A gravity-driven film flow on a slippery inclined plane is considered within the framework of long-wave and boundary layer approximations. Two coupled depth-averaged equations are derived in terms of the local flow rate $q(x, t)$ and the film thickness $h(x, t)$. Linear stability analysis of the averaged equations shows good agreement with the Orr–Sommerfeld analysis. The effect of a slip at the wall on the primary instability has been found to be non-trivial. Close to the instability onset, the effect is destabilising whereas it becomes stabilising at larger values of the Reynolds number. Nonlinear travelling waves are amplified by the presence of the slip. Comparisons to direct numerical simulations show a remarkable agreement for all tested values of parameters. The averaged equations capture satisfactorily the speed, shape and velocity distribution in the waves. The Navier slip condition is observed to significantly enhance the backflow phenomenon in the capillary region of the solitary waves with a possible effect on heat and mass transfer.
A gravity-driven falling film on a saturated porous inclined plane is studied via a continuum approach, where the liquid and porous layers are considered as a single composite layer. Using a weighted residual technique, a two-equation model is derived in terms of the local flow rate $q(x, t)$ and the entire layer thickness $H(x, t)$. Its linear stability analysis has been satisfactorily compared to the results of the Orr–Sommerfeld problem. The principal effect of the porous substrate on the film flow is to displace the liquid–porous interface to an effective liquid–solid interface located at the lower boundary of the upper momentum boundary layer in the porous medium. The stability and dynamics of the film is thus only weakly affected by the presence of a permeable substrate. In both the linear and the nonlinear regimes, the spatial response of a falling film on a porous medium is not very different from that observed on an impermeable inclined wall. However, the wavy motion of the film triggers a significant exchange of mass at the liquid–porous interface.
Direct numerical simulations of the fully developed turbulent flow through a porous square duct are performed to study the effect of the permeable wall on the secondary cross-stream flow. The volume-averaged Navier-Stokes equations are used to describe the flow in the porous phase, a packed bed with porosity ε c = 0.95. The porous square duct is computed at Re b 5000 and compared with the numerical simulations of a turbulent duct with four solid walls. The two boundary layers on the top wall and porous interface merge close to the centre of the duct, as opposed to the channel, because the sidewall boundary layers inhibit the growth of the shear layer over the porous interface. The most relevant feature in the porous duct is the enhanced magnitude of the secondary flow, which exceeds that of a regular duct by a factor of four. This is related to the increased vertical velocity, and the different interaction between the ejections from the sidewalls and the porous medium. We also report a significant decrease in the streamwise turbulence intensity over the porous wall of the duct (which is also observed in a porous channel), and the appearance of short spanwise rollers in the buffer layer, replacing the streaky structures of wall-bounded turbulence. These spanwise rollers most probably result from a Kelvin-Helmholtz type of instability, and their width is limited by the presence of the sidewalls.
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