An estimation of turbulent secondary flow resistance in rectangular ducts from a flow model

Abstract

Incompressible fluid flow in smooth rectangular ducts at working streamwise velocities is turbulent because of resistance of the four side-walls. A second feature of such flows is generation of secondary currents, though weak, in transverse cross sections. These secondary currents cause additional flow resistance, resulting in pressure drop in the direction of the primary flow. A number of experimental studies have been reported on the turbulence structure and consequent geometrical structures of the flow. In particular, the two diagonals and the pair of bisectors of the side walls divide a cross section into eight cells, in each of which vortical patches of motion take place. In this paper, it is shown that the vortical motion in a cell is kinematically analogous to the torsion problem of a prismatic isotropic elastic beam. Based on experimental results, the patch vortex in a cell is modeled to have elliptic shape with the major axis thrust toward a corner of the duct, giving a mathematical model of the flow field. Using the expressions for the transverse velocity components in the total momentum equation, with 1/pth power law where p ≈ 7 for the streamwise velocity, an equation is obtained between the side-wall resistance due to the secondary flow and the vorticity in each cell of division of the duct. Two particular cases are considered in numerical detail when the duct is square and when the height of the duct is one-half of the base length. For experimental validation of the side-wall resiatance formulae, additional experimental research is needed.