**Journal of Applied Mathematics and Physics**

Vol.04 No.04(2016), Article ID:65454,7 pages

10.4236/jamp.2016.44073

Numerical Investigation on Downstream Increase in Peak Discharge of Hyperconcentrated Floods in the Lower Yellow River

Zhijing Li^{1,2}, Zhongwu Jin^{1,2}

^{1}Changjiang River Scientific Research Institute, Wuhan, China

^{2}Collaborative Innovation Center for Geo-Hazards and Eco-Environment in Three Gorges Area, Yichang, China

Received 11 January 2016; accepted 6 April 2016; published 13 April 2016

ABSTRACT

Hyperconcentrated floods in the Yellow River usually accompanied with some peculiar phenomena that cannot be explained by general conceptions of ordinary sediment-laden flow
^{3}/s larger than the value at Xiaolangdi station at upstream. It is found that a fully coupled model that incorporates the contribution of bed evolution to the mass conservation of the water-sediment mixture, can reasonably well capture the characteristics of peak discharge rise and severe bed scour, while separate numerical experiment using a decoupled model, which ignores the feedback effects of bed evolution, shows no rise in the peak discharge. This leads us to comment, if only briefly, that the entrainment of sediment due to bed erosion is the main reason for causing peak discharge increase along downstream course.

**Keywords:**

Hyperconcentrated Flood, Abnormal Phenomena, Peak Discharge Increase, Sediment Transport, Mathematical Model

1. Introduction

Hyperconcentrated flow is identified as a kind of complex sediment laden flow carrying a large amount of sediments with high concentration and certain fine particles, which is a typical solid and liquid two-phase flow [1]. The mechanism and hydraulic features of hyperconcentrated floods are complicated and are different from those of low sediment concentration floods [2] [3]. In the Yellow River and its tributaries on the Loess Plateau, hyperconcentrated flood occurs frequently during the rainy seasons [4]. Large amount of sediment load is transported during hyperconcentrated floods. Both the historical maximum concentration of
^{3} and the mean annual amount for sediment transport of 1600 million tons are the highest values worldwide [5]. The fluvial process is extremely rapid with the river morphology altered more at a faster rate by one hyperconcentrated flood than by normal flow and low sediment concentration floods over a decade. In turn, the rapid fluvial process also changes the propagation of flood waves and induces peculiar phenomena.

When flood propagates downstream, peak discharge decreases usually due to head loss or channel storing. However, this situation is altered when the duration of hyperconcentrated flood is fairly long [6]. In August 1992
^{3}/s which was
^{3}
^{3}/s) at upstream with a length of

This paper has made a numerical investigation of hyperconcentrated floods, the typical flood

2. Model Formulations

2.1. Governing Equations

Consider 1D flow in an open channel with rectangular cross-sections of constant width. The complete governing equations for the flow, sediment transport and morphological evolution are built upon the mass and momentum conservation equations for the water-sediment mixture and the mass conservation equations respectively for sediment carried in the flow and bed material [9], which in general read

(1)

(2)

where t = time; x = streamwise coordinate; h = flow depth; u = depth-averaged streamwise velocity; z = bed elevation; C = flux-averaged volumetric sediment concentration; g = gravitational acceleration; = bed slope; = friction slope; p = bed sediment porosity; E, D = sediment entrainment and deposition fluxes across the bottom boundary of flow, representing the sediment exchange between the water column and bed;

= the density of water-sediment mixture; = the density of the saturated bed;, = densities of water and sediment, respectively; and.

2.2. Auxiliary Relationships

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To close the governing equations of models for fluvial sediment transport, auxiliary relationships are required. For the friction slope, the conventional empirical relation is used, which involves the Manning roughness n

Sediment exchange between the flow and bed involves two distinct mechanisms, i.e., bed sediment entrainment due to turbulence and possibly interaction between sediment particles, and also sediment deposition due to gravitational action. The determination of the entrainment and deposition fluxes continues to be one of the pivotal components of computational models for sediment transport and morphological evolution. The traditional and extensively used approach to specifying bed sediment entrainment flux is based on the premise that entrainment occurs always at the same rate as it does under equilibrium conditions. Here, the entrainment and depositions fluxes are estimated by

where is the settling velocity of a single sediment particle in tranquil water, which can be calculated using for instance Zhang’s formula [1], is the exponent denoting the effects of hindered settling due to high sediment concentrations, is the particle Reynolds number, d is the sediment diameter, and is the kinematic viscosity of water.

is the sediment transport capacity exclusively determined by local flow conditions. In general, sediment transport capacity of hyperconcentrated flow is more complicated than that of common sediment-laden flows [10]. The high sediment concentration induces an increase of the fluid viscosity and a decreasing sediment settling velocity due to hindered settling, these effects result that less energy is required to keep sediments in suspension. At the same time, the erosion rate probably decreases with the high sediment concentration flow where less water available to penetrate into the bed and erode bed sediments [11]. However, as the density of the eroding fluid increases, the density difference between the bed and the fluid decreases, the lift force required to erode sediments decreases.

Here, the well-tested and widely used semi-empirical formula of Zhang and Xie

(8)

is a coefficient denoting the difference between near-bed concentration and depth-averaged concentration of sediment. The following relationship which has a similar approach of depth integration of the Einstein sediment transport formula is introduced [2] [13] [14].

where is the roughness height, and

where a = reference level, is Rouse number (i.e., suspension index), defined as the ratio of sediment settling velocity to the product of von Karman constant and bed shear velocity.

It is appreciated that existing formulations of sediment entrainment and deposition fluxes remain empirical and the roughness parameter is hard to establish for natural rivers, especially as it is influenced by sediment. Thus Equations

2.3. Numerical Algorithm

The complete governing equations are first rewritten in conservative form, and then computed following a second-order TVD scheme that is an extension of the first-order centered FORCE scheme. The SLIC approximate Riemann solver is employed in this study [8]. The present capacity and non-capacity models are essentially adapted from a coupled model for turbidity currents over erodible sediment bed, and have been compared with that based on TVD-WAF along with HLLC approximate Riemann solver.

3. Numerical Case Study

In August 1992
^{3}/s which was 1690 m^{3}/s larger than the value at Xiaolangdi station (4570 m^{3}/s) at upstream. In this study, numerical experiments have been conducted on this typical hyperconcentrated flood. Numerical cases are summarized in Table 1. Case 1 employs the fully coupled model (Equations (1) to (4)) as to simulate the abnormal increase of the peak flow rate along the downstream course. Case 2 uses a decoupled model with the right hand side

other relevant parameters are specified as:, , , , ,.

4. Results and Discussion

Figure 2 illustrates the comparison between computed results from Case 1 and measured data at Huayuankou station which is

Figure 1. Measured discharge and sediment concentration used as boundary conditions.

Table 1. Summary of numerical cases.

Figure 2. Computed results for case 1 at Huayuankou station in comparison with measured data.

The evolutional characteristics of hyperconcentrated floods are very complicated owing to its physical and propagation characteristics, and it is very difficult to explain it theoretically yet. When hyperconcentrated flood propagates through braided channel like the Yellow River and its tributaries on the Loess Plateau, extremely rapid fluvial process happens, in turn, the fluvial process also changes the propagation of flood waves and induces peculiar phenomena. From Figure 2(a), we can see that the measured peak discharge lagged behind the peak of computed hydrograph. This is owing to the flow storage and friction resistance by brained channel at the early stage of the flood, and it must be noted that the flood propagation speed is related to the channel configuration, channel shape changes from wide and shallow to narrow and deep during flood period, thus propagation speed changes.

The abnormal increase of the peak flow rate along the downstream course during the hyperconcentrated flood of “

Figure 3 presents the computed results for Case 2 which uses a decoupled model with RHS of Equation

5. Conclusion

The typical hyperconcentrated flood

Figure 3. Computed results for case 2 at Huayuankou station in comparison with measured data.

that rapid fluvial evolution process happens during hyperconcentrated flood period, large amount of sediment is entrained by the hyperconcentrated flow due to bed erosion, and this is the main reason for causing peak discharge increase along downstream course.

Acknowledgements

The research is funded by Natural Science Foundation of China (Grants No. 51479009 and 51339001), and the special fund for basic scientific research business of central public research institutes (CKSF2016011/HL).

Cite this paper

Zhijing Li,Zhongwu Jin, (2016) Numerical Investigation on Downstream Increase in Peak Discharge of Hyperconcentrated Floods in the Lower Yellow River. *Journal of Applied Mathematics and Physics*,**04**,641-647. doi: 10.4236/jamp.2016.44073

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