On the origin of turbulent secondary flows in non-circular ducts

The cause of turbulent secondary flows in straight ducts of noncircular cross-section is examined from a theoretical standpoint. It is proven that the necessary and sufficient condition for the development of such secondary flows is for the axial mean velocity to give rise to some non-zero difference in the transverse normal Reynolds stresses. By making use of this simple condition, it is demonstrated that turbulent secondary flows in noncircular ducts arise from the third-order diffusion correlation, the pressure gradient-velocity correlation, and the dissipation rate correlation. However, in all of the currently popular turbulent closure models the dissipation rate correlation does not lead to the generation of secondary flows. A modification in the closure model for the dissipation rate correlation is proposed that remedies this deficiency.