A hyperbolic two-fluid model for compressible flows with arbitrary material-density ratios

• Published in: Journal of fluid mechanics Vol. 903
• Main Authors: Fox, Rodney O., Laurent, Frédérique, Vié, Aymeric
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 25.11.2020; Cambridge University Press (CUP)
• Subjects: 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; particle/fluid flow; multiphase flow; disperse multiphase flow; compressible flow; hyperbolic two-fluid model; material-density ratio
• DOI: 10.1017/jfm.2020.615
• ISSN: 0022-1120; 1469-7645

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.