A low dimensional model to predict secondary mean flow in a square duct*

Turbulent flows through square and rectangular ducts have mean flowfields that include secondary (non-streamwise) mean velocity components, typically consisting of pairs of counterrotating streamwise vortices near each corner (Prandtl’s secondary flow of the second kind). While such secondary flows are typically small in magnitude, they are important for accurate characterization of turbulent statistics, yet are also typically difficult for turbulence models to accurately predict. This project aims to determine the simplest possible model that can correctly characterize the qualitative features of such secondary mean flows. To do this, we develop a nonlinear, physics-based Galerkin projection model, using basis functions obtained from analysis of the linearized Navier-Stokes equations about a laminar baseflow (using either stability or resolvent modes). We demonstrate that a model of this type obtained using only streamwise-constant modes is sufficient for predicting a physically realistic secondary mean, provided a white noise forcing term is included when simulating the model. We further discuss what these findings suggest about the physical origins of such secondary mean flows.