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.
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Abbreviations
- C :
-
Volumetric concentration of particles
- \(C_D\) :
-
Drag coefficient
- d :
-
Particles diameter
- \(D_0\) :
-
Fractal, Hausdorff or box-counting dimension of the bedload flux
- \(d^+\) :
-
Non-dimensional diameter
- e :
-
Coefficient of restitution
- \(F^{col}\) :
-
Force due to inter-particles collisions
- \(F_i\) :
-
Feedback force from the particles to the flow
- Fr :
-
Froude number
- \(f_i\) :
-
External pressure gradient applied to the flow
- \(f_{i,\eta }^{p}\) :
-
Hydrodynamic drag force felt by a particle
- \(f(\alpha )\) :
-
Singularity spectrum
- g :
-
Gravity acceleration
- h :
-
Length scale of a domain
- k :
-
Shape parameter of the Gamma distribution function
- \(M(q,r\Delta t)\) :
-
q-th Order statistical moment of \(\mu _i\)
- \(N(r\Delta t)\) :
-
Number of time windows of size \(r\Delta t\) contained in the total time when transport occurs
- P :
-
Pressure field of the flow
- Re :
-
Reynolds number based on bulk velocity
- \(Re_p\) :
-
Particle Reynolds number based on the particle velocity relative to the flow
- \(Re_p^*\) :
-
Particle Reynolds number based on the friction velocity
- \(Re_\tau \) :
-
Friction Reynolds number
- r :
-
Integer for computing different sampling time scale of sediment transport
- \(r^+_{inactive}\) :
-
Dimensionless radius of the inactive particles
- S(t):
-
Cumulative amount of particles that have crossed the control plane until time t
- \(St_k\) :
-
Stokes number
- s(t):
-
Number of particles crossing the control plane in a given time step \(\Delta t\)
- \(T_{max}\) :
-
Total time of simulation
- \(T(r\Delta t)\) :
-
Total width of periods with no transport that are larger than \(r\Delta t\)
- t :
-
Time
- U :
-
Bulk velocity of the flow
- \(u_{f_i}\) :
-
Velocity of the fluid at the particle location (\(i=1,2,3\))
- \(u_j\) :
-
Instantaneous velocity components of the flow (\(j=1,2,3\))
- \(u_p\) :
-
Particles velocity in the streamwise direction
- \(u^+\) :
-
Dimensionless bulk velocity
- \(u^*\) :
-
Friction velocity
- \(v_{cr}\) :
-
Arbitrary critical velocity which delineate active motions
- \(v_p\) :
-
Particles velocity in the spanwise direction
- \(v_{p_i}\) :
-
Instantaneous particle velocity component (\(i=1,2,3\))
- \(w_p\) :
-
Particles velocity in the vertical direction
- \(w_{\eta }^{j}\) :
-
Linear geometric weight for the projection of the feedback forces from particles to flow
- \(x_i\) :
-
Cartesian coordinates (\(i=1,2,3\)). Also written as x, y, z
- \(x_L\) :
-
Streamwise dimension of the channel
- Y :
-
Young’s modulus
- \(y_L\) :
-
Spanwise dimension of the channel
- \(z_L\) :
-
Vertical dimension of the channel
- \(z^+\) :
-
Dimensionless vertical coordinate
- \(\alpha \) :
-
Hölder exponent
- \(\beta \) :
-
Scale parameter of the Gamma distribution function
- \(\Delta t\) :
-
Time step used by the flow solver
- \(\Delta t_p\) :
-
Time step used by the particles solver
- \(\Delta V_{\eta }\) :
-
Volume of a computational cell
- \(\delta \) :
-
Boundary layer thickness
- \(\zeta \) :
-
Scaling exponent function
- \(\theta \) :
-
Shields parameter
- \(\vartheta \) :
-
Poisson’s ratio
- \(\kappa \) :
-
von Kármán constant
- \(\lambda \) :
-
Distance traveled downstream by the particles
- \(\mu _f\) :
-
Coefficient of friction
- \(\mu _i\) :
-
Mass fraction of sediment that cross the reference plane between two successive plateaus
- \(\nu \) :
-
Kinematic viscosity of the fluid
- \(\rho _f\) :
-
Density of the fluid
- \(\rho _p\) :
-
Density of the particles
- \(\tau _0\) :
-
Bed shear stress
- \(\mathcal {X}\) :
-
Random variable distributing Gamma
- \(\forall _p\) :
-
Volume of a particle
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Acknowledgments
This work has been supported by Fondecyt Project 1130940, ONR-G NICOP N622909-11-1-7041, and Conicyt/Fondap Grant 15110017. C. González acknowledges the funding from the Ph.D. National Grant Conicyt-21120939. S. Bateman and J. Calantoni were supported under base funding to the Naval Research Laboratory from the Office of Naval Research.
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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 (\(F_t\)):
The normal force along the line of center between two colliding particles is computed as:
whereas the tangencial force is calculated as follows:
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:
The parameters in Eqs, 18–21 are defined as follows:
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.
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.
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].
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].
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González, C., Richter, D.H., Bolster, D. et al. Characterization of bedload intermittency near the threshold of motion using a Lagrangian sediment transport model. Environ Fluid Mech 17, 111–137 (2017). https://doi.org/10.1007/s10652-016-9476-x
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DOI: https://doi.org/10.1007/s10652-016-9476-x