On the flow and coherent structures generated by a circular array of rigid emerged cylinders placed in an open channel with flat and deformed bed

Abstract

The flow and the turbulence structure generated by a circular porous cylinder of diameter $D$ containing solid cylinders of diameter $d$ placed in an open channel of depth $h\approx 0.5D$ are investigated using eddy-resolving simulations which resolve the wakes past the individual solid cylinders in the array. The solid cylinders extend from the bed through the water surface. This geometrical set-up is directly relevant to understand the physics of flow past an emerged patch of aquatic vegetation developing in a river channel or over its floodplain. Simulations are conducted with different solid volume fractions (SVFs) of the porous cylinder ($0.034<\text{SVF}<0.23$), relative diameters of the solid cylinders ($d/D=0.03$ and 0.06) and with flat and equilibrium scour bathymetry corresponding to the start and respectively the end of the erosion and deposition process. Comparison with the limiting case of a solid cylinder ($\text{SVF}=1$) is also discussed. The bed shear stress distributions and the turbulent flow fields are used to explain the sediment erosion mechanisms inside and around the porous cylinder. Simulations of the flat-bed cases reveal that for sufficiently large SVF values ($\text{SVF}>0.2$), necklace vortices form around the upstream face of the cylinder, the downflow penetrates partially inside the porous cylinder and a region of strong flow acceleration forms on the sides of the porous cylinder. These flow features are used to explain the development of scour around high-SVF porous cylinders. The effects of the SVF and $d/D$ on generating ‘corridors’ of strong flow acceleration in between the solid cylinders and energetic eddies in the wake of these cylinders are discussed, as these flow features control the amplification of the bed shear stress inside the porous cylinder. Simulations results are also used to quantify the time-averaged drag forces on the cylinders in the array, to identify the regions where these forces are comparable to those induced on an isolated cylinder and the percentage of cylinders in the array subject to relatively large mean drag forces. A logarithmic decrease of the mean time-averaged streamwise drag coefficient of the solid cylinders, $\overline{C}_{d}$, with increasing non-dimensional frontal area per unit volume of the porous cylinder, $aD$, is observed. Behind the cylinder, the eddies shed in the separated shear layers (SSLs) of the porous cylinder, and, for sufficiently large SVFs, the von Kármán wake billows are the main coherent structures responsible for the amplification of the bed shear stress and sediment entrainment. This paper also analyses the vertical non-uniformity of the mean flow and turbulent kinetic energy, and discusses how the SVF and bathymetry affect the spatial extent of the wake region (e.g. length of the SSLs and steady wake, total wake length) and other relevant variables (e.g. strength of the bleeding flow, dominant wake frequencies, turbulence amplification in the near wake). For the relatively shallow flow conditions ($D/h\approx 2.0$) considered, the simulation results show that the antisymmetric (von Kármán) shedding of wake billows behind the porous cylinder is greatly weakened once equilibrium scour conditions are approached. Comparison with data from laboratory experiments and from 3-D and 2-D simulations conducted for long porous cylinders (no bed) is also discussed.