Evolution Rules of Rolling Waves on Slopes Based on Artificial Flat Slopes of Loess

Yang, Miao; Gong, Jiaguo; Zhao, Yong; Wang, Hao; He, Xiaohui; Zhao, Cuiping

doi:10.1061/(ASCE)HE.1943-5584.0001899

Open access

Technical Papers

Feb 27, 2020

Evolution Rules of Rolling Waves on Slopes Based on Artificial Flat Slopes of Loess

Authors: Miao Yang [email protected], Jiaguo Gong https://orcid.org/0000-0002-5064-0976 [email protected], Yong Zhao [email protected], Hao Wang [email protected], Xiaohui He [email protected], and Cuiping Zhao [email protected]Author Affiliations

Publication: Journal of Hydrologic Engineering

Volume 25, Issue 5

https://doi.org/10.1061/(ASCE)HE.1943-5584.0001899

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Abstract

We used a fixed-bed flume test and ultrasonic sensor measurement technology to study the evolution of the characteristics of thin-layer flow roll waves on a sloped surface. The results indicated that there was a positive correlation between the velocity of the roll wave and the Reynolds number when the energy slope and roughness were small. In addition, the amplitude of the increase gradually decreased with increases in roughness, and the height of the roll waves showed a single-hump trend of change. When the roughness was relatively small, the frequency of the roll waves increased as the Reynolds number increased, and the ratio of the wavelength to the water depth gradually decreased. As the energy slope increased, both the wave velocity and frequency of the roll waves gradually increased; they peaked when the energy slope was 0.3420. We found that the dominant factors that affected different roll-wave characteristic parameters varied. The roughness of the bed had a relatively prominent impact on the velocity of the roll waves, but the energy slope had a greater effect on the height of the roll waves.

Introduction

Flow on a slope is the primary stage of surface runoff and the main dynamic factor of slope soil erosion. Many factors affect overland flow under natural conditions; in addition, mass and momentum sources continuously join the flow path, making it difficult to quantitatively describe the flow on slopes. Because of different experimental conditions, researchers (Woolhiser et al. 1970; Kirkby 1978; Govers 1992; Nearing et al. 1997; Zhao et al. 2015) still disagree about the basic hydrodynamic characteristics (flow state, mean velocity, and resistance characteristics) of the overland flow. The erosion of sandy slopes in the loess plateau of the northern Shaanxi area of China is of serious concern, and the slopes are relatively flat in the ecotone between the areas that experience wind erosion and water erosion. This is also the area where the slope flow is prone to instability and roll waves. Our ability to select characteristic parameters for roll waves and our understanding of the factors affecting the evolution of the flow on slopes need to be strengthened.

The roll wave of flow on slopes is initially produced by surface tension. As the water depth and flow velocity increase, the surface tension effect weakens and a “capillary roll wave” is gradually transformed into an “inertial roll wave” (Campomaggiore et al. 2016). The inertial roll wave originates from the inertial instability (Brock 1969; Logan and Iverson 2007). It is produced when the inertial force is greater than the balance of the viscous forces in a water body, as is the case in a roll wave found in a drainage chute or spillway. Roll waves are likely to occur under both a uniform laminar and turbulent flows. Lighthill’s motion wave theory predicts that the shape of the roll wave will change under various flow conditions.

Roll-wave formation can involve non-Newtonian fluids, such as debris flows (Wang et al. 2006). In surface runoff situations, the roll-wave movement is complex because of the presence of surface precipitation and soil infiltration. For simplification, the roll wave is generally regarded as a periodic travelling wave. The wave characteristics of the roll wave are affected by three factors: average flow, slope of the bed, and the coefficient of resistance. It is generally quantified by the wave velocity, wave height, wavelength, and frequency (Liu et al. 2005; Zhang 2011).

Because the roll wave has significant impacts on the origination and transportation of sediments on the slope, its prediction is of great interest to the field of environmental engineering (Campomaggiore et al. 2016). Harold (1925) and Cornish (1934) were the first to study the roll wave. Subsequently, Thomas (1940) discussed the necessary role of the bed resistance in the formation of a roll wave. Because of the characteristics of the motion form of the roll wave, actual measurement is difficult. Owing to the improvements in science and technology, the research on model optimization and measurement device development has made great progress (Cao et al. 2015; Maciel et al. 2017). Before this, few experimental studies have been conducted.

The coarse bed flume test conducted by Brock in 1969 is a classic work in this field (Brock 1969). Brock found that the roll wave formed after the uniform flow lost stability, and the wave velocity and wave shape changed continuously. Since kinematic waves commonly have a constant wave shape and velocity, the roll wave was characterized as a dynamic wave. Benjamin (1957) and Yih (1963, 1977) investigated the critical conditions required to produce a roll wave using the linear stability of the oblique lower flow and determined that its critical Froude number was about 0.5. Julien and Hartley (1986) also found the roll wave in the laminar critical flow with a Froude number as low as 0.74 (Pan and Shangguan 2009). In recent years, the impact of the roll wave on the erosion of slope soils has received increasing attention (Zhao et al. 2015; Ivanova et al. 2017). Pan and Shangguan (2009) indicated that the rainfall and slope can affect the development of roll waves. Zhang (2011) found, through fixed-bed flume tests, that with the increased discharge per unit width (i.e., the increase in the water depth), the instability of the water surface and the quenching of roll waves occur in flows on slopes. Zhao et al. (2015) investigated the response of the roll wave to the sediment concentration and surface hydraulics on a steep slope. Ivanova et al. (2017) compared simulation results with experimental results and found that the resulting free surface profiles matched well; additionally, the transmission of energy formed a regular roll-wave train.

In summary, most of the existing analyses of the roll-wave problem were based on model tests and numerical models. The influence of changes in the underlying surface on the evolution of the roll wave has not been systematically considered. Both the velocity and depth of the water flow greatly change after a roll wave is produced by the instability of the flow on slopes. These changes affect both the slope soil erosion and watershed runoff calculations; despite this, neither the traditional hydrological model nor the soil erosion model take it into consideration. In this paper, we examine the evolution of the roll wave in the flow on slopes and discuss the factors that most influence its characteristics. Using artificial modeling (the fixed-bed flume test) and an ultrasonic wave measurement system, we determined the response relationship between the characteristic parameters of a thin-layer flow roll wave on a flat slope surface of loess and the Reynolds number, energy slope, and roughness. Our results provide a theoretical basis for the sediment transport calculations in river basins. Because of the limitations imposed by the experimental conditions, we did not consider the effects of rainfall and infiltration in this study.

Method

It is difficult to delineate the boundary conditions and implement a site survey for testing at natural sites. In this study, we carried out an artificial simulation on an indoor flume. Although this type of test neglects the seepage problem, the fixed-bed flume is easy to observe and eliminates the impact of changes in the bed shape and bottom sediment exchange on the flow turbulence (Zhang 2001). The experimental device consisted of a testing flume, a reservoir, a frequency-varying pump, an electromagnetic flowmeter, and an outlet pool. The cross section of the test flume was rectangular, with dimensions of

11 \times 0.5 \times 0.5 m

. The bottom plate and walls of the flume were made of 10-mm-thick glass, and the slope was adjustable between 0° and 30°. The monitoring range of the electromagnetic flowmeter was

0 - 6 m^{3} / h

, with a measurement error of 0.4%. A honeycomb rectifier was installed at the outlet of the discharge pool to decrease the turbulence of the flow.

According to the characteristics of wind-water erosion interlaced area in loess plateau, a water sand cloth with a relatively uniform roughness was used as the underlying surface at the bottom of the test flume. The size of the cloth was #180 mesh, with a roughness size

k_{s}

of 0.09 mm, close to the medium-sized sand loess particles in that area. Considering the impact of the roughness on the experimental results, two other water sand cloths (#80 and #40 mesh) and 2 sand-binding beds (with quartz sand particle sizes of 1–2 mm and 3–5 mm) were also used. The roughness sizes

k_{s}

of these water sand cloths were converted, respectively, to 0.09, 0.19, 0.38, 1.50, and 4.00 mm, in accordance with the international standard. Sheet erosion and rill erosion mainly occurred at 5°–35° of the slope. Channel erosion tended to occur when the slope exceeded 25°; this experiment did not consider the slope range for the channel occurrence. The experimental bottom slopes were set to 5°, 10°, 15°, 20°, and 25° (0.0872, 0.1736, 0.2588, 0.3420, and 0.4226 rad, respectively). After the flow range of the roll wave was determined pretest under different energy slope conditions, the discharges per unit width were set to 0.1670, 0.2000, 0.2330, 0.2670, 0.3000, 0.3330, and

0.3670 L / (s \cdot m)

. We set up six observation cross sections with 1 m of observation distance. To avoid the experimental error associated with an unstable flow at the beginning cross section of the flume, the first observations were made 1.5 m away from the flume inlet; the remaining observation sections were defined, in order, as 2-2, 3-3, 4-4, 5-5, and 6-6. The following results only focus on the 6-6 observation section, which exhibited better experimental characteristics than the others.

An ultrasonic water level meter system was used to measure the roll wave. Two sensors were installed at each observation section, one in front and one behind. The spacing was 5 cm. The measurement of the parameters, such as the velocity and height of the roll wave on the slope surface, was implemented through the combination of the two sensors; the measurement device is shown in Fig. 1(a). The measured data in this laboratory test mainly included the flow rate and water depth. The flow rate was measured by an electromagnetic flowmeter, and the relative error was 0.4%. The final result was the average of multiple records. The water depth was measured by an ultrasonic water-level meter with a measurement error of 0.3 mm. The detailed measurement process and principles were obtained from the literature (Yang et al. 2017). Compared with the traditional measurement method, the average relative error of the data obtained by this measurement system was 0.23%, and the average variation coefficient was 0.66%. The trough bottom data measurements and water surface surveys were carried out in each experiment. The measurement duration was 30 s, and the water depth data were collected 2,000 times, with an average interval of 15 ms. At present, there are no criteria for a quantitative hydraulic parameter assessment for the roll wave. We found from pretest observations that when the flow reached a certain level, there would be one wave after another on the flow surface. The wave front traversed the whole section, and the velocity of the wave was higher than the average flow velocity, so that small waves merged to form large waves, which propagated forward like a snowball rolling downhill [as shown in Fig. 1(b)]. This observation was used as the adjudging criterion of the emergence of roll wave.

Fig. 1. (a) Experimental measurement system; and (b) test observation.

The Manning roughness coefficient was used to describe the underlying surface of the flume. It was determined using the Chezy formula [Eq. (1)] and the Manning formula [Eq. (2)], as follows:

v = c \sqrt{R J}

(1)

c = \frac{1}{n} R^{\frac{1}{6}}

(2)

where

v

= average flow velocity of the cross section (

m / s

);

c

= Chezy coefficient;

R

= hydraulic radius (m), which is equivalent to the water depth in the wide shallow canals;

J

= hydraulic slope; and

n

= Manning roughness coefficient. The formula for the Manning roughness coefficient was determined by Eq. (3)

n = \frac{1}{v} R^{\frac{2}{3}} J^{\frac{1}{2}}

(3)

When the fixed-bed roughness is constant, the drag condition of the water flow can change under different slope conditions because of the change of the force angle between the particles and the water flow. The effect of the flow rate and gradient on the Manning roughness coefficient was relatively low. For the sake of analysis, the average values of the Manning roughness coefficients for five types of bed were 0.0107, 0.0159, 0.0435, 0.0512, and 0.0591.

Results and Discussion

Wave Shape and Types of Roll Waves

Liu et al. (2005) studied the roll-wave shape of the flow on slopes; their research outcomes are shown in Fig. 2(a). The roll wave is characterized by a steep front part and a gentle back part. The roll wave forms a periodic wave train under the application of gravity and friction forces. Fig. 2(b) shows the relationship between the water depth of a fixed cross section and time under the experimental conditions of #40-mesh water sand cloth, 5° of slope, and

5.87 L / \min

of flow. The figure shows that the roll wave exhibits apparent periodicity, and when the roll wave passed through the monitoring cross section, it took less time for the roll wave to reach the peak value of the water depth, but a longer time to drop to a lower value. This observation indicates that the roll wave propagating down the flume had a steeper front part and a gentler back part. The roll-wave shape is also not a smooth curve—the water depth fluctuated. Fig. 2(a), which is for reference only, describes the roll-wave form; the approximate shape is basically consistent with experimental observations. The wave velocity, period, wavelength, and other parameters were used to study the wave phenomenon. Because the depth of the roll wave is directly related to the shear force, the height of the roll wave is also listed as a characteristic parameter of the roll wave.

Fig. 2. Spatial and temporal morphological changes of rolling waves: (a) longitudinal profile of the roll waves and the definition of the related parameters; and (b) relationship between the water depth of the section and measuring time.

In an open-channel flow, the unsteady flow describes a change in the water surface or an increase or decrease in the flow somewhere in the channel; such a flow results in a wave phenomenon with a free surface (Lu et al. 2002). This wave is a gravity wave, also known as a running wave, a translation wave, or a displacement wave. Wherever the running wave goes, it will cause a change in the discharge and water level at the cross section. Such waves produced by open channels are different from roll waves. In a roll wave, when the water layer in the open-channel flow becomes thinner, the effects of the bottom-bed resistance and surface tension on the flow increase. The bottom-bed resistance is a necessary condition for the formation of a roll wave (Pan and Shangguan 2009). Roll waves in open channels are common in the flood discharge chutes of hydropower stations and in dam spillways, as well as in the flood channels of other bodies of water, such as a river or diversion canal. The boundary conditions of the flow on slopes are more complex than those of open-channel flow, but at present, only fixed-bed flume tests have been used to study the flow on slopes. During the propagation of the roll wave observed in the flume test, both the wave shape and the wave velocity evolved and could not be attenuated, resulting in a continuous waveform coarsening process in which the wave number continuously decreased and the wave steepness continuously increased, leading to the formation of a dynamic wave. The one-dimensional Saint-Vignan equation is usually used to describe flow on steep slopes (Liu et al. 2005).

Variation in the Roll-Wave Velocity

The velocity of a roll wave is the velocity at which it propagates down a slope. In the past, most studies have only considered the roll-wave velocity from the viewpoint of the average flow velocity of the slope; they seldom considered the influence of the roll-wave velocity on the slope confluence calculation and slope erosion. It is generally believed that, with increases in the discharge per unit width and slope, the average velocity increases according to a power function. However, as the roughness increases, the resistance increases, and the average velocity tends to decrease. The effect of the slope on the mean velocity is being debated. Velocity is defined as the power function of the discharge and the gradient, whether the Chezy formula for laminar flow or the Manning formula for turbulent flow is used. However, recent research conducted by Govers (1992) and Nearing et al. (1997) revealed that the flow velocity was independent of the slope.

The Reynolds number characterizes the ratio of the fluid inertial force to the viscous force; it is also the main index for measuring the turbulence intensity of a water body. However, the turbulence of the water flow is directly related to the erosion intensity of the thin-layer flow. For a wide and shallow open flow, the Reynolds number is more suitable for research than the unit discharge value because the Reynolds number can eliminate the disturbance caused by the variation of the water temperature. The relationship between the roll-wave velocities and the Reynolds numbers of selected sections is shown in Fig. 3.

Fig. 3. Relationship between roll-wave velocity and Reynolds number: (a) $S = 0.0872$ ; (b) $S = 0.1736$ ; (c) $S = 0.2588$ ; (d) $S = 0.3420$ ; and (e) $S = 0.4226$ .

Fig. 3 shows that when the energy slope and the roughness coefficient are different, the relationship between the wave velocity and the Reynolds number is different. When the energy slope and the roughness coefficient are both small, there is a positive correlation between the wave velocity and the Reynolds number. This is mainly because, when the Reynolds number increases, the weight of the flow inertial force and viscous force increases. The gravity component promotes the increase in the wave velocity, which results in an increase of the roll-wave velocity. However, when the roughness coefficient is greater than 0.0512, as the Reynolds number increases, the reaction force is applied to the resistance; that is, energy is provided and the resistance is consumed, and the increase of the wave velocity will take the second place. Further, we found that when the energy slope is relatively large, the roll-wave velocity fluctuates with the increase in the unit discharge, which is related to the roll waves merging. When the energy slope is greater, the flow state of the flow is more unstable, so that turbulent flow caused by a converging roll wave is more severe and the wave velocity becomes unstable. We found that the larger the energy slope, the more unstable the water flow and the more intense the turbulence, and therefore the greater the erosion intensity.

Fig. 4 shows that with increases in the energy slope, the roll-wave velocity gradually increases before stabilizing. The peak value appears at an energy slope of 0.3420. The main reason for this result is that with the increase of slope, the work from gravity component force increases. Because the fixed-bed condition cannot offset the increase of energy through the bed surface deformation, the gravity potential energy is mainly converted into kinetic energy. At the same time, the increasing rate of wave velocity decreases with the increasing roughness. When the roughness coefficient is 0.0107, and the discharge per unit width is

0.3113 L / (s \cdot m)

, the increase in the wave velocity reaches

0.45 m / s

. For a roughness coefficient of 0.0591 and a discharge per unit width of

0.3291 L / (s \cdot m)

, the increase in the wave velocity is the smallest at

0.05 m / s

. At this time, the effect of roughness on the velocity of the roll wave increases, and the effect of the energy slope on the velocity of the roll wave is weakened; when the roughness coefficient is greater than 0.0591, the energy slope has little impact on the velocity of the roll wave.

Fig. 4. Relationship between roll-wave velocity and energy slope: (a) $n = 0.0107$ ; (b) $n = 0.0159$ ; (c) $n = 0.0435$ ; (d) $n = 0.0512$ ; and (e) $n = 0.0591$ .

Fig. 5 presents the variation relationship between the velocity of the roll wave and the Manning roughness. The velocity of the roll wave decreases with increasing Manning roughness. With the increase of the energy slope, the decreasing rate of the roll-wave velocity gradually increases. Under a small energy slope, when the roughness coefficient increases from 0.0107 to 0.0159, the variation of the roll-wave velocity is apparent. When the roughness coefficient increases from 0.0159 to 0.0435, a threefold increase, the variation of the roll-wave velocity is minor. This observation indicates that 0.0159 might represent a critical value, after which the roughness coefficient has little impact on the roll-wave velocity.

Fig. 5. Relationship between roll-wave velocity and roughness coefficient: (a) $S = 0.0872$ ; (b) $S = 0.1736$ ; (c) $S = 0.2588$ ; (d) $S = 0.3420$ ; and (e) $S = 0.4226$ .

Variation of the Roll-Wave Height

During the propagation of the roll wave in the flow on slopes, as it is with uniform flow, the roll wave’s turbulent kinetic energy comes from the bottom friction; therefore, the turbulent kinetic energy can be deduced from the friction loss caused by the propagation of the wave energy. The calculation of the friction loss involves not only the wave velocity but also the wave height. Therefore, it is very important to study the variation of the roll-wave height to understand the sediment transport by a roll wave on a slope surface. The relationship between the wave height and the Reynolds number for the selected section is shown in Fig. 6.

Fig. 6. Relationship between roll-wave height and Reynolds number: (a) $S = 0.0872$ ; (b) $S = 0.1736$ ; (c) $S = 0.2588$ ; (d) $S = 0.3420$ ; and (e) $S = 0.4226$ .

Fig. 6 shows that when the roughness and the energy slope are constant and with increases in the Reynolds number, the water depth, effect of gravity on the roll wave, viscous resistance, and wave height all increase, while the force of the wall resistance decreases. When the Reynolds number reaches a certain value, the effect of the viscous resistance and friction on the roll wave is strengthened, and the wave height gradually decreases. Overall, the wave height exhibits a single hump-type of relationship relative to the Reynolds number. At the same time, with a small energy slope, as the roughness increases, the resistance increases, and the height of the roll wave corresponding to the same Reynolds number also gradually decreases. For instance, when the energy slope is 0.0872, the Reynolds number is 243, the roughness coefficient is 0.0107, and the wave height reaches 1.5 mm; when the roughness coefficient is 0.0591, the wave height is only 0.6 mm. This observation indicates that roughness mainly inhibits the height of the roll wave. However, for a larger energy slope with the increased roughness coefficient, the roll-wave height did not exhibit a monotonous decrease. This was primarily because the effect of gravity on the roll wave increased. The energy slope can also affect the variation of the roll-wave height. Fig. 7 presents the relationship between the roll-wave height and the energy slope under various roughness conditions.

Fig. 7. Relationship between roll-wave height and energy slope: (a) $n = 0.0107$ ; (b) $n = 0.0159$ ; (c) $n = 0.0435$ ; (d) $n = 0.0512$ ; and (e) $n = 0.0591$ .

Fig. 7 shows that when the roughness coefficients are 0.0107 and 0.0591, with an increasing energy slope, the height of the roll wave first exhibits a trend of increasing and then decreasing. The peak values of the roll-wave height corresponding to the conditions of the two roughness coefficients are 1.65 and 1.35 mm, respectively, and the range of the energy slope values corresponding to the peak values is between 0.1736 and 0.2588. Under other roughness conditions, the roll-wave height decreases monotonously with the increasing energy slope. We found that the greater the roughness, the smaller the influence of the energy slope on the height of the roll wave, the lower the decline rate, and the smaller the difference of the wave heights between different energy slopes. When the roughness is 0.0512, the impact of the discharge per unit width on the roll-wave height is relatively small for a constant energy slope.

Variation of the Roll-Wave Frequency

The frequency of the roll wave characterizes the number of roll waves per unit time passing through an observed section. For a constant roll energy, the higher the frequency, the more times the surface soil is impacted by the water flow, resulting in enhanced erosion. The relationship between the roll-wave frequency and the Reynolds number for a selected study section is presented in Fig. 8. We found that the variation of the roll-wave frequency with the Reynolds number is different under the same roughness condition. When roughness is 0.0107, the roll-wave frequency under various energy slope conditions gradually increases with the increases in the Reynolds number; however, with increases in roughness, the resistance of the bottom bed gradually increases, and the frequency fluctuates with increases in the Reynolds number.

Fig. 8. Relationship between roll-wave frequency and Reynolds number: (a) $S = 0.0872$ ; (b) $S = 0.1736$ ; (c) $S = 0.2588$ ; (d) $S = 0.3420$ ; and (e) $S = 0.4226$ .

Table 1 provides the values of the frequency and the Reynolds number for a roughness coefficient of 0.0107. The roll-wave frequency shows an increasing tendency with increases in the Reynolds number. With the increase of the energy slope, the amplitudes of the roll-wave frequency are 0.66, 0.10, 0.13, 0.29, and 0.33, and the amplitude exhibits a tendency of decreasing first and then increasing. The Reynolds number has the largest impact on the frequency when the energy slope is 0.0872.

Table 1. Relationship between the roll wave frequency and Reynolds number under a roughness coefficient of 0.0107

Energy slope	Hydraulic parameters	Average discharge per unit width, $L / (s \cdot m)$
Energy slope	Hydraulic parameters	0.1712	0.2240	0.2719	0.3113	0.3476	0.3799
0.0872	Reynolds number	126	160	192	218	243	265
0.0872	Frequency	1.10	1.24	1.33	1.56	1.58	1.76
0.1736	Reynolds number	122	167	203	232	260	285
0.1736	Frequency	1.82	1.87	1.88	1.88	1.85	1.92
0.2588	Reynolds number	161	203	239	272	300	328
0.2588	Frequency	2.05	2.19	2.13	2.13	2.12	2.18
0.3420	Reynolds number	142	194	236	271	305	328
0.3420	Frequency	2.27	1.93	2.20	2.25	2.34	2.56
0.4226	Reynolds number	144	185	236	273	306	338
0.4226	Frequency	2.11	2.25	2.38	2.38	2.57	2.44

Fig. 9 shows the relationship between the roll-wave frequency and the energy slope. Under various roughness conditions, with the increase of energy slope, the roll-wave frequency increases at first and then decreases. This observation is consistent with the research outcomes achieved by Pan and Shangguan (2009) in his flume test. The energy slope value corresponding to the peak value of the roll-wave frequency is around 0.3420. It is mainly related to the variation of the gravity component and resistance. When the roughness coefficient increases to 0.0591, the roll-wave frequency exhibits a decreasing tendency, with the energy slope ranging from 0.0872 to 0.1736; this result was mainly related to the variations in the water depth. Benjamin (1957) found that greater water depths resulted in more superimposed waves than the surface water flow could carry. Therefore, when the roughness coefficient is between 0.0107 and 0.0512, as the energy slope increases from 0.0872 to 0.1736, the decrease in the water depth is small. The roll wave is not impacted by the bottom-bed resistance but is apparently affected by the energy slope. Therefore, the roll-wave frequency increases. When the roughness coefficient is 0.0107, as the energy slope increases from 0.0872 to 0.1736, the change in the water depth is large. The roll wave is significantly impacted by the bottom-bed resistance, and the roll-wave frequency exhibits a decreasing tendency. After that, the effect of the energy slope on the frequency of the roll wave is strengthened, and the frequency of the roll wave begins increasing again.

Fig. 9. Relationship between roll-wave frequency and energy slope: (a) $n = 0.0107$ ; (b) $n = 0.0159$ ; (c) $n = 0.0435$ ; (d) $n = 0.0512$ ; and (e) $n = 0.0591$ .

Variation in the Roll Wavelength

When the ratio of the wavelength to the water depth is greater than 20, the whole body of water can be disturbed by waves. In this case, the water waves are called shallow waves. Unsteady flow in many rivers and channels is a long wave in shallow water. The fluctuation of the shallow water wave affects the entire water column. Therefore, the impact of the resistance must be considered. The resistance is nonlinear, and the water flow carrying sediments must also be considered when determining the impact of the roll wave (Ivanova et al. 2017).

The roll wave produced by the flow instability on the slope surface is also a shallow water wave. The study of the roll wavelength of the thin-layer flow on the slope surface is the basis for the study of the wave steepness of the roll wave. The roll-wave steepness is the basis for calculating the roll-wave energy; it provides a theoretical foundation for the study of the sediment-carrying mechanism on a slope. We took into consideration the influence of the wavelength on the water depth and analyzed the ratio of the wavelength to the depth. The average value of the ratio of the wavelength to the water depth under the conditions of six different discharges per unit width is shown in Table 2. We found that the ratio varies between 39 and 342; the wave motions under the experimental flow condition belong to the shallow-water long-wave category. Additionally, we found that under equivalent roughness conditions, this ratio gradually increases as the slope increases. The aforementioned results indicate that with a smaller roughness and a larger slope, the larger the ratio, the larger the impact of the roll wavelength on the water depth, and the larger the impact on the sediment transport. The relationship between the ratio of the wavelength to the water depth and the Reynolds number for Section 6-6 under various energy slope conditions is presented in Fig. 10.

Table 2. Distribution table for the ratio of the wavelength to the water depth

Energy slope, $S$	Roughness coefficient, $n$
Energy slope, $S$	0.0107	0.0159	0.0435	0.0512	0.0591
0.0876	218	201	99	70	39
0.1736	227	201	103	74	56
0.2588	238	227	146	74	58
0.3420	277	295	168	101	47
0.4226	342	303	192	106	71

Fig. 10. Relationship between the ratio of the wavelength to the water depth and the Reynolds number: (a) $S = 0.0872$ ; (b) $S = 0.1736$ ; (c) $S = 0.2588$ ; (d) $S = 0.3420$ ; and (e) $S = 0.4226$ .

Fig. 10 shows that when the roughness coefficient is 0.0107 and 0.0159, the ratio is greatly affected by the Reynolds number. With an increasing Reynolds number, the ratio of the wavelength to the water depth gradually decreases at a decreasing rate. When the roughness coefficient is greater than 0.0159, the ratio of the wavelength to the water depth varies little with the Reynolds number, especially for energy slopes ranging between 0.0876 and 0.1736. This result indicates that with a larger roughness and a smaller energy slope, the Reynolds number has little impact on the ratio of the wavelength to the water depth.

Under natural conditions, many factors affect the roll wave of the flow on slopes, and the process of energy conversion during water flow becomes complicated. For instance, during infiltration, the shear force on the near-bottom bed will be increased. The resistance of the flow on the slopes will be increased, which will intensify the instability of the water surface and result in a more dramatic evolution of the roll wave. Because of the limitations imposed by the experimental conditions in this study, the characteristics of the flow evolution and factors associated with the slope surface were discussed only under fixed-bed conditions. In the future, we will research on the quantitative description of the roll-wave evolution and the sediment-driving mechanism.

Conclusions

The following conclusions were obtained:

1.

When the energy slope and roughness are small, there is a positive correlation between the wave velocity and the Reynolds number. The wave velocity increased first with the increase of the energy slope, and then tended to be stable. It reached a peak value at an energy slope of 0.3420. With the increasing bed roughness, the impact of the Reynolds number and the energy slope on the velocity of the roll wave gradually decreased.

2.

Increasing values of the Reynolds number resulted in the wave height of the roll wave exhibiting a single-hump type of change. When the energy slope was small, the roughness inhibited the increase of the roll-wave height. When the roughness ranged between 0.0159 and 0.0512, the roll-wave height presented a monotonous decrease with the increasing energy slope, and the effect of the energy slope on the roll-wave height was apparent.

3.

When the roughness coefficient was 0.0107, the roll-wave frequency increased with the increasing Reynolds number; however, under conditions of a high roughness, the roll-wave frequency fluctuated with increases in the Reynolds number. As the energy slope increased, the roll-wave frequency first increased, then decreased. It reached its peak value at an energy slope of 0.3420.

4.

The ratio of the roll wavelength to the water depth produced under the experimental conditions ranged between 39 and 342, and it belonged to shallow-water long waves. The ratio increased with the increasing slope and decreased with the increasing roughness. Under low-roughness conditions, the ratio gradually decreased with increases in the Reynolds number. Under high roughness conditions, the ratio varied little with changes in the Reynolds number.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant No. 51209222) and the indigenous programs of the National Key Laboratory of Simulation and Regulation of Water Cycle in River Basins (Grant Nos. 2015ZY01 and 2015TS01).

References

Benjamin, T. B. 1957. “Wave formation in laminar flow down an inclined plane.” J. Fluid Mech. 2 (6): 554–573. https://doi.org/10.1017/S0022112057000373.

Abstract

Introduction

Method

Results and Discussion

Wave Shape and Types of Roll Waves

Variation in the Roll-Wave Velocity

Variation of the Roll-Wave Height

Variation of the Roll-Wave Frequency

Variation in the Roll Wavelength

Conclusions

Acknowledgments

References

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