Influence of submergence ratio on flow and drag forces generated by a long rectangular array of rigid cylinders at the sidewall of an open channel

Abstract

This paper discusses how the submergence ratio, defined as the ratio between the flow depth, D, and the height, h, of the solid rigid cylinders forming the array affects flow and turbulence structure inside and around a rectangular array of cylinders placed adjacent to one of the channel sidewalls. As the array becomes submerged, a vertical shear layer develops in between the top face of the array and the free surface, which strongly increases flow three-dimensionality and modifies how the momentum exchange between the array and the surrounding open water regions occurs with respect to the case of an emerged array where only a horizontal shear layer forms as part of the incoming flow approaching the array is deflected laterally. Eddy-resolving simulations are conducted for several values of the solid volume fraction, ϕ, and of the submergence ratio, 1.0 ≤ D/h ≤ 4.0. Similar to the limiting case of an emerged array (D/h = 1.0), the width- and depth-averaged streamwise velocity inside the array reaches a constant value after an initial adjustment region in the submerged-array cases. For D/h ≥ 1.33, the mean normal velocities through the top and side faces of the array do not become equal to zero downstream of the initial adjustment region. The flow inside the array reaches an equilibrium regime where the local flux of fluid leaving the array through its side face is balanced by the local flux of fluid entering the array through its top face. This regime is observed until close to the end of the array. For D/h ≥ 2.0, the horizontal shear layer vortices do not generate successive regions of high and low streamwise velocity and bed friction velocity inside the array, as is observed in the emerged cases. With respect to the emerged case, the size of the shear layer vortices and the shear layer width increase for low submergence ratios before decreasing rapidly for D/h ≥ 1.33. Significant three-dimensional effects are present inside the array and the horizontal shear layer for cases with both emerged and submerged arrays. In the ϕ = 0.08 cases, strong upwelling and downwelling motions are observed inside the array for D/h ≥ 1.33, while the circulation of the streamwise-oriented cell of secondary flow forming close to the lateral face of the array peaks when the submergence ratio is close to 1.33. For constant ϕ, the total streamwise drag force normalized with the height and width of the array increases with increasing submergence ratio. As the array submergence increases, the cylinders near the front of the array contribute less to the total force acting on the cylinders forming the array. For constant D/h, the total streamwise drag force acting on the array increases with ϕ for the submerged and emerged cases.