Laminar, Turbulent, and Inertial Shear-Thickening Regimes in Channel Flow of Neutrally Buoyant Particle Suspensions

The aim of this Letter is to characterize the flow regimes of suspensions of finite-size rigid particles in a viscous fluid at finite inertia. We explore the system behavior as a function of the particle volume fraction and the Reynolds number (the ratio of flow and particle inertia to viscous forces). Unlike single-phase flows, where a clear distinction exists between the laminar and the turbulent states, three different regimes can be identified in the presence of a particulate phase, with smooth transitions between them. At low volume fractions, the flow becomes turbulent when increasing the Reynolds number, transitioning from the laminar regime dominated by viscous forces to the turbulent regime characterized by enhanced momentum transport by turbulent eddies. At larger volume fractions, we identify a new regime characterized by an even larger increase of the wall friction. The wall friction increases with the Reynolds number (inertial effects) while the turbulent transport is weakly affected, as in a state of intense inertial shear thickening. This state may prevent the transition to a fully turbulent regime at arbitrary high speed of the flow.

Figure 1

Time series of the streamwise velocity fluctuation ${\tilde{v}}_{\mathrm{rms}}$ for (a) $\mathrm{\Phi }=0.001$ and (b) $\mathrm{\Phi }=0.3$.

Figure 2

(a) Contour map of the streamwise velocity fluctuations $v_{\mathrm{rms}}$ and (b) relative viscosity ${\nu }_r$ in the ($\mathrm{Re},\mathrm{\Phi }$) plane. In total, 96 points have been simulated in the ($\mathrm{Re},\mathrm{\Phi }$) plane.

Figure 3

Budget of the momentum transport for (a) $\mathrm{Re}=500$ and $\mathrm{\Phi }=0.05$, (b) $\mathrm{Re}=2500$ and $\mathrm{\Phi }=0.3$, and (c) $\mathrm{Re}=5000$ and $\mathrm{\Phi }=0.1$. Red triangle represents Reynolds stress; blue square, viscous stress; and green circle, particle stress.

Figure 4

(a) Phase diagram for particle laden channel flow in the $\mathrm{Re}-\mathrm{\Phi }$ plane (blue, red, and green colors indicate laminarlike, turbulentlike, and inertial shear-thickening regime). The solid black lines show the boundary of the regions where each term in the stress budget is over 50% of the total stress. The dashed purple lines are isolevels of Bagnold number. (b) Contributions to the total wall friction from the viscous stresses, Reynolds stresses, and particle stresses versus the Reynolds number Re for $\mathrm{\Phi }=0.025$ and $\mathrm{\Phi }=0.3$.