Density-weighted mean ($\tilde{u}=⟨\rho u⟩/⟨\rho ⟩$)

For positive mean density, this average weights a velocity by local mass rather than volume. It satisfies the exact identity $⟨\rho u⟩=⟨\rho ⟩\tilde{u}$, so it naturally represents mass flux in averaged conservation laws. The ordinary volume mean generally fails that identity when density and velocity are correlated. Applied inside a phase, the same construction uses density times that phase's indicator as its weight.