed to investigate a wide variety of physical phenomena, ranging from bubbly-gas and solid-loaded liquid flows to detonation waves in granular explosives. The set of governing equations consists of mass, momentum and energy balance for each phase, and an equation for the evolution of volume fraction of either phase. The set is closed by providing an equation of state for each phase and by specifying the inter-phase interaction terms. Examination of the time scales associated with the interaction terms shows that in many regions of the flow, the pressures and velocities equilibrate rapidly, which has led to the development of a single-pressure and single-velocity reduced, 5-equation model. Such a reduction is incomplete in that it omits thin shock-like regions (inner layers) where O(1) pressure and velocity changes can remain. To regularise the reduced model, a number of studies have added dissipative terms which smoothen and thereby resolve these thin shock-like regions. However, the sensitivity of the solution to the choice of regularisation has not been investigated. Further, although the 5-equation model has been used extensively in the literature for numerical computations, its regularisation is often left to the numerical method employed. This study carries out a systematic analysis of the structure of this thin shock-like inner layer, with emphasis on the dependence of the structure of the layer and of the jumps across it on system parameters. The results are compared with those obtained via computations with the parent, 7-equation model.

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A compressible two-phase model for dispersed particle flows with application from dense to dilute regimes

Cited by 24 publications

References 24 publications

Numerical simulation of particle jet formation induced by shock wave acceleration in a Hele-Shaw cell

Numerical simulation of particle jet formation induced by shock wave acceleration in a Hele-Shaw cell

A kinetic-based hyperbolic two-fluid model for binary hard-sphere mixtures

Shock structure for the seven-equation, two-phase continuum-mixture model

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