# Modelling Intense Combined Load Transport in Open Channel

## Abstract

**:**

## 1. Introduction

_{0}is the bed shear stress, ρ

_{f}is the fluid density, g is the gravitational acceleration and R

_{h0}is the hydraulic radius of the bed associated area of a flow cross section [5]. An ability of flow to transport a solid particle is evaluated using the bed Shields parameter θ

_{0}, which the ratio of the shear force exerted by fluid on the particle and the submerged weight of the particle, ${\mathsf{\theta}}_{0}={\mathsf{\tau}}_{0}/\left({\mathsf{\rho}}_{s}-{\mathsf{\rho}}_{f}\right)gd$, where ρ

_{s}is the solids density and d is the particle size. In pressurized flows, transport of solids is typically very intense, with the maximum volumetric flow rate of solids exceeding 25 per cent of the total flow rate of mixture [6]. For laboratory pipes, various measuring techniques to sense distributions in a pipe cross section have been available, for instance, a radiometric profiler for a concentration distribution [5,6,7]. Results of pipe experiments were used primarily to evaluate the friction condition at the top of the erodible bed at high bed shear and resulted in relationships between the bed friction coefficient and the bed roughness modified to include the interaction of transported sediment with the bed surface [5,8,9]. Furthermore, the pipe experiments, which typically cover a broad range of bed shear conditions, were exploited to define different modes of sediment transport. The widely used approximate criterion, based on a pressurized-pipe experiment, requires that the suspension ratio, i.e., the ratio of the bed shear velocity, ${u}_{*0}=\sqrt{{\mathsf{\tau}}_{0}/{\mathsf{\rho}}_{f}}$, and the terminal settling velocity of a particle, exceeds 1 (or 1.25) to enable transport of particles in the turbulent suspension mode [8]. For lower values of the suspension ratio, the non-suspension mode (basically contact-load transport) dominates.

## 2. Materials and Methods

#### 2.1. Combined-Load Model

#### 2.1.1. Modelled Conditions

- The stationary bed (granular deposit with the plane surface expressed as the 0-boundary);
- the permanent-contact (dense sliding) layer (its top is the d-boundary);
- the combined-load (collision and turbulent suspension) layer (its top is the c-boundary);
- the suspended-load (turbulent suspension) layer (its top is the water surface, the w-boundary).

#### 2.1.2. Model Assumptions

_{0}, here and below an index indicates the position at which the quantity is evaluated) being larger than the concentration at the top of the layer c

_{d}. This lower concentration at the top is responsible for a predominantly collisional contact among particles at this interface. Hence, the support of particles is assumed to be purely collisional at the d-boundary and this boundary is the lowest elevation at which the collisional contact applies.

#### 2.1.3. Model Principles and Applied Theoretical Concepts

_{f}).

#### 2.1.4. Model Features

_{d}(y is the distance from the top of the bed in the direction perpendicular to the bed). Above the d-boundary, the applied assumptions (including experiment-based ones) enable to abandon the constitutive relations and to simulate the flow exclusively using a combination of momentum balance and particle-fluid interactions.

_{f}and dynamic viscosity μ

_{f}) and sediment (density ρ

_{s}, equivalent diameter d and terminal settling velocity w) and on flow (the depth H and the longitudinal slope ω). It employs constants related to constitutive relations (bed concentration c

_{0}, concentration at d-boundary c

_{d}, effective coefficient of restitution of particles in dry conditions ε, particulate friction coefficient at the bed β

_{0}, constant for solids stress ratio at the d-boundary C

_{bd}) and to turbulent-diffusivity relations (von Karman constant κ, constant for initial mixing-length position C

_{ini}, particle diffusivity constant C

_{η}). The model predicts elevations of the d-boundary, c-boundary and distributions of concentration and velocity as major outputs.

- Inputs: Solid/liquid properties: d, ρ
_{s}, w, ρ_{f}, μ_{f} - Flow: H, ω
- Constants: c
_{0}, c_{d}, ε, β_{0}, κ, C_{bd}, C_{ini}, C_{η} - Major outputs: Positions of interfaces between layers: y
_{d}, y_{c} - Distributions of velocity and solids concentration in layers: u(y), c(y)

#### 2.1.5. Model Equations and Computational Procedure

_{s}, ρ

_{f}and the specific gravity S = ρ

_{s}/ρ

_{f}is summarized in Appendix A. The notation of dimensionless quantities and corresponding dimensional quantities is the same for sake of simplicity.

_{d}(subject to iteration in the procedure) is introduced. The coefficient e differs from ε by the additional damping effect of lubrication forces on collisions expressed through the fluid dynamic viscosity, μ

_{f}, and the granular temperature, T (a measure of local particle velocity fluctuations due to intergranular collisions) [18]. In the procedure, the introduction of e

_{d}enables to express T

_{d}at the d-boundary as in [19]

_{cr}is the critical concentration and c

_{f}is the freeze concentration (we assume c

_{d}= c

_{f}).

_{d}can be isolated from the constitutive relation expressing the balance of fluctuation energy of colliding particles for the assumption of negligible diffusion flux in the energy balance as in [19]

_{d}= 1. For calculating the solids shear stress at the d-boundary using the stress ratio β

_{d}, it appeared necessary to expand the relation with the empirical constant C

_{βd}which in a simplistic way compensated for effects not included in the constitutive relation. This compensation was required to match a predicted velocity gradient with an experimentally determined velocity gradient across the combined-load layer which was considered constant across the entire layer. The effects not covered by the used constitutive relation may include the effect of the diffusion flux (neglected in the balance above) and/or the effect of inhomogeneity (non-uniform distribution of sediment and only partial collisional suspension above the interface) on the solids shear stress. Thus

_{s}, is also related to local c and T in a constitutive relation for the shear stress which relates τ

_{s}to the local solids shear rate γ

_{s}(the gradient of longitudinal velocity u) [18]. This information enables to express the shear rate at the d-boundary

_{f}(f-index for fluid) is related to the local fluid shear rate through the local fluid effective viscosity. Note that the shear rates are assumed to be equal for liquid and solids at any elevation in the flow (γ = γ

_{f}= γ

_{s}). Following [29], the local fluid effective viscosity at the d-boundary is split into two components

_{d},

_{0}= 0) to determine the velocity at the top of the layer

_{0}= 0) and so is the fluid shear stress (τ

_{f}

_{0}= 0). The solids shear stress is related to the solids normal stress through the yielding value given by the coefficient β

_{0}. The normal stress is obtained from the momentum balance

_{d}for which values of other quantities at the d-boundary are determined.

_{ini}is estimated as a multiple of the thickness of the d-layer, ${y}_{ini}={C}_{ini}\mathsf{\cdot}{y}_{d}$, and a value of C

_{ini}is calibrated by experiments. The liquid shear stress at the c-boundary

_{c}.

_{c}, is obtained from the turbulent diffusion balance (Schmidt–Rouse equation). Its use is justified by the assumption that all particles are suspended by turbulence at this elevation,

_{c}. The velocity at the top of the c-layer is determined simply from the assumption of the linear distribution of velocity across the c-layer

_{−1}, y

_{−1}being values at the previous elevation).

#### 2.2. Experimental Work

#### 2.2.1. Experimental Set-Up

#### 2.2.2. Measuring Techniques

#### 2.2.3. Tested Solids

#### 2.2.4. Experimental Flow Conditions

#### 2.2.5. Experimental Data Set

## 3. Results

#### 3.1. Experimental Results

_{0,flow}is obtained from the measured H and ω using Equation (17) and u

_{*0,flow}is based on θ

_{0,flow}.

#### 3.1.1. Concentration Profiles

_{*0,flow}/w (Test 1: u

_{*0,flow}/w = 1.57, see Table 2) where it spans a distance between the assumed top of a thin combined-load layer and some position very near the water surface, i.e., the interval 5 < y < 20 approximately (Figure 4). If the u

_{*0,flow}/w value decreases (Tests 2 to 5), then the region of full suspension narrows, its top departs from the water surface and leaves an increasingly thick particle-free zone below the water surface. At the lowest u

_{*0,flow}/w (equal to 0.86 in Test 5), the full-suspension region is marginal and can be neglected. Note also that the permanent-contact layer tends to become thinner if the applied bed shear stress (the bed Shields parameter) decreases. In general, this layer is considerably thinner than the other layers at all bed shear conditions covered by Tests 1–5.

#### 3.1.2. Velocity Profiles

#### 3.2. Model Predictions

_{0}= 0.65 are employed for the SUN25 sediment.

#### 3.2.1. Concentration Profiles

_{d}, y

_{c}) above the bed (y

_{0}= 0), from the chosen values of the local concentrations at the interfaces (c

_{0}, c

_{d}) and from the calculated c

_{c}at y

_{c}. The values of c

_{0}= c

_{cr}(c

_{cr}= 0.62) and c

_{d}= 0.45 were chosen for all test runs to be consistent with the measured values of c at the positions of the abrupt change in a concentration gradient as discussed above. At the d-boundary, c

_{d}was considered the freeze concentration (c

_{d}= c

_{f}) to satisfy the kinetic-theory based condition for dense limit (L

_{d}= 1). The c

_{f}is sensitive to the shape of particles and this justifies its chosen value which is slightly lower for our tested particles than for particles of simple shapes (e.g., 0.49 for cylindrical particles). In the suspended-load layer, a predicted shape of the concentration profile is sensitive to the particle diffusivity constant, C

_{η}= 2 was used for all tested flow conditions.

#### 3.2.2. Velocity Profiles

_{0}= 0) and it is calculated by a constitutive relation at the d-boundary (Equation (7)). At the c-boundary, γ

_{c}= γ

_{d}. The predicted elevations y

_{d}and y

_{c}determine u

_{d}and u

_{c}. In the suspended-load layer, a predicted shape of the velocity profile is logarithmic and affected by the local mixing length (κ = 0.41) and by the initial position determined empirically using the constant C

_{ini}, which is the same for all tested flow conditions (C

_{ini}= 1.65).

## 4. Discussion

_{bd}can be removed. Furthermore, modelling of interactions of suspended particles with turbulent flow in the sediment-rich region just above the top of the combined-load layer requires a refinement to reproduce more closely the characteristic shape of a concentration profile observed experimentally in this region.

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACVP | Acoustic Concentration and Velocity Profiler |

DPT | Differential Pressure Transmitter |

ERT | Electrical Resistivity Tomography |

UVP | Ultrasonic Velocity Profiler |

## Appendix A

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**Figure 1.**Layered structure of modelled flow including schematic concentration profile projected onto photograph of combined-load transport in tilting flume.

**Figure 2.**Experimental set-up in Water Engineering Laboratory: (

**a**) Lay-out of the recirculating system with tilting flume; (

**b**) Overall view of recirculating system [26].

**Figure 3.**Schematic layout of tilting flume and measuring equipment in Water Engineering Laboratory [26].

**Figure 4.**Profiles of velocity (left panels) and concentration (right panels) in Tests 1–5 for flow with transported SUN25 particles in tilting flume (all quantities are dimensionless). Legend: Black circle—stereoscopic measurement; green circle—Prandtl tube measurement; black square—laser-stripe measurement; red line—model prediction in permanent-contact layer; magenta line—prediction in combined-load layer; blue line—prediction in suspended-load layer; horizontal blue line—water surface.

**Table 1.**Properties of tested solids: Particle equivalent diameter d, specific gravity S, terminal settling velocity w, particle Reynolds number R.

SUN25 | |
---|---|

d [mm] | 2.8 |

S [-] | 1.28 |

w [mm/s] | 76.5 |

w [-] | 0.987 |

R [-] | 217 |

**Table 2.**Combined-load experiment in tilting flume: Experimental conditions for Tests 1 to 5 (longitudinal slope ω, flow depth H, flow Reynolds number Re, bed Shields parameter θ

_{0,flow}, suspension ratio u

_{*0,flow}/w; all quantities are dimensionless).

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | |
---|---|---|---|---|---|

ω [-] | 0.0257 | 0.0142 | 0.0154 | 0.0098 | 0.0070 |

H [-] | 20.4 | 25.5 | 20.0 | 25.1 | 22.7 |

Re [-] | 4.0 × 10^{4} | 6.1 × 10^{4} | 4.1 × 10^{4} | 5.1 × 10^{4} | 4.0 × 10^{4} |

θ_{0,flow} [-] | 1.88 | 1.30 | 1.10 | 0.88 | 0.57 |

u_{*0,flow}/w [-] | 1.57 | 1.31 | 1.20 | 1.08 | 0.86 |

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Matoušek, V.
Modelling Intense Combined Load Transport in Open Channel. *Water* **2022**, *14*, 572.
https://doi.org/10.3390/w14040572

**AMA Style**

Matoušek V.
Modelling Intense Combined Load Transport in Open Channel. *Water*. 2022; 14(4):572.
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**Chicago/Turabian Style**

Matoušek, Václav.
2022. "Modelling Intense Combined Load Transport in Open Channel" *Water* 14, no. 4: 572.
https://doi.org/10.3390/w14040572