Transitional and turbulent flows in rectangular ducts: budgets and projection in principal mean strain axes

We carry out Direct Numerical Simulation (DNS) of flows in closed rectangular ducts with several aspect ratios. The Navier–Stokes equations are discretised through a second-order finite difference scheme, with non-uniform grids in two directions. The duct cross-sectional area is maintained constant as well as the flow rate, which allows to investigate which is the appropriate length scale in the Reynolds number for a good scaling in the laminar and in the fully turbulent regimes. We find that the Reynolds number based on the half length of the short side leads to a critical Reynolds number which is independent on the aspect ratio ((Formula presented.)), for ducts with (Formula presented.). The mean and rms wall-normal velocity profiles are found to scale with the local value of the friction velocity. At high friction Reynolds numbers, the Reynolds number dependence is similar to that in turbulent plane channels, hence flows in rectangular ducts allow to investigate the Reynolds number dependency through a reduced number of simulations. At low Re, the profiles of the statistics differ from those in the two-dimensional channel due to the interaction of flow structures of different size. The projection of the velocity vector and of the Reynolds stress tensor along the eigenvectors of the strain-rate tensor yields reduced Reynolds stress anisotropy and simple turbulence kinetic energy budgets. We further show that the isotropic rate of dissipation is more difficult to model than the full dissipation rate, whose distribution does not largely differ from that of turbulence kinetic energy production. We expect that this information may be exploited for the development of advanced RANS models for complex flows.