Phenomenological friction equation for turbulent flow of Bingham fluids

Most discussions in the literature on the friction coefficient of turbulent flows of fluids with complex rheology are empirical. As a rule, theoretical frameworks are not available even for some relatively simple constitutive models. In the present work, a formula is proposed for the evaluation of the friction coefficient of turbulent flows of Bingham fluids. The developments combine a fresh analysis for the description of the microscales of Kolmogorov and the phenomenological turbulence model of Gioia and Chakraborty [G. Gioia and P. Chakraborty, Phys. Rev. Lett. 96, 044502 (2006)]. The resulting Blasius-type friction equation is tested against some experimental data and shows good agreement over a significant range of Hedstrom and Reynolds numbers. Comments on pressure measurements in yielding fluids are made. The limits of the proposed model are also discussed.

Figure 1

Generalized Blasius equation (16), as compared with the results of the Darby-Melson formula, the laminar friction equation, and the experimental data with (a) $\text{He}=2\times {10}^6$ from [16], (b) $\text{He}=7\times {10}^5$ from [17], and (c) $\text{He}=5\times {10}^5$ from [18].

Figure 2

Comparison of experimental results from [16] and the final friction equation (16), the friction equation without subtracting the yield stress effect (14), and the Newtonian Blasius equation.

Figure 3

Plot of $H\equiv {\delta }_{\nu }^{\left(n\right)}/{\ell }_d^{\left(n\right)}$ versus $\text{Re}$, for some fixed values of $\text{He}$. Notice that $H<1$ for $\text{Re}\sim 3.5\times {10}^5$, which is compatible with Blasius's range of validity. Notice also that for $\text{He}=7.25\times {10}^6$, the ratio $H$ is never greater than 1.

Figure 4

Level curve $H\equiv 1$. We expect the domain of validity of Eq. (16) to be in a region similar to the region where $H<1$.