# Simulated Flow Velocity Structure in Meandering Channels: Stratification and Inertia Effects Caused by Suspended Sediment

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}and p

_{0}= reference mixture density and pressure, and $\overrightarrow{g}$ = acceleration of gravity), the momentum equation can be expressed as

## 2. Model Setup

#### 2.1. Hydrodynamic Model

_{h}and ν

_{z}= eddy viscosity in the horizontal (x and y) and vertical (z) directions. The density of the mixture is determined by

_{w}= water density; and ρ

_{s}= sediment density.

_{b}= bed elevation; and w

_{zb}is the vertical velocity at the bed. At the water surface,

_{a}= pressure at the water surface which is set to zero in the present simulations. The pressure gradient is obtained as follows

_{*}= friction velocity.

#### 2.2. Sediment Transport Model

_{f}= sediment settling velocity; and ε

_{h}and ε

_{z}= eddy diffusivity in the horizontal (x and y) and vertical (z) directions.

#### 2.3. Boundary Condition

_{b}and v

_{b}= near-bed components of horizontal velocity; and C

_{D}= drag coefficient to be calibrated. Besides, the near-bed net flux of sediment is

_{su}= sediment entrainment rate; and c

_{b}= near-bed volumetric concentration.

#### 2.4. Domain Discretization

_{0}. The top/bottom sublayer is combined with its adjacent sublayer, producing a layer with a thickness between h

_{0}and 2h

_{0}. Then, all the rest inner sublayers are combined to form an inner layer with its thickness being multiple of h

_{0}. Therefore, there will be three vertical layers throughout the whole calculation domain. All these three layers are variable according to water surface and bed changes during calculation process. Thickness thresholds z

_{min}and z

_{max}are imposed over top and bottom layers. When the thickness of the top/bottom layer is out of range (>z

_{max}or <z

_{min}), after the calculation of water surface/bed elevation at a certain time step, the layer interface will move a distance of h

_{0}inward or outward to ensure the thickness stays in the range. Let h

_{0}= (z

_{max}− z

_{min})/2, after adjustment, its thickness will be close to (z

_{max}− z

_{min})/2. Then, there is no need to readjust the thickness for small surface/bed changes. Every thickness adjustment is accompanied by local redistribution of physical quantities.

#### 2.5. Solution of Generalized Equation by SEM

_{f}is set only for sediment transport; σ = the Prandtl–Schmidt number for ϕ; and f = other terms. The top and bottom boundaries are chosen according to the specific equation to be solved. At each vertical layer i with thickness L

_{i}= z

_{i}

_{+1}− z

_{i}, generalized equation is conducted under local vertical coordinate ζ = 2 (z − z

_{i})/L

_{i}− 1. We set an approximation such that ϕ belongs to the finite dimensional space of degree N. Choosing the Legendre polynomials as the local basis functions, the approximate solution is defined by

_{k}and h

_{k}(ζ) = the coefficients and the Lagrange form of Legendre polynomials, respectively; interpolation point ζ

_{k}= Legendre–Gauss–Lobatto point with its weight ω

_{k}; and ${L}_{N}\left(\zeta \right)$ and ${L}_{N}^{\prime}\left(\zeta \right)$ = the Legendre polynomials and their derivatives. Generalized equation is multiplied by the test function and integrated over ζ (between [–1, 1]) and time step. We choose the local basis function as the test function. Then, the weak formulation after including boundary condition, can be derived as

_{j}. The depth integrated continuity and horizontal momentum equations are solved in the coupled way.

_{h}for simplification; σ = Schmidt number; and u

_{m}

_{*}

^{n}= the backtracked value of u

_{m}at time step n. Momentum equation for v has a similar form. Underlined terms are zero at the outlet for imposing water level condition for subcritical flows. In the momentum and sediment transport equations, advection terms are incorporated in the total derivatives. By backtracking characteristic lines accurately under Eulerian–Lagrangian context [35], the backtracked value at the foot of the characteristic lines at time step n can be approximated by interpolation. The final set of momentum equations can be solved by Conjugate Gradient. Equation (21) is semi-implicit with respect to v. Hence, v is firstly taken as an explicit quantity in the horizontal momentum for u, and then one more iteration is applied after the solution of the horizontal momentum equations. Once the horizontal velocities are known, water surface level and vertical velocity can be computed using the depth integrated continuity equation.

## 3. Model Verification and Results

#### 3.1. Uniform and Steady Open Channel Flows

^{3}/s and ~0.169 m, respectively. The velocity and sediment concentration were measured with a Pitot-static tube. Forty runs were performed with sediments of three different grain diameters and different concentrations. Runs 9, 20, 25, 31, and 40 are selected for comparisons with numerical results. Diameter of the sand grains were about 0.105, 0.210, and 0.420 mm, respectively, in runs {9 and 20}, {25 and 31}, and {40}. The Schmidt number for sediment vertical diffusivity is closely related to sediment diameter. Hence, we use the Schmidt numbers of 0.7, 1.2, and 2, respectively, for those runs.

^{3}/s. The flume was composed of a 10 m inflow section with a rigid bed, a 16 m test section with a perforated bed and a 4 m outflow section. Sediment was released at upstream section through the water surface. Nearly uniform sediment was used with settling velocity of w

_{f}= 0.007 m/s. The velocity was measured with a micropropellor and the sediment concentration was estimated by using the siphon method. The sediment profile was calculated at the inflow section as in an equilibrium state under given near-bed condition of q

_{su}= 1.19 × 10

^{−6}m/s. In the test section, q

_{su}was set to zero. According to Wang and Ribberink [37], fluctuations were observed in the measured profile of the sediment concentration.

#### 3.2. Steady Flows in Meandering Channels

^{3}/s.

#### 3.3. Effects of Suspended Sediment on Velocity Structure

_{b}. Runs BAL and N0L are those with and without the Boussinesq approximation for c

_{b}of 0.015; and runs BAH and N0H are also with and without applying the Boussinesq approximation but for a higher concentration, i.e., c

_{b}of 0.1. Run CL, which is the clear water case presented in Section 3.2, is also included for comparison.

_{b}= 0.015), i.e., the runs BAL and N0L, in comparison with the clear water (CL) case. As can be seen, the streamwise velocity near the inner bank is smaller in the BAL and N0L cases compared to the CL case, while it is larger near the outer bank for BAL and N0L runs. That means the secondary flow is stronger in the BAL and N0L runs with respect to the CL case. Besides, it is clear that the suspended sediment has negligible inertia effect on the flow structure in present condition, since the velocity and sediment profiles almost overlap in runs BAL and N0L.

_{n}

^{2}> and <f

_{s}f

_{n}> were adapted as characteristic parameters which represent the secondary flow strength and advecting flow momentum [39]. The sediment size is varied from 0.01 to 0.1 mm with depth averaged volumetric concentration from 0.01 to 0.2 at the inlet. The sediment buoyancy effect on turbulence is excluded here as the simple profile presented in Equation (12) for eddy viscosity was employed in this analysis. The results are summarized in Figure 9. The secondary flow strength decreases as the concentration increases. The finer sediment has a nearly uniform profile and only slightly affects the secondary flow strength. For 0.05- and 0.10-mm sediments, the non-uniformity of concentration profile at low concentration is much higher, vertical density gradient is large and the secondary flow strength declines efficiently to ¼ with concentration larger than 0.075. At higher concentration, the group hinder settling velocity is much smaller and the profile approaches to uniformity. Thus, the decrease retards and an exponentially decaying rate is found for the secondary flow here.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Equilibrium condition: estimated profiles of velocity (

**a**) and sediment concentration (

**b**) in comparison with the experimental data.

**Figure 2.**Non-equilibrium condition: calculated sediment concentration profiles in comparison with the experimental data.

**Figure 5.**Calculated streamwise velocity profiles in comparison with the measurements: (

**a**) Cross-section 90°; (

**b**) cross-section 180°. η is the position in the lateral direction from the central line towards the outer bank.

**Figure 6.**Calculated transverse velocity profiles in comparison with the measurements: (

**a**) Cross-section 90°; (

**b**) cross-section 180°. η is the position in the lateral direction from the central line towards the outer bank.

**Figure 7.**Velocity and sediment concentration profiles at the cross-section 90° for lower concentration case: (

**a**) Streamwise velocity; (

**b**) transversal velocity; (

**c**) sediment volumetric concentration. (velocity unit: m/s).

**Figure 8.**Velocity and sediment concentration profiles at the cross-section 180° for higher concentration case: (

**a**) Streamwise velocity; (

**b**) transversal velocity; (

**c**) sediment volumetric concentration. (velocity unit: m/s).

**Figure 9.**<f

_{n}

^{2}> and <f

_{s}f

_{n}> variation with suspended sediment at the center of cross-section 90°. The dot line represents the clear water (CL) case.

Run | BAL | N0L | CL | BAH | N0H |
---|---|---|---|---|---|

D (mm) | 0.05 | 0.05 | - | 0.05 | 0.05 |

Boussinesq approximation | Yes | No | - | Yes | No |

c_{b} | 0.015 | 0.015 | 0 | 0.10 | 0.10 |

ρ_{s}/ρ_{w} − 1 | 1.65 | 1.65 | - | 1.65 | 1.65 |

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**MDPI and ACS Style**

Yang, F.; Shao, X.; Fu, X.; Kazemi, E. Simulated Flow Velocity Structure in Meandering Channels: Stratification and Inertia Effects Caused by Suspended Sediment. *Water* **2019**, *11*, 1254.
https://doi.org/10.3390/w11061254

**AMA Style**

Yang F, Shao X, Fu X, Kazemi E. Simulated Flow Velocity Structure in Meandering Channels: Stratification and Inertia Effects Caused by Suspended Sediment. *Water*. 2019; 11(6):1254.
https://doi.org/10.3390/w11061254

**Chicago/Turabian Style**

Yang, Fei, Xuejun Shao, Xudong Fu, and Ehsan Kazemi. 2019. "Simulated Flow Velocity Structure in Meandering Channels: Stratification and Inertia Effects Caused by Suspended Sediment" *Water* 11, no. 6: 1254.
https://doi.org/10.3390/w11061254