Virtual mass in two-phase bubbly flow

Abstract

The two-phase flow equations in their usual form are unstable. It is known that the inclusion of the virtual mass of the gas bubbles greatly improves the stability of numerical computations. Since there is some confusion concerning the proper form of the virtual-mass terms, we first present a systematic derivation of the two-fluid equations for a liquid/gas mixture from a generalised form of Hamilton's variational principle. The two-fluid theory of superfluid 4He has been derived in a similar way. The resulting equations do not seem to have been presented before in the literature on two-phase flow. The derivation demonstrates how the pressure of the liquid/gas mixture can be defined in a natural way. Two independent vorticities can be distinguished, each having its own law of transportation (Kelvin theorem). A subsequent stability analysis shows that at neutral stability the virtual-mass coefficient m(α) takes the form m(α)= 1 2 α(1−α)(1−3α) in the case of spherical gas bubbles with void fraction α. The corresponding local distribution of gas bubbles is anisotropic. The vanishing of the virtual mass at α= 1 3 is interpreted as the breakdown of bubbly flow. Dissipative terms are introduced and analysed in an appendix.