Superelevation and runup height of debris flows in bends based on typical rectangular cross-sections and gravity center offset

Abstract

Owing to the superelevation in the bend, debris flow rushed out of the channel and destroyed the engineering facilities near the bend, which brings challenges to the design of disaster prevention and mitigation projects. Under the assumption of a continuous homogeneous medium and rectangular channel section, the debris flow velocity in the bend is decomposed into a tangential velocity that causes superelevation and a normal velocity that generates runup. The calculation equations of superelevation, runup, and total superelevation of debris flow in the bend are derived based on the gravity center offset of the flow section in the debris flow movement. The proposed equation can determine and depict the evolution of total superelevation value along the bend, which is useful for engineers to carry out the targeted design in debris flow prevention and control engineering. Meanwhile, using the equation in this paper and two other representative equations, the total superelevation, superelevation, and runup height of debris flow in 37 groups of test data and bends in the case study are computed to compare with the test and measured results. The test findings showed that the runup height accounts for about 40% of the total superelevation, and the runup height of the five bend points in the case accounts for 43–48%, indicating that the runup is an essential part of the total superelevation and needs to be calculated in the design of the bend of the debris flow prevention and control project.

Appendix: Derivation of correction coefficient for superelevation and runup

As shown in Fig. 4, from the flow section AB to HI, the range is 0 < θ < arccosR₂/R₁. Let RF be arbitrary flow section from AB to HI. S_AEJ (representing the mass involved in superelevation and runup in this process)/S_AEFB (representing the total mass of all flow sections from AB to HI) is the correction coefficient when reaching the flow section E_iF_i.

$$\begin{aligned}\xi&=S_{AEJ}/S_{AEJ} \\ &= (S_{AEJO}-S_{AOE})/(S_{AOE}-S_{BOF}) \\ &=\frac{(\frac{1}{2}(R_{1}-R_{1}\cos\theta)+R_{1})R_{1}\sin\theta-R_{1}^{2}\;\theta/2)}{\theta(R_{1}^{2}-R_{2}^{2})/2} \\ &=\frac{(R_{1}^{2}\sin\theta-0.5R_{1}^{2}\sin\theta\cos\theta-R_{1}^{2}\;\theta/2}{\theta(R_{1}^{2}-R_{2}^{2})/2}\end{aligned}$$

(24)

where S represents the area of subscript surface, R₁ and R₂ are the outer radius and inner radius of the bend, and θ is the included angle between arbitrary flow section and OA.

From the flow section HI to MN, the range is arccosR₂/R₁ < θ < π/2. Let E’F’ be arbitrary flow section from HI to MN. The velocity direction of debris flow in AB section of the initial flow section all changes. Under the constraint of the channel wall, the general flow trend changes to the circular movement trend. It is assumed that the uniform rate of change of the angle between the velocity direction and the tangent direction is:

$$d\alpha=\frac{\Delta \theta}{\angle\;AOH+\Delta \theta}$$

(25)

where Δθ is the included angle between arbitrary flow section E_i’F_i’ and HI.

At the flow section HI, the included angle between the movement direction of debris flow and the tangent direction is α (indicated by point P), and the included angle between the debris flow movement direction of arbitrary flow section E_i’F_i’ and the tangent direction is α-αdα (indicated by point Q). Multiply (α-αdα)/α by the correction coefficient at the time of the flow section HI to be the correction coefficient at arbitrary position of the flow section E_i’F_i’. The correction coefficient of the flow section HI to MN is:

$$\xi=\xi_{CH} \cdot (1-d\alpha)$$

(26)

Substituting Eq. (25) into Eq. (26):

$$\begin{aligned}\xi&=\xi_{CH}\cdot\frac{\angle\;AOH}{\angle\;AOH+\Delta\theta} \\ &=\frac{S_{AHK}}{S_{AHIB}}\cdot\frac{\angle\;AOH}{\angle\;AOE^{\prime}}\\ &=\frac{S_{AHK}}{S_{AEFTB}}\cdot\frac{S_{AHIB}}{S_{AEFB}} \\ &=\frac{S_{AHK}}{S_{AEFTB}}\\ &=\frac{0.5\;\left(R_{1}-R_{2}+R_{1})\right.\sqrt{R_{1}^{2}R_{2}^{2}}-\pi R_{1}^{2}\;arcoss\;\left(R_{2}/R_{1})/2\right.}{\theta\left(R_{1}^{2}-R_{2}^{2})/2\right.}\\ &=\frac{\sqrt{R_{1}^{2}-R_{2}^{2}}\left(R_{1}-0.5\;R_{2})\right. -R_{1}^{2}\;arcoss\; \left(R_{2}/R_{1})\right. /2}{\theta\;\left(R_{1}^{2}-R_{2}^{2})\right. /2}\end{aligned}$$

(27)

where angle is in radian. The final correction coefficient is expressed as a piecewise function:

$$\xi=\begin{cases}{\frac{R_{1}^{2}\;sin\;\theta-0.5\;R_{1}^{2}\;sin\;\theta\;cos-\;\theta\;R_{1}^{2}\;\theta/2}{\theta\;\left(R_{1}^{2}-R_{2}^{2})\right./2}},0<\theta < arccos\;\frac{R_{2}}{R_{1}} \\ \frac{\sqrt{R_{1}^{2}-R_{2}^{2}}\;\left(R_{1}-0.5\;R_{2})\right.-R_{1}^{2}\;arccos\;\left(R_{2}/R_{1})\right./2}{\theta\;\left(R_{1}^{2}-R_{2}^{2})\right./2},\;arccos\;\frac{R_{2}}{R_{1}}<\theta<\frac{\pi}{2}\end{cases}$$

(28)