Fully Turbulent Mean Velocity Profile for Purely Viscous non-Newtonian Fluids

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The characteristic near wall behavior of turbulent flow of purely-viscous non-Newtonian fluids is discussed for both power-law (P.-L.) and Herschel-Bulkley (H.-B.) rheological models. A proper scaling is presented for H.-B. fluids to establish an analogy with power-law fluids with same flow index. To provide reference data for turbulent flow of non-Newtonian fluids, DNS simulations of power-law fluids are conducted in a rectangular channel for a large range of power-law indices ($n$ = 0.5, 0.69, 0.75, 0.9, 1, 1.2). The DNS data show that the mean velocity profile in the viscous and logarithmic layers follow expressions of the form $u^{+}=y^{+}$ and $u^{+}=2.5\,log(y^{+})+B_{n}$ respectively, where $B$ shows a logarithmic dependency on the flow index.Comparison with some experimental data shows the above formulation to be valid for Reynolds numbers (based on shear velocity) as high as 1000.