An extended volume of fluid method is developed for two-phase direct
numerical simulations of systems with one viscoelastic and one Newtonian phase.
A complete set of governing equations is derived by conditional
volume-averaging of the local instantaneous bulk equations and interface jump
conditions. The homogeneous mixture model is applied for the closure of the
volume-averaged equations. An additional interfacial stress term arises in this
volume-averaged formulation which requires special treatment in the
finite-volume discretization on a general unstructured mesh. A novel numerical
scheme is proposed for the second-order accurate finite-volume discretization
of the interface stress term. We demonstrate that this scheme allows for a
consistent treatment of the interface stress and the surface tension force in
the pressure equation of the segregated solution approach. Because of the high
Weissenberg number problem, an appropriate stabilization approach is applied to
the constitutive equation of the viscoelastic phase to increase the robustness
of the method at higher fluid elasticity. Direct numerical simulations of the
transient motion of a bubble rising in a quiescent viscoelastic fluid are
performed for the purpose of experimental code validation. The well-known jump
discontinuity in the terminal bubble rise velocity when the bubble volume
exceeds a critical value is captured by the method. The formulation of the
interfacial stress together with the novel scheme for its discretization is
found crucial for the quantitatively correct prediction of the jump
discontinuity in the terminal bubble rise velocity