Universal friction law at granular solid-gas transition explains scaling of sediment transport load with excess fluid shear stress

A key interest in geomorphology is to predict how the shear stress $\tau$ exerted by a turbulent flow of air or liquid onto an erodible sediment bed affects the transport load $M\tilde{g}$ (i.e., the submerged weight of transported nonsuspended sediment per unit area) and its average velocity when exceeding the sediment transport threshold ${\tau }_t$. Most transport rate predictions in the literature are based on the scaling $M\tilde{g}\propto \tau -{\tau }_t$, the physical origin of which, however, has remained controversial. Here we test the universality and study the origin of this scaling law using particle-scale simulations of nonsuspended sediment transport driven by a large range of Newtonian fluids. We find that the scaling coefficient is a universal approximate constant and can be understood as an inverse granular friction coefficient (i.e., the ratio between granular shear stress and normal-bed pressure) evaluated at the base of the transport layer (i.e., the effective elevation of energetic particle-bed rebounds). Usually, the granular flow at this base is gaslike and rapidly turns into the solidlike granular bed underneath: a liquidlike regime does not necessarily exist, which is accentuated by a nonlocal granular flow rheology in both the transport layer and bed. Hence, this transition fundamentally differs from the solid-liquid transition (i.e., yielding) in dense granular flows even though both transitions are described by a friction law. Combining this result with recent insights into the nature of ${\tau }_t$, we conclude that the transport load scaling is a signature of a steady rebound state and unrelated to entrainment of bed sediment.

Figure 1

Visualization of Bagnold interface properties. Vertical profiles of (a) the fraction $Q_{>z}/Q$ of sediment transport occurring above elevation $z$, (b) the friction coefficient $\mu$, and (c) the ratio $-P_{zx}/\tau$ between the particle shear stress $-P_{zx}$ and fluid shear stress $\tau$. The solid lines correspond to data obtained from direct sediment transport simulations (see Sec. 2) for two representative cases: turbulent bedload (turquoise) and saltation transport (brown). The black, dashed lines mark the Bagnold interface $z=z_r$.

Figure 2

Exemplary vertical profiles of quantities associated with the (cross-correlation) fluctuation energy balance. (a, b) Vertical profiles relative to the rebound location $z_r$ of $P_{zz}\dot{\gamma }/\left({\rho }_p\tilde{g}\sqrt{\tilde{g}d}\right)$. (c, d) Vertical profiles of $-P_{zx}\dot{\gamma }/{\mathrm{\Gamma }}_{ii}^{\mathrm{coll}}$. Symbols correspond to viscous bedload transport [$s=2.65$, $\mathrm{Ga}=0.1$, $\mathrm{\Theta }/{\mathrm{\Theta }}_t=(1.9,6.2)$], turbulent bedload transport [$s=2.65$, $\mathrm{Ga}=20$, $\mathrm{\Theta }/{\mathrm{\Theta }}_t=(2.0,7.2)$], and turbulent saltation transport [$s=2000$, $\mathrm{Ga}=5$, $\mathrm{\Theta }/{\mathrm{\Theta }}_t=(2.3,49)$], where the data with smaller values of the rescaled Shields number $\mathrm{\Theta }/{\mathrm{\Theta }}_t$ are shown in (a) and (c) and those with larger values of $\mathrm{\Theta }/{\mathrm{\Theta }}_t$ in (b) and (d). For the bedload transport conditions, the restitution coefficient has been varied to mimic the minimal ($e=0.9$) and nearly maximal ($e=0.01$) possible effect of lubrication forces.

Figure 3

Test of Bagnold interface properties. Test of (a) Property 1, (b) Property 2, and (c) Property 3 of the Bagnold interface with data from our direct transport simulations for various combinations of the particle-fluid-density ratio $s$, Galileo number $\mathrm{Ga}$, Shields number $\mathrm{\Theta }$, and thus impact number $\sqrt{s}\mathrm{Ga}$. For conditions with $s\le 2.65$ (corresponding to bedload transport), the restitution coefficient has been varied to mimic the minimal ($e=0.9$) and nearly maximal ($e=0.01$) possible effect of lubrication forces. The vertical bars indicate the range of values the quantities cover with varying $\mathrm{\Theta }$ above about $2{\mathrm{\Theta }}_t$. This lower limit is imposed to separate the random variability due to bad statistics when $\mathrm{\Theta }$ is close to ${\mathrm{\Theta }}_t$ [e.g., see Fig. 4] from the actual variability. Indications that the Bagnold interface properties are obeyed: (a) the sediment transport rate ratio $Q_r/Q$ is near unity, (b) the bed friction coefficient ${\mu }_b$ is approximately constant with $\mathrm{\Theta }$ (relatively small vertical bars), and (c) the quantity $-(\tau -{\tau }_t)/P_{zx}\left(z_r\right)$ is near unity.

Figure 4

Exemplary trends of quantities associated with Bagnold interface properties. (a) Sediment transport rate ratio $Q_r/Q$, (b) bed friction coefficient ${\mu }_b$, and (c) $-(\tau -{\tau }_t)/P_{zx}\left(z_r\right)$ versus rescaled Shields number $\mathrm{\Theta }/{\mathrm{\Theta }}_t$. (d) Rescaled surface fluid shear stress ${\tau }_f\left(z_r\right)/{\tau }_t$ versus Shields number $\mathrm{\Theta }$. The interface $z=z_r$ is calculated by Eq. (6) if not otherwise stated in the legends. Symbols correspond to viscous bedload transport ($s=2.65$, $\mathrm{Ga}=0.1$), turbulent bedload transport ($s=2.65$, $\mathrm{Ga}=20$) and turbulent saltation transport ($s=2000$, $\mathrm{Ga}=5$). For the bedload transport conditions, the restitution coefficient has been varied to mimic the minimal ($e=0.9$) and nearly maximal ($e=0.01$) possible effect of lubrication forces.

Figure 5

Failure of dense rheology interpretation. (a) Particle volume fraction $\varphi \left(z_r\right)$ at the Bagnold interface versus Shields number $\mathrm{\Theta }$. (b, c) Friction coefficient $\mu$ versus (b) particle volume fraction and (c) viscoinertial number $K$. Symbols in (a) correspond to data from our direct transport simulations for various combinations of the particle-fluid-density ratio $s$, Galileo number $\mathrm{Ga}$, and $\mathrm{\Theta }$. For symbol legend, see Fig. 3. For conditions with $s\le 2.65$ (corresponding to bedload transport), the restitution coefficient has been varied to mimic the minimal ($e=0.9$) and nearly maximal ($e=0.01$) possible effect of lubrication forces. The turquoise and brown lines in (b) and (c) correspond to the conditions $(s,\mathrm{Ga},e)=(2.65,20,0.9)$ and $(s,\mathrm{Ga},e)=(2000,5,0.9)$, respectively, which are representative for turbulent bedload and saltation transport, respectively.

Figure 6

Dense rheology interpretation for very viscous bedload transport. (a) Viscous number $J\left(z_r\right)$ at the Bagnold interface versus rescaled Shields number $\mathrm{\Theta }/{\mathrm{\Theta }}_t$. Symbols correspond to data from our direct transport simulations for those combinations of the particle-fluid-density ratio $s$, Galileo number $\mathrm{Ga}$, and Shields number $\mathrm{\Theta }$ that obey $\sqrt{s}\mathrm{Ga}\le 1$. The two conditions $(s,\mathrm{Ga},e)=(2.65,20,0.9)$ (turbulent bedload transport, turquoise circles) and $(s,\mathrm{Ga},e)=(2000,5,0.9)$ (turbulent saltation transport, brown triangles) from Figs. 5 and 5 are also shown for comparison. (b, c) Effective friction coefficient $\tau /P_{zz}$ versus $J$ for the case $(s,\mathrm{Ga},e)=(2.65,0.5,0.01)$ and several $\mathrm{\Theta }/{\mathrm{\Theta }}_t$ in (b) log-linear and (c) log-log scale.

Figure 7

Sketch of the trajectory of a particle hopping along a granular bed. Driven by the flow, a transported particle (blue) hops along the solidlike or liquidlike granular bed (yellow particles). Instants of particle contacts are colored deep blue, and the ones for which the center of mass of the transported particle is above the Bagnold interface ($z>z_r$) are numbered consecutively (for illustrating the mathematical derivation in the Appendix).

Figure 8

Approximate equality of friction coefficients. Friction ratios (a) ${\mu }_b^t/{\mu }_b$ and (b) ${\mu }_b^c/{\mu }_b$ and pressure ratio (c) $[P_{zz}^t/P_{zz}]\left(z_r\right)$ versus impact number $\sqrt{s}\mathrm{Ga}$. Symbols correspond to data from our direct transport simulations for various combinations of the particle-fluid-density ratio $s$, Galileo number $\mathrm{Ga}$, and Shields number $\mathrm{\Theta }$. For symbol legend, see Fig. 3. For conditions with $s\le 2.65$ (corresponding to bedload transport), the restitution coefficient has been varied to mimic the minimal ($e=0.9$) and nearly maximal ($e=0.01$) possible effect of lubrication forces. The vertical bars indicate the range of values the quantities cover with varying $\mathrm{\Theta }$ above about $2{\mathrm{\Theta }}_t$. This lower limit is imposed to separate the random variability due to bad statistics when $\mathrm{\Theta }$ is close to ${\mathrm{\Theta }}_t$ [e.g., see Fig. 4] from the actual variability. Inset of (a): friction ratio ${\mu }_b^t/{\mu }_b$ versus rescaled Shields number $\mathrm{\Theta }/{\mathrm{\Theta }}_t$ for very viscous bedload transport ($s=2.65$, $\mathrm{Ga}=0.1$, $e=0.9$).