Onset of turbulence in particle-laden pipe flows

We propose a scaling law for the onset of turbulence in pipe flow of neutrally buoyant suspensions. This scaling law, based on a large set of experimental data, relates the amplitude of the particle-induced perturbations $\epsilon$ to the critical suspension Reynolds number ${\text{Re}}_{s,c}$. Here $\epsilon$ is a function of the particle-to-pipe diameter ratio and the volume fraction of the suspended particles, $\epsilon ={(d/D)}^{1/2}{\varphi }^{1/6}$. ${\text{Re}}_{s,c}$ is found to scale as ${\epsilon }^{-1}$. Furthermore, the perturbation amplitude allows a distinction between classical, intermediate, and particle-induced transitions.

Figure 1

Friction factor $f$ as a function of the suspension Reynolds number ${\text{Re}}_s$. The single-phase case is shown for reference. For the particle-laden cases the concentration $\varphi$ is fixed at 0.05 to highlight the diameter-ratio ($d/D$) effect.

Figure 2

Regime map $\varphi$ vs $d/D$, where each marker represents one complete transition curve. The transition curves indicated with the triangles have a local minimum, whereas the transition curves with a monotonically decreasing friction factor are represented by squares. The marker color represents the critical suspension Reynolds number ${\text{Re}}_{s,c}$.

Figure 3

(a) ${\text{Re}}_{s,c}$ as a function of $d/D$ for three representative volume fractions and (b) ${\text{Re}}_{s,c}$ as function of $\varphi$ for three representative diameter ratios. The dashed curves are based on a regression analysis, where the slope is rounded to the nearest common fraction.

Figure 4

Critical suspension Reynolds number ${\text{Re}}_{s,c}$ as a function of particle perturbation amplitude $\epsilon$ for all experiments. All ${\text{Re}}_{s,c}$ collapse on one single curve with a slope of $-1$. The marker on the ordinate axis represents the critical Reynolds number for a single-phase flow ${\text{Re}}_{c,0}$.