Flow and axial dispersion in a sinusoidal-walled tube: Effects of inertial and unsteady flows

• We simulate Navier–Stokes flow in a sinusoidal-walled tube.
• We compare dispersion from volume averaging with direct numerical simulation.
• We evaluate effects of inertial and unsteady flows on effective dispersion.
• Reduced dispersion at high Re is caused by mixing due to unsteady flow.

In this work, we consider a sinusoidal-walled tube (a three-dimensional tube with sinusoidally-varying diameter) as a simplified conceptualization of flow in porous media. Direct numerical simulation using computational fluid dynamics (CFD) methods was used to compute velocity fields by solving the Navier–Stokes equations, and also to numerically solve the volume averaging closure problem, for a range of Reynolds numbers (Re) spanning the low-Re   to inertial flow regimes, including one simulation at Re=449Re=449 for which unsteady flow was observed. The longitudinal dispersion observed for the flow was computed using a random walk particle tracking method, and this was compared to the longitudinal dispersion predicted from a volume-averaged macroscopic mass balance using the method of volume averaging; the results of the two methods were consistent. Our results are compared to experimental measurements of dispersion in porous media and to previous theoretical results for both the low-Re, Stokes flow regime and for values of Re   representing the steady inertial regime. In the steady inertial regime, a power-law increase in the effective longitudinal dispersion (DLDL) with Re   was found, and this is consistent with previous results. This rapid rate of increase is caused by trapping of solute in expansions due to flow separation (eddies). One unsteady (but non-turbulent) flow case (Re=449Re=449) was also examined. For this case, the rate of increase of DLDL with Re was smaller than that observed at lower Re. Velocity fluctuations in this regime lead to increased rates of solute mass transfer between the core flow and separated flow regions, thus diminishing the amount of tailing caused by solute trapping in eddies and thereby reducing longitudinal dispersion. The observed tailing was further explored through analysis of concentration skewness (third moment) and its assymptotic convergence to conventional advection–dispersion behavior (skewness = 0). The method of volume averaging was applied to develop a skewness model, and demonstrated that the skewness decreases as a function of inverse square root of time. Our particle tracking simulation results were shown to conform to this theoretical result in most of the cases considered.