Pipe flow with large particles and their impact on the transition to turbulence

The classical transition from laminar to turbulent flow is affected if solid particles are added. The transition behavior is a function of particle size $d$ and solid volume fraction $\varphi$ and the flow undergoes a smooth transition, as opposed to intermittent, if $\varphi$ exceeds a certain threshold. In this work we show that, for particle-laden pipe flows with large particle-to-pipe diameter ratios $d/D$, the $\varphi$ threshold for altering the transition is much lower than previously reported for smaller particles. Magnetic resonance velocimetry reveals that particles introduce turbulent-like fluid velocity fluctuations in laminar flow. Factors that might control the limits between “classical” and “smooth” transition in the state space spanned by $d/D$ and $\varphi$ are discussed based on scaling analyses.

Figure 1

(a) Friction factor $f$ as a function of ${\text{Re}}_e$ for all cases and pure water, as indicated in the legend. The inset shows relative increase in $f$ at constant Reynolds number based on carrier fluid viscosity (constant Re). The increase in the inset is the increase experienced by adding particles to a fluid at constant total flow rate. (b) The data in panel (a) is plotted as $f-16/{\text{Re}}_e$ to highlight deviations from laminar single-phase flow. Conservative estimates of the maximum errors are illustrated as gray regions [in panel (b), the error depends on ${\text{Re}}_e]$. Symbols and lines are the same in panels (a) and (b).

Figure 2

(a), (d), (g) Mean axial velocity at $\text{Re}=700$. (b), (e), (h) Variance of velocity at $\text{Re}=700$. (c), (f) Variance of velocity at $\text{Re}=350$, 700, and 2000. (i) Variance of velocity at $\text{Re}=350$ and 700. Top row $SS$, middle row $LS$, and bottom row $C$. A flattening of the mean profiles at the center of the pipe can be seen with increasing volume concentration, indicating a turbulent-like flow. Peak in the variance indicate aggregation of particles at given $r/R$. The particle size in each case is indicated with gray schematics.

Figure 3

State space showing classical and smooth transitions where experiments from Hogendoorn and Poelma [13], Agrawal et al. [14], and our own work are plotted. Filled markers indicate smooth transition and empty markers indicate classical transition. The red curve shows constant Bagnold number. The blue curve indicates that the limit between the two regions is derived from a scaling analysis based on particle-agitation to fluid-dissipation ratio. The black dashed line represents the size of the smallest Kolmogorov length scales at a Reynolds number of 1000.