On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

Abstract

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible twofluid model, applied to gas-liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column.

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

“…As mentioned above, the bubbly flow model in table 6 can be augmented with a viscous-stress tensor (including pseudo-turbulence) for the fluid phase. Unlike in most other hyperbolic formulations for bubbly flows (see, for example, Panicker et al 2018), it is not necessary to add a turbulent-dispersion term to enforce hyperbolicity. As shown in appendix B, the model for P p ensures global hyperbolicity.…”

“…Also, note that the partial derivative of the second term in (B 8) with respect to α p has the form of the 'turbulent-dispersion' force that is used to make bubbly flow models hyperbolic (see, e.g. Panicker et al 2018).…”

A quadrature-based moment method for polydisperse bubbly flows

“…3 are used in these simulations. The dispersion coefficient was modeled according to Panicker et al [9], drag and lift forces are described by the model of Tomiyama [40,58], the wall-pressure model of Antal et al [42] was used, and a constant virtual-mass coefficient of 0.5 was used.…”

“…In addition, the basic EE formulation also suffers from not being hyperbolic and, therefore, cannot reach a converged solution with grid refinement [7][8][9].…”

“…The interface coupling terms are also modified to include contributions arising from the polydisperse nature of the flow and by its description with QBMM. In addition, the hyperbolicity of the model is enforced by using a dispersion term [9], proportional to the gradient of the gas volume fraction.…”