Fully developed turbulent flow in a square duct is numerically predicted with two nonlinear low-Reynolds-number ${kappa}-{varepsilon}$ models. Typical predicted quantities such as axial and secondary velocities, turbulent kinetic energy and Reynolds stresses are compared in detail with each other. It is found that the nonlinear low-Reynolds-number ${kappa}-{varepsilon}$ model adopted in a commercial code is unable to predict accurately duct flows involving turbulence-driven secondary motion with the prediction level of secondary flows one order less than that of the experiment.