This article presents a new turbulence closure based on the k-ω SST model for predicting turbulent flows of Herschel-Bulkley fluids, including Bingham and power-law fluids. The model has been calibrated with direct numerical simulations (DNS) data for fully-developed pipe flow of shear-thinning and viscoplastic fluids. The new model shows good agreement in the mean velocity, average viscosity, mean shear stress budget and friction factor. The latter compares well also against correlations from the literature for a wide range of Reynolds numbers. With the new model, improvements are also observed in the iterative convergence, which is often difficult for calculations with yield-stress fluids. Additionally, three eddy-viscosity models for Newtonian fluids, namely the k-ω SST, k-√kL and Spalart-Allmaras model, have been tested on turbulent Herschel-Bulkley flows. Results show that (i) the new model produces the best prediction; (ii) the standard SST model may be considered for simulations of weakly shear-thinning/viscoplastic fluids at high Reynolds numbers; (iii) the k-√kL and the Spalart-Allmaras models appear to be unsuitable for turbulent Herschel-Bulkley flows. The new model is simple and appealing for engineering applications concerned with turbulent wall-bounded flows and is presented in a formulation that can be easily adapted to other generalised Newtonian fluids.