Characterization of bedload intermittency near the threshold of motion using a Lagrangian sediment transport model

Abstract

At the smallest scales of sediment transport in rivers, the coherent structures of the turbulent boundary layer constitute the fundamental mechanisms of bedload transport, locally increasing the instantaneous hydrodynamic forces acting on sediment particles, and mobilizing them downstream. Near the critical threshold for initiating sediment motion, the interactions of the particles with these unsteady coherent structures and with other sediment grains, produce localized transport events with brief episodes of collective motion occurring due to the near-bed velocity fluctuations. Simulations of these flows pose a significant challenge for numerical models aimed at capturing the physical processes and complex non-linear interactions that generate highly intermittent and self-similar bedload transport fluxes. In this investigation we carry out direct numerical simulations of the flow in a rectangular flat-bed channel, at a Reynolds number equal to Re = 3632, coupled with the discrete element method to simulate the dynamics of spherical particles near the bed. We perform two-way coupled Lagrangian simulations of 48,510 sediment particles, with 4851 fixed particles to account for bed roughness. Our simulations consider a total of eight different values of the non-dimensional Shields parameter to study the evolution of transport statistics. From the trajectory and velocity of each sediment particle, we compute the changes in the probability distribution functions of velocities, bed activity, and jump lengths as the Shields number increases. For the lower shear stresses, the intermittency of the global bedload transport flux is described by computing the singularity or multifr actal spectrum of transport, which also characterizes the widespread range of transport event magnitudes. These findings can help to identify the mechanisms of sediment transport at the particle scale. The statistical analysis can also be used as an ingredient to develop larger, upscaled models for predicting mean transport rates, considering the variability of entrainment and deposition that characterizes the transport near the threshold of motion.

Appendices

Appendix 1

The discrete element method (DEM) used for the computation of the particles dynamics is the open source LIGGGHTS, available on http://www.liggghts.com. The equations solved by this algorithm have been previously explained by Chand et al. [16] and Schmeeckle [45], but following their work and for the clarity of the reader, we repeat them here.

For each particle, collisions with other grains are computed when the distance between their center is less or equal than the sum of their radii, i.e. they overlap, or at least, they are in contact. For two particles, the force product of their collision is divided into a normal force ($F_n$) and a tangencial force ($F_t$):

$$\begin{aligned} F^{col} = F_n + F_t \end{aligned}$$

(15)

The normal force along the line of center between two colliding particles is computed as:

$$\begin{aligned} F_n = K_n \delta _n - \gamma _n v_n \end{aligned}$$

(16)

whereas the tangencial force is calculated as follows:

$$\begin{aligned} F_t = K_t \delta _t - \gamma _t v_t \end{aligned}$$

(17)

where $K_n$ and $K_t$ are the stiffness coefficients, $\delta _n$ and $\delta _t$ are the overlap distance between two grains, $\gamma _n$ and $\gamma _t$ are the viscoelastic damping constants, $v_n$ and $v_t$ the relative velocity and the subscripts n and t correspond to the normal and tangencial components. If necessary, the tangential overlap is truncated in order to satisfy the condition $F_t \le \mu _f F_n$, where $\mu _f$ is the coefficient of friction. In turn, the previous variables are computed as follows:

$$\begin{aligned} K_n = \frac{4}{3} Y^* \sqrt{R^* \delta _n} \end{aligned}$$

(18)

$$\begin{aligned} \gamma _n = -2 \sqrt{\frac{5}{6}} B \sqrt{S_n m*} \ge 0 \end{aligned}$$

(19)

$$\begin{aligned} K_t = 8 G^* \sqrt{R^* \delta _n} \end{aligned}$$

(20)

$$\begin{aligned} \gamma _t = -2 \sqrt{\frac{5}{6}} B \sqrt{S_t m^*} \ge 0 \end{aligned}$$

(21)

The parameters in Eqs, 18–21 are defined as follows:

$$\begin{aligned} S_n = 2 Y^* \sqrt{R^* \delta _n} \end{aligned}$$

(22)

$$\begin{aligned} S_t = 8 G^* \sqrt{R^* \delta _n} \end{aligned}$$

(23)

$$\begin{aligned} B = \frac{\ln (e)}{\sqrt{\ln ^2(e)+\pi ^2}} \end{aligned}$$

(24)

$$\begin{aligned} \frac{1}{Y^*} = \frac{1-\vartheta ^2_1}{Y_1} + \frac{1-\vartheta ^2_2}{Y_2} \end{aligned}$$

(25)

$$\begin{aligned} \frac{1}{G^*} = \frac{2(2+\vartheta _1)}{Y_1} + \frac{2(2+\vartheta _2)}{Y_2} \end{aligned}$$

(26)

$$\begin{aligned} \frac{1}{R^*}=\frac{1}{R_1}+\frac{1}{R_2} \end{aligned}$$

(27)

$$\begin{aligned} \frac{1}{m^*}=\frac{1}{m_1}+\frac{1}{m_2} \end{aligned}$$

(28)

where e is the coefficient of restitution, $\vartheta$ is the Poisson’s ratio, Y the Young’s modulus, R the radius of a particle, m the mass of a particle and the subscripts 1 and 2 are the identifiers for two particles in contact.

Appendix 2

In this appendix we present a validation of the coupling between the fluid and solid phases. The numerical method for the fluid solver has already been validated before [see for example 43]. Also, the dispersed phase solver has been broadly used and validated in several previous investigations [16, 26, 45].

In order to carry out the validation, we perform additional simulations of a flow over a flat bed channel, different than those presented in this article. The setup reproduces the simulations carried out by Schmeeckle [45]. The configuration of the system is a rectangular channel of 0.12 m long, 0.06 m wide and 0.04 m deep. The mean velocities of the considered cases for comparisons are U = 0.2, 0.3, 0.4 and 0.5 m/s. Let us note that given these high velocities, here our algorithm works as a LES model with numerical dissipation instead of a DNS one (for this approach, see for example [33]). Following the reference case, our flow is driven by a pressure gradient, with periodic lateral boundary conditions, a free-slip rigid lid at the top, a solid wall at the bottom, and a two-way coupling approach.

Schmeeckle [45] used 115, 728 spherical particles with a diameter $d = 0.5$ mm, which corresponds to 7.57 cm$^3$ of solid material. We consider 28, 800 hemispherical particles stuck to the bed in order to have static particles at the bottom. In this way, we fulfill the condition used by [45] of particles with almost zero velocity at the bottom. Furthermore, we use 101, 328 spherical mobile particles. The diameter that we use is the same that the reference case. According to these conditions, the solid material of our simulations is 7.57 cm$^3$, coinciding with the value employed by [45]. More parameters of these simulations are shown in Table 4.

Table 4 Particles parameters of comparison cases

A comparison of the bulk velocity profiles of the flow in the downstream direction is shown in Fig. 12a. The concentration of particles in the vertical direction is also compared in Fig. 12b. These results show that our model captures the region of slower flow velocity in and near the sediment bed, the development of the logarithmic velocity profile, and the distribution of particles in the vertical direction, which is directly related to the fluid-particle interactions.

Fig. 12

Comparison of bulk velocity profiles of the flow (U) and volumetric concentration of particles (C). The horizontal axis corresponds to the non-dimensional vertical coordinate by the channel depth. Data extracted from [45] are plotted as black square, soft gray diamond, white circle and gray triangle and they represent U = 0.2, 0.3, 0.4 and 0.5 m/s respectively. Continuous lines are our simulated results. a In the original plot from [45], the bulk velocity was divided by the friction velocity. We extract it from the plot in order to directly compare the flow field. b Concentration is in volume of particles divided by volume of fluid

Additionally, we compare the velocity profile for a bulk velocity U = 0.2 m/s using a one-way and two-way coupling approach (shown in Fig. 13). For the one-way coupling case, the profile matches theoretical curves (see 29). From the perspective of the flow, this is similar to the boundary layer flow without sediments. Even though 29 can be used to calculate separately every region of a velocity profile without sediment particles [49, 54], the equation for the buffer layer can represent the whole inner layer accurately [48]. Here, the inner layer is composed by the viscous, buffer, and logarithmic layers. This equation is also known as the Spalding’s law of the wall [54] and we use it for the range $0 \le z^+ \le 350$. We use the outer-layer equation from 29 outside this range [13].

Fig. 13

(color online) Bulk velocity profile for U = 0.2 m/s. Red line corresponds to simulations with a one-way coupling approach, whereas the blue line corresponds to simulations with a two-way coupling approach. In black line the log-law is plotted and in dashed line the Spalding’s law is plotted [48, 49, 54]. The consideration of the two-way coupling approach yields to a completely different development of the bulk velocity profile

$$\begin{array}{llll} \frac{U}{u^*} = \left\{ \begin{array}{lll} z^+ & \text{if } \,z^+ \le 5 & {\rm Viscous sublayer} \\ z^+ - e^{-\kappa B} [e^{\kappa u^+} - 1 - \kappa u^+ -\frac{(\kappa u^+)^2}{2} - \frac{(\kappa u^+)^3}{6}] &{} \text{if }\, 5 z^+ \le 30 & {\rm Buffer layer} \\ \frac{1}{\kappa } \ln z^+ + B & \text{if }\, 30< z^+ \le 350 & \qquad {\rm Log. layer} \\ \frac{1}{\kappa } \ln z^+ + B + \frac{2 \Pi }{\kappa } \sin ^2(\frac{\pi }{2} \frac{z}{\delta }) & \text{if } \,350 < z^+ \le 1750 & {\rm Outer layer} \\ \end{array} \right. \end{array}$$

(29)

In 29, $u^*$ is the friction velocity, $z^+ = \frac{z u^*}{\nu }$, $\nu$ is the kinematic viscosity, $u^+ = \frac{U}{u^*}$, $\delta$ is the boundary layer thickness, $\kappa = 0.41$ is the von Kármán constant, $\Pi = \frac{\kappa A}{2}$ and A and B are experimental coefficients with $A = 2.5$ for flat plates and $B = 5.0$ [14]. In the outer-layer we use the approximation $\delta \approx 1750 \frac{\nu }{u^*}$ [54].