Universal Scaling Laws for Dense Particle Suspensions in Turbulent Wall-Bounded Flows

The macroscopic behavior of dense suspensions of neutrally buoyant spheres in turbulent plane channel flow is examined. We show that particles larger than the smallest turbulence scales cause the suspension to deviate from the continuum limit in which its dynamics is well described by an effective suspension viscosity. This deviation is caused by the formation of a particle layer close to the wall with significant slip velocity. By assuming two distinct transport mechanisms in the near-wall layer and the turbulence in the bulk, we define an effective wall location such that the flow in the bulk can still be accurately described by an effective suspension viscosity. We thus propose scaling laws for the mean velocity profile of the suspension flow, together with a master equation able to predict the increase in drag as a function of the particle size and volume fraction.

Figure 1

Mean streamwise flow velocity, $u/u_{\tau }$, versus the wall-normal distance in inner scaling $y/{\delta }_v^e$. Vertical dashed lines depict a wall-normal distance of 1 particle diameter ($y=D_p$) for cases D10 (closest to $y=0$) and D20; see Table 1. Maximum statistical error within 95% confidence interval is $\pm 0.9\%$. The inset shows the same velocity profile but with the wall-normal distance scaled with ${\delta }_v$.

Figure 2

(a) Mean particle number density $n$ divided by its bulk value $N$ versus $y/h$ in the main panel and $y/D_p$ in the inset (maximum statistical error within 95% confidence interval is $\pm 0.8\%$). (b) Mean streamwise particle and fluid velocity. The inset shows the fluid velocity, normalized with $u_{pw}$ (definition in the text) versus $y/R_p$. The data from [11] pertain to the case of $\mathrm{\Phi }=20\%$. Vertical dashed lines depict a wall-normal distance of 1 particle diameter ($y=D_p$) for cases D10 (closest to $y=0$) and D20; see Table 1.

Figure 3

Profiles of mean streamwise fluid velocity $u/u_{\tau }^{*}$ versus the wall-normal coordinate $(y-{\delta }_{pw})/{\delta }_v^{e*}$. The inset shows the defect law, $(U_c-u)/u_{\tau }^{*}$, versus the distance to the wall in outer units $(y-{\delta }_{pw})/(h-{\delta }_{pw})$ (definitions in the text). Maximum statistical error is the same as in Fig. 1.

Figure 4

Relative difference to the theoretical prediction of friction Reynolds number (the shaded area corresponds to a difference of $\pm 4\%$). Filled symbols correspond to values that were not corrected for the presence of the particle-wall layer (i.e. ${\delta }_{pw}=0$). The maximum statistical error in the computation of the overall drag (with 95% confidence interval) from the DNS is below 1%. The corresponding (shifted) error bar is also shown on the left-hand side of the plot legend.