**3.1.1 Wall shear stress and friction Reynolds number**

In table 8, the relative magnitude of the wall shear stress *W W* <sup>0</sup> and the friction Reynolds number Re Re 0 are shown for the bubbly and droplet flows. Note that these two quantities are related with the wall units as

$$\frac{\text{Re}\_{\tau}}{\text{Re}\_{\tau 0}} = \frac{l\_{\tau 0}}{l\_{\tau}} = \frac{u\_{\tau}}{u\_{\tau 0}}, \quad \frac{\tau\_{\text{IV}}}{\tau\_{\text{IV0}}} = \frac{t\_{\tau 0}}{t\_{\tau}} = \left(\frac{\text{Re}\_{\tau}}{\text{Re}\_{\tau 0}}\right)^{2} \tag{23}$$

when and *<sup>c</sup>* are fixed. The wall shear stress is increased by the factor of 1.65 (or 1.37) for the bubbly (or droplet) flow.


Table 8. Wall shear stress and friction Reynolds number. The values in the parentheses represent those for the bubbly flow with the lower grid resolution.
