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This paper presents a series of numerical simulations of non-Newtonian fluids in high Reynolds number flows in circular pipes. The fluids studied in the computations have shear-thinning and yield stress properties. Turbulence is described using the Reynolds-Averaged Navier–Stokes (RANS) equations with the Boussinesq eddy viscosity hypothesis. The evaluation of standard, two-equation models led to some observations regarding the order of magnitude as well as probabilistic information about the rate of strain. We argue that an accurate estimate of the rate of strain tensor is essential in capturing important flow features. It is first recognised that an apparent viscosity comprises two flow dependant components: one originating from rheology and the other from the turbulence model. To establish the relative significance of the terms involved, an order of magnitude analysis has been performed. The main observation supporting further discussion is that in high Reynolds number regimes the magnitudes of fluctuating rates of strain and fluctuating vorticity dominate the magnitudes of their respective averages. Since these quantities are included in the rheological law, the values of viscosity obtained from the fluctuating and mean velocity fields are different. Validation against Direct Numerical Simulation data shows at least an order of magnitude discrepancy in some regions of the flow. Moreover, the predictions of the probabilistic analysis show a favourable agreement with statistics computed from DNS data. A variety of experimental, as well as computational data has been collected. Data come from the latest experiments by Escudier et al. [1], DNS from Rudman et al. [2] and zeroth-order turbulence models of Pinho [3]. The fluid rheologies are described by standard power-law and Herschel–Bulkley models which make them suitable for steady state calculations of shear flows. Suitable regularisations are utilised to secure numerical stability. Two new models have been proposed taking as a basis a κ-[variant_greek_epsilon] κ-ω roaches. They both incorporate modification of of strain calculation according to the points outlined above. The preliminary runs show improvements in results includign accurate mass flow rate prediction. [ABSTRACT FROM AUTHOR]