A logarithmic law of the wall for upward, turbulent, bubbly two-phase flow

In this paper, a logarithmic law of the wall is proposed for steady, upward, turbulent, bubbly two-phase flow in the earth's gravitational force field. The single-phase logarithmic law of the wall is extended for modeling the near-wall area for low void-fraction bubbly two-phase flows. In this model, the interfacial forces, in particular the drag force, are explicitly accounted for in the model formulation. The final nondimensionalized form of the proposed logarithmic law of the wall is similar to that of the corresponding law for single-phase turbulent flows. However, in the proposed wall law, the Von-Karman term is calculated from an expression which relates its value to the two-phase flow properties. The proposed model matches satisfactorily available experimental results for upward, turbulent, bubbly two-phase flow over a vertical flat plate.