Rainfall and roughness simultaneously affect the rainfall runoff of a high steep slope, which results in a change in the movement process of the rainfall runoff on high steep slopes, thus affecting the analysis and calculation of the movement process^{25}. Therefore, based on the “theory of high steep slope double turbulent thin layer flow” and the existing dispersion wave model, the calculation process of runoff dynamic parameters with different rainfall intensities and slopes is given.

### Variations caused by double turbulence sources

The hydraulic characteristics of rainfall runoff on high and steep slopes were obviously different from those on gentle slopes. Figure6 illustrates the runoff generation processes on slopes with different gradients published by this experiment and different scholars^{21}. Figure6 shows that the net discharge per unit width was positively correlated with rainfall duration during the process of runoff generation and in the confluence of slopes with different gradients. Moreover, the time of slope runoff generation varied with slope gradient. The larger the slope was, the smaller the starting time and confluence time were, and the larger the corresponding peak flow was.

For the same slope, the larger the rainfall intensity was, the smaller the starting time and confluence time were, and the larger the corresponding peak discharge was. For example, under the condition of 1.221 rad, the starting time of the 135 mm·h^{−1} rainfall intensity was approximately 7% of the 30 mm·h^{−1} rainfall intensity time, and the net flow per unit width of the 135 mm·h^{−1} rainfall intensity was approximately 520% of the 30 mm·h^{−1} rainfall intensity time.

Under the same rain intensity conditions, the greater the slope was, the shorter the starting time was, the shorter the equilibrium time was, and the larger the net flow per unit width of the slope surface was. For example, under the condition of 135 mm·h^{−1}, the starting time of 1.22 rad was approximately 16% of the 0.087 rad condition. The net discharge per unit width of the 1.22 rad slope was approximately 144% of the 0.087 rad slope.

The runoff generation process of high and steep slopes was similar to that of gentle slopes, but it was more severely affected by the slope and rainfall intensity.

#### Variations in the Reynolds number with different rainfall intensities and slopes

The Reynolds number is a parameter used to distinguish the flow pattern. Figure7 shows that the Reynolds number on a high-steep slope was significantly positively correlated with rainfall intensity (60~120 mm·h^{−1}). Compared with gentle slopes, the Reynolds number increased rapidly after runoff generation on steep slopes. With an increase in simulated rainfall intensity in the same soil bulk density and slope gradient conditions, the Reynolds number increased gradually, which indicated that an increase in rainfall intensity would change the slope flow pattern and would increase the degree of flow turbulence.

With the same rainfall intensity, when the slope was smaller than the critical slope, the Reynolds number did not change much; however, when the slope exceeded the critical slope, the Reynolds number increased rapidly. The empirical relationship between the Reynolds number and slope was as follows:

$${\rm{Re}}=376{{\rm{\theta }}}^{2}-80{\rm{\theta }}+35$$

With different rainfall intensities, the critical slope gradient with the obvious variation in the Reynolds number was approximately 0.42 rad.

#### Variation in the Froude number with different rainfall intensities and slopes

The Froude number, which is the ratio of inertial force and gravity in fluid, is a dimensionless number that is used to judge the state of the water flow. Figure8 shows the Froude numbers with different rainfall intensities and slopes. With an increase in simulated rainfall intensity with the same soil bulk density, the Froude number increased gradually, and the Froude number after stabilization was greater than 1, which showed that the flow pattern would change with the increase in simulated rainfall intensity.

After runoff generation on steep slopes, the Froude number increased rapidly when the gradient did not exceed the critical gradient, and the runoff on slopes was rapid. When the critical gradient was exceeded, the Froude number tended to decrease with increasing slope. The empirical relationship between the Froude number and slope was as follows:

$${\rm{Fr}}=1.45\,\mathrm{ln}\,{\rm{\theta }}+297$$

With different rainfall intensities, the critical slope gradient at which the growth rate of Froude number obviously slowed was approximately 0.93 rad.

### Prediction of the hydraulic parameters

In practice, the calculation of hydrodynamic parameters of high and steep slopes directly affected the design of erosion and corrosion prevention engineering. Using the method given by Ligget, J.A. and Woolhiser, D.A^{24}, Gao *et al*. calculated the hydrodynamic parameters of slope flow of 0.0174 rad and 0.0872 rad^{25}. The calculated results were in accordance with the *in situ* data and met the actual production requirements. To determine the application of this method on high and steep slopes, the rainfall runoff on high and steep slopes was calculated. The calculation results are shown in Figs.9–11.

#### The equilibrium time

Figure9 shows that the calculation results of the slope flow equilibrium times of 0.26–1.22 rad were smaller, and the difference between the observed and calculated values was approximately 50-fold.

#### Water surface line

After calculating the runoff of high and steep slopes by the dimensionless method, the runoff water surface lines on high and steep slopes with different rainfall intensities could be calculated. Figure10 shows that under the condition of high and steep slopes, the calculated value of the water surface line with different rainfall conditions was larger than the observed value, and the error along the slope length increased gradually; furthermore, the error at the foot of the slope was the largest, which was approximately 30% smaller than the calculated value.

#### Runoff and drainage on the slope surface

After calculating the runoff of high and steep slopes by the dimensionless method, the runoff outflow process of high and steep slopes with different rainfall intensities could be calculated. According to the runoff calculation process and actual outflow process of slopes, Fig.11 was obtained. Figure11 shows that there was a great difference in time between the calculated flow process and the observed flow process.

These verification results showed that when the original calculation method for the slope flow hydrodynamic parameters was used to calculate the high and steep slope, the calculated values of the slope flow equilibrium time, the slope flow water surface line and the slope outflow process were quite different from the observed values. Therefore, the equations must be modified when applying this method to the calculations of high and steep slopes.

### Model building

Compared with open-channel flows, the slope flows had the following characteristics: there was no fixed boundary for the slope flow; the slope flow depth was far less than the open-channel flow; and the influences of rainfall, infiltration and roughness on the slope flow were more obvious than those in the open-channel flow. In most cases, the slope flow movement was approximately described by the motion wave model^{23}. Because of the large slope and the previously described influence of rainfall, the high and steep slope had runoff and was considered “sheet flow influenced by double turbulence sources”. Therefore, the surface flow movement on the high and steep slopes had a significant variation compared with the gentle slope. In the calculation of the high and steep slope surface flow, we had to consider the effects of slope gradient and rainfall at the same time.

On the slope, if the flow direction was the x-axis and the slope length was L, the Saint-Venant equation group of a one-dimensional unstable flow controlling the movement of slope flow was as follows:

$$\frac{\partial ({{\boldsymbol{V}}}_{{\boldsymbol{y}}})}{\partial {\boldsymbol{x}}}+\frac{\partial {\boldsymbol{y}}}{\partial {\boldsymbol{t}}}=i(x,t)-f(x,t)=r(x,t)$$

(4)

$${{\boldsymbol{S}}}_{0}-{{\boldsymbol{S}}}_{{\boldsymbol{F}}}=\frac{\partial {\boldsymbol{y}}}{\partial {\boldsymbol{x}}}+\frac{1}{{\boldsymbol{g}}}\frac{\partial {\boldsymbol{V}}}{\partial {\boldsymbol{t}}}+\frac{{\boldsymbol{V}}}{{\boldsymbol{g}}}\frac{\partial {\boldsymbol{V}}}{\partial {\boldsymbol{x}}}+\frac{{\boldsymbol{V}}}{{\boldsymbol{gy}}}[{\boldsymbol{i}}({\boldsymbol{x}},{\boldsymbol{t}})-{\boldsymbol{f}}({\boldsymbol{x}},{\boldsymbol{t}})]$$

(5)

When the international standard units were used in the equations, y was the flow depth of the slope, m; V was the flow velocity of the slope, m·s^{−1}; i(x, t) was the rainfall intensity, and the function of x and t, m·s^{−1}; f (x, t) was the infiltration intensity, and the function of x and t, m·s^{−1}; r(x, t) was the net rainfall intensity, and the function of x and t, m·s^{−1}; r(x, t) = i(x, t) − f(x, t), m·s^{−1}; S_{0} was the slope gradient; S_{F} was the friction gradient; and g was the gravity acceleration.

In a classical study of the slope flow problem^{27,28,29}, the slope gradient was small (less than 5°) when considering that slope gradient i_{0} and resistance slope gradient i_{f} were approximately equal:

$$\,{i}_{0}-{i}_{f}=0$$

(6)

In this case, Eqs. (4) and (5) were simplified, and the slope flow was regarded as a uniform flow. In the motion wave model, the slope was equal to the resistance slope. With the small slope, the motion wave model analyzed the slope flow more accurately^{25}. When the actual slope exceeded 5°, Eq. (6) was no longer valid, as the original motion wave model calculation had errors. Therefore, the influences of slope and rainfall on the accuracy of the model should be considered in the analytical calculation of a large slope flow.

#### Correction of equilibrium time

An equation for calculating the equilibrium time was obtained based on the motion wave theory (7):

$${t}_{e}=\frac{1}{{r}^{0.4}}{\left(\frac{n{L}_{0}}{{{S}_{0}}^{0.5}}\right)}^{0.6}$$

(7)

Where t_{e} is the equilibrium time, s; r is the net rainfall intensity, m·s^{−1}; n is roughness; L_{0} is the slope length, m; S_{0} is the slope, rad.

On a high and steep slope, M_{1} stands for the influences of the bed surface and rainfall on the variation in equilibrium time, and then Eq. (7) becomes:

$${t}_{em}={M}_{1}\frac{1}{{r}^{0.4}}{\left(\frac{n{L}_{0}}{{{S}_{0}}^{0.5}}\right)}^{0.6}$$

(8)

Based on the observed data, the relationship expression of M_{1} was obtained as follows:

$${M}_{1}=\frac{{S}_{0}}{{i}^{0.4}\theta }$$

(9)

Equation (9) shows that the influences of slope and rainfall on the equilibrium time are important. By substituting Eq. (8), a unified relationship of the equilibrium time with different gradients was obtained:

$${t}_{e}=\frac{{S}_{0}}{{i}^{0.4}\theta }\frac{1}{{r}^{0.4}}{\left(\frac{n{L}_{0}}{{{S}_{0}}^{0.5}}\right)}^{0.6}$$

(10)

Equation (10) was an equation for calculating the runoff equilibrium time on slopes, and the equation showed that the runoff equilibrium time on high and steep slopes was affected by the bed roughness, rainfall intensity and slope.

#### Correction of water surface line

Similarly, based on the motion wave theory, the equation for calculating the flow surface line on slopes was obtained (11):

$$y={\left(\frac{nxr}{{{S}_{0}}^{0.5}}\right)}^{0.6}$$

(11)

Where y is the water depth on the slope, x is the coordinate along the journey, and the other parameters have the same meanings as described above.

On a high and steep slope, M_{2} stands for the influence of the bed surface and rainfall on the variation in the flow surface line on the slope surface, and Eq. (11) becomes:

$$y={M}_{2}{\left(\frac{nxr}{{{S}_{0}}^{0.5}}\right)}^{0.6}$$

(12)

Based on the observed data, the relationship expression of M_{2} was obtained as follows:

$${M}_{2}=\frac{{\theta }^{0.6}}{{S}_{0}^{1.5}{i}^{0.1}}$$

(13)

Equation (13) shows that the influences of slope and rainfall on the water surface line are important. By substituting Eq. (11), the unified relationship of the flow surface lines on slopes with different gradients was obtained:

$$y=\frac{{\theta }^{0.6}}{{S}_{0}^{1.5}{i}^{0.1}}{\left(\frac{nxr}{{{S}_{0}}^{0.5}}\right)}^{0.6}$$

(14)

Equation (14) is an equation for calculating the runoff surface line on slopes, and the equation shows that the distribution of the rainfall runoff water surface line on high and steep slopes is affected by bed roughness, rainfall intensity and slope.

#### Outflow process modification

Figure11 shows that the calculated flow process of slopes under different rainfall conditions is very short on a high and steep slope. The observed flow process time was approximately 50 times higher than the calculated time. Being affected by the slope gradient and rainfall, the equilibrium time and water surface line of the steep slope had to be modified. The steep slope flow process was obtained from the corrected equilibrium time and equilibrium flow at the foot of the slope.

Based on our analysis, in considering the influences of the slope gradient and rainfall intensity on the runoff flow, the original calculation process was modified. The modified equations and the original equations are shown in Table1, including the equilibrium time, the slope surface line, and the rising and retreating processes of the slope flow.

In Table1, K is the motion parameter; t_{e} is the slope flow equilibrium time, s; y is the depth of the flow, m; q_{e} is the single wide flow when slope flow reached equilibrium, m^{2}·s^{−1}; t is rainfall duration, s; n is roughness; L_{0} is the length of slope, m; r is the net rainfall intensity, m·s^{−1}; m is empirical constant; i is the rainfall intensity, m·s^{−1}; θ is the slope gradient.

Table1 shows the original calculation process and modified calculation process of flow on slopes. In Table1, the motion parameter(K_{1} and K_{2}) is same and is used as the discriminant number of flow. Balanced flow at foot of slope(q_{e1} and q_{e2}) is just affected by rainfall intensity under the certain slope length. Equilibrium time, water surface line, rising and receding process at foot of slope are all affected by the double-turbulent, and the influences of the bed surface and rainfall on the variation are shown in the new equations.

### Performances of the modified methods

#### Equilibrium time verification

Based on the observed data of indoor and field slopes with different rainfall intensities of 0.087 rad, 0.26 rad, 0.44 rad, 0.65 rad and 1.22 rad, the runoff equilibrium time on slopes was calculated by using the modified equations. The calculated runoff equilibrium time on slopes was compared with the observed values as shown in Fig.12.

Figure12 shows that using the modified methods, the difference between the calculation equilibrium time and the measured equilibrium time under different conditions is small, and the effect is good.

#### Verification of the water surface line

According to the observed water surface line in the test and the adjusted calculated water surface line, the water surface line of the high and steep slope was obtained as shown in Fig.13. As shown in Fig.13, the adjusted calculated water surface line along the slope length was essentially the same as the observed water surface line.

Figure13 shows that using the modified methods, the shape of the calculated water surface line is similar to the observed ones, and the depth of the calculated value and the measured value does not exceed 10%.

#### Outflow process verification

According to the observed outflow process indoor test, field observations and the adjusted outflow process, the high and steep slope outflow process was obtained as shown in Figs.14 and 15. From Figs.14 and 15, the calculated flow process of the adjusted steep slope was similar to the results of the laboratory tests and field observations.

Figures14 and 15 show that using the modified methods, the duration of the calculated rising process and the calculated receding process are similar to the observed ones, and the flow rates at different times are not much different, and the balance flow at the foot of the slope is basically the same.

These verification results showed that the calculated values of the parameters such as the equilibrium time, water surface line and the slope flow outflow process were in accordance with the observed values. The modified equations were able to apply in the calculation of the high and steep slope flow.