The effect of permeability on the flow past permeable disks at low Reynolds numbers

Abstract

The viscous flow over a thick permeable circular disk in the Reynolds number (Re) range of 10 to 130 and in the Darcy number (Da) range of 10−9 to 1 is examined. Direct numerical simulations are performed on a 2D grid with axisymmetric boundary conditions. Three flow regimes are observed: I, II, and III. In regime I (effectively impervious; Da<10−6), the wake is characterized by the presence of a toroidal vortex whose length is approximately equal to that of an impervious disk. In regime II (transition; 10−6<Da<10−3), the increase in Da causes the vortex to shorten and move downstream and eventually vanishes at a critical Darcy number Dac. Regime III (Da>10−3) is the highly permeable regime, in which there is no recirculation. In I, good agreement with existing experimental data for impervious disks is found. In III, an analytical expression for the drag force on the disk is derived, showing good agreement with the numerical results. A global upper limit of Dac=Damax above which the disk is unable to maintain a recirculating wake for any Re is identified. Finally, in regime II, it is demonstrated that increasing the permeability can lead to large variations in the length of the recirculating wake but with minimal effect on the drag coefficient even when Da>Damax. This has important implications in our understanding of the locomotive strategies adopted by organisms that use porous bodies for movement.

Suppression of von Kármán vortex streets past porous rectangular cylinders

“…Increasing the Darcy number, the C D slightly decreases first and a significant reduction is successively visible for Da > 1 × 10 −3 . In this range of Da, the drag coefficient follows the scaling C D ∼ Re −1 Da −1 , as reported in[3].…”

“…Increasing the porosity, and so the permeability, the vortex street formation moves further downstream. More recently, the problem of the flow around porous square cylinders [8,9] and porous disks [3] has been approached numerically. In the latter case, by increasing the disk permeability three different flow regimes have been recognized: (i) first an effectively impervious regime, which is characterized by the presence of a toroidal vortex recirculation region located close to the disk, is observed at low permeability; (ii) Subsequently a transition regime in which the recirculation region shortens and moves downstream occurs for intermediate permeability;…”

A separated vortex ring underlies the flight of the dandelion

“…We adopted the modelling approach of ref. 25, where we considered the steady, axisymmetric flow past the permeable disk. In the fluid domain, the steady-state Navier-Stokes equations are solved and inside the permeable disk, the steady-state Darcy-Brinkmann equations are solved.…”

“…However, our numerical modelling and experimental measurements revealed that the flow through the seed entails strong interactions between neighboring filaments 19,20 , causing the pappus to behave as a permeable membrane. It has been suggested that changing the permeability of a body could be useful to control or suppress the vortex shedding 25 . A recent study has confirmed that the motion of freely falling disks (with > 10 3 ) can be stabilized by a hole in the center of the disk 26 .…”

Environmental morphing enables informed dispersal of the dandelion diaspore

“…The velocity deficit behind porous disks is associated with negative pressure (12). A larger vortex with a greater velocity deficit may therefore increase the magnitude of the negative pressure, enhancing drag on the pappus.…”