Front dynamics and macroscopic diffusion in buoyant mixing in a tilted tube

The buoyancy driven interpenetration of two fluids of different densities has been studied in a long tilted tube in the strong mixing regime for which the mean concentration profile along the tube length satisfies a macroscopic diffusion equation. Variations of the corresponding macroscopic diffusion coefficient $D$ and of the front velocity $V_f$ are studied as a function of the Atwood number $At$, the viscosity $\nu$, the tube diameter $d$ and the tilt angle $\theta$. Introducing the characteristic inertial velocity $V_t$ and the Reynolds number $Re_t$, the normalized front velocity $V_f/ V_t$ and dispersion coefficient $D/ (V_t d)$ are observed to scale respectively as $Re_t^{-3/4}$ and $Re_t^{-3/2}$ for $Re_t \alt 1000$. Also, $V_f$ increases linearly with $\tan \theta$ and the ratio $(D/V_f^2)$ remains of the order of ($35 \pm 10) d/V_t$ in a wide range of values of the tilt angle and of the other control parameters. This close relation observed between the variations of $D$ and $V_f^2$ will be discussed in terms of the characteristic time for transverse mixing across the flow channel.