Abstract

Overland flow is the initial driver of slope surface erosion. To discover resistance characteristics of overland flow influenced by rainfall intensity and roughness, indoor simulated rainfall experiments with six kinds of roughness, five flow discharges, and five rainfall intensities were investigated. Results showed that overland flow over rough surfaces could be considered as laminar and turbulent flow when using flow Reynolds number. According to roll waves observed, flow regimes belonged to the laminar transitional zone based on the viscosity-to-depth ratio. A critical water depth formula for overland flow was re-derived, and it showed that this test water flow consisted of supercritical flow in most cases, and subcritical flow in only a few cases. The flow resistance coefficient increased with increasing roughness, whereas it decreased as rainfall intensity increased. Considering the ‘increasing resistance’ phenomenon, this study focused on frictional resistance, thickness of the viscous sublayer, pressure drag and roll waves. Finally, a formula for sheet flow resistance was proposed based on resistance segmentation and multi-element linear regression. These findings are of significance both for understanding the characteristics and development of overland flow and overland flow dynamics.

INTRODUCTION

Overland flow, which is formed by rainfall and snowmelt, is considered as a shallow open-channel flow moving along a slope under the influence of gravity. It is the initial driver for soil erosion evolution and an important factor causing soil destruction, transport, and deposition (Zhang et al. 2017). The depth of overland flow is generally less than a few millimeters, and the flow direction is unstable as there are continuous mass and momentum sources along the flow path (Dunkerley 2010; Ali et al. 2012). Moreover, overland flow is highly sensitive to many factors, including underlying surface conditions, splashing raindrops, and slope gradient, making it more complex and worthy of further study.

Currently, research on overland flow has focused on flow patterns and resistance characteristics. In previous studies, three flow regimes have been identified: laminar, transitional, and turbulent flow (Abrahams et al. 1986; Li et al. 1996; Wong & Chen 1997). However, there are still no accurate conclusions about flow patterns. Peng et al. (2011, 2013) adopted a creative numerical method named the lattice Boltzmann method to study shallow flows. Woolhiser et al. (1970) proposed that laminar flow is the main flow pattern of overland flow by analyzing data for a small pasture watershed during several rainstorms. The results of Pan & Shangguan (2006) and Dunkerley (2010) showed that flow patterns of overland flow could be classified into laminar subcritical flow when the underlying surface was covered with vegetation. However, overland flow is known to have the capacity to transport sediment, meaning that applying laminar flow theory may be inappropriate. Therefore, in order to distinguish overland flow from open-channel laminar flow, Horton (1945) and Selby (1993) put forward the concept of ‘mixed laminar flow’. They agreed that slope sheet flow is a mixed flow zone with laminar flow and turbulent flow. However, Emmett (1978) found that overland flow is not completely consistent with laminar flow, transitional flow, and turbulent flow in the traditional sense, defining it as ‘disturbing flow’. Wang et al. (2012) agreed that it is difficult for overland flow to maintain laminar flow, but this is due to the transition of rolling wave flow to transitional flow in advance. Wang et al. (2018) further stated that sheet flow starts with transitional flow at the surface and finishes with turbulent flow underneath.

Regarding the resistance laws of sheet flow, flow resistance is one of the most important aspects in the study of overland flow (Abrahams & Li 1998; Smith et al. 2007). It is sensitive to multiple factors including flow velocity, flow pattern, surface conditions, sediment deposition, and tillage (Parsons et al. 1994; Lawrence 1997; Lane 2010). The effect of rain intensity on resistance is particularly controversial and has been considered as a critical factor with regard to runoff and erosion (Cerda 2002; Ziadat & Taimeh 2013). Similarly, Savat (1980) found that the ratio of rainfall resistance to total resistance is as high as 1:5 when flow on a slope is gentle and laminar, which is clearly an important constituent part. Parsons et al. (1994) proposed that rainfall resistance may provide a substantial source of resistance due to low water depth and velocity near the drainage watershed. Yoon & Wenzel (1971) and Shen & Li (1973) stated that resistance of sheet flow increased with rainfall intensity when the Reynold number (Re) < 2,000, whereas rainfall exerted little influence when Re > 2,000. Emmett (1978) argued that flow resistance would be double under rainfall conditions. However, Dunne & Dietrich (1980) found that rainfall resistance constitutes a negligible proportion of total resistance, so it can be neglected.

With respect to the derivation of resistance formula, it has been widely accepted that resistance of different kinds equals to the sum of individual components of resistance (Rauws 1988; Weltz et al. 1992; Parsons et al. 1994; Hu & Abrahams 2005). Li (2009) considered surface objects (e.g., stone or pebbles) and rainfall as the roughness elements, and applied a linear superposition approach to define composite resistance. Similarly, Abrahams & Parsons (1994) stated that resistance to overland flow could be partitioned into several categories including grain resistance, form resistance, wave resistance, and rain resistance. Hu & Abrahams (2006, 2010) proposed that resistance should be broken up into four components without simulated rainfall: surface resistance fs, form resistance ff, wave resistance fw, and bed-mobility resistance fm. In Shen & Li (1973)'s study, the Darcy–Weisbach drag coefficient when Re < 900 was expressed as the sum of the drag coefficient at ‘no rainfall’ conditions (λ0) and the drag coefficient at ‘simulated rainfall’ conditions (λR). However, there are no consistent explanations of an appropriate formula. Nevertheless, exploring overland flow resistance is critical for better understanding the hydrodynamic mechanisms of soil erosion and the characteristics of overland flow.

The flow hydraulics experiments reported here used six kinds of granular surfaces with varying slope gradient and simulated rainfall characteristics. The objectives of the study were: (1) to find a better way of identifying overland flow regimes by comparing different methods; (2) to partition the contributions of rainfall intensity and roughness to total resistance; and (3) to derive a formula for overland flow resistance based on resistance segmentation.

MATERIALS AND METHODS

Experimental apparatus

This study was designed on fixed beds, which not only ensured that water flow had no effect on the bed surface, but also controlled surface roughness easily. In order to make the roughness of the bed surface uniform, artificial water sand cloth and screened quartz sand were pasted onto it. This experimental set-up consisted of a test flume, water supplier, vortex stabilizer, rainfall system, and flow-measuring devices (Figure 1). The experimental flume area was 6.0 m long, 0.5 m wide, and 0.25 m deep, with plexiglass sides. The adjustable range of the water flow feeder was 0–12, 5–15, 5–20, and 10–40 L/min. A stationary, manual rainfall device (QYJY-503) was used in this study. The placement height of the rain sprinkler was 18 meters, and the diameter of raindrops could be adjusted between 0.4 and 6 mm. At the same time, the uniformity of simulated rain was greater than 0.80, and the precision range of adjustable rain intensity was 30–300 mm/h. This experiment was operated with a SX402 digital probe tester (Chongqing Hydrological Instrument Factory, Chongqing, China) to measure the water depth in different conditions, with 0.01-mm precision.

Figure 1

Schematic of the experimental set-up.

Figure 1

Schematic of the experimental set-up.

Experimental design

Two series of trials on granular surfaces and smooth surfaces were conducted under artificial simulated rainfall conditions, for a total of 145 tests. The experimental flume was adjusted at 15° to reflect a common gradient in the Loess Plateau region. Six kinds of bed conditions were used including: a smooth surface (equivalent roughness = 0.009 mm); a 240 mesh, 120 mesh, and 24 mesh gauze bed (particle sizes = 0.061, 0.120, and 0.700 mm, respectively); and two kinds of sand-pasted surfaces (particle size = 1.770 and 3.680 mm, respectively). In addition, five unit discharges (discharge per unit width) were tested (0.167, 0.333, 0.500, 0.667, and 1.000 L/(s·m)) to reflect the range of rainfall intensity associated with erosion on the Loess Plateau. Also, these designed unit discharges ensured that the water depth was tested within a wide range. Based on the natural rainfall intensities in the Loess Plateau region, rainfall was simulated as: Ri = 0 mm/h, Ri = 60 mm/h, Ri = 90 mm/h, Ri = 120 mm/h, and Ri = 150 mm/h. Since the roughness of the slope did not function under smooth conditions, and was significantly different from natural slope conditions, the rain intensity of 150 mm/h was excluded. There were five longitudinal cross sections for observation in the flume, positioned at 1.0, 2.0, 3.0, 4.0, and 5.0 m along the slope from top to bottom. For each cross section, five measurement points were set transversely to observe the surface velocity using the high-definition photography method with KMnO4 which has a higher precision than traditional methods (Roels 1984; Gilley et al. 1990). An SX402 digital probe tester with 0.01-mm precision was used to measure water depth.

Data measurement and analysis

  1. According to the continuity equation of flow, the average velocity of a cross section (u) can be calculated as: 
    formula
    (1)
    where u is the average velocity of the cross section (m/s); q is unit discharge (m2/s); and h is average water depth (m).
  2. The hydraulics dual flow Reynolds number Reh can be expressed as: 
    formula
    (2)
    where R is the hydraulic radius (m); h is depth; and v is the viscosity coefficient of flow (cm2/s).
    ν can be calculated by the Poiseuille equation as follows: 
    formula
    (3)
    where t is water temperature.
  3. The Froude number is often used to judge the flow pattern in open-channel flow, and can be expressed as follows: 
    formula
    (4)
    where g is the acceleration due to gravity.
  4. Friction velocity u* is calculated as: 
    formula
    (5)
    where g is the acceleration of gravity (=9.81 m/s2); R is the hydraulic radius, approximated as the depth h; and J is the hydraulic gradient approximated as sin θ, and θ was the gradient of the sink.
  5. The flow Reynolds number Red is calculated as: 
    formula
    (6)
    where ks is the bed surface roughness (mm). For this, when the bed surface is composed of uniform sand, the average particle size is taken.
  6. The thickness of the viscous sublayer δ is calculated as: 
    formula
    (7)
  7. The Darcy–Weisbach resistance factor λ can be calculated as: 
    formula
    (8)
    where h is water depth (m); u is mean velocity (m/s); and g is acceleration due to gravity (m/s2).

RESULTS AND DISCUSSION

Flow regime

Flow zone

In this study, we considered overland flow as open-channel flow, and compared a traditional flow regime discriminant method with the roll-wave discriminant method proposed by Zhang et al. (2011). First, 145 sets of test points could be separated from the lower Reynolds number (580) and the upper Reynolds number (6,500), and the relationship between Reynolds number and resistance coefficient of sheet flow was plotted in Figure 2.

Figure 2

The relationship between Reynolds number and resistance coefficient.

Figure 2

The relationship between Reynolds number and resistance coefficient.

As can be seen from Figure 2, most experimental data were distributed on the left side of the laminar boundary. A few sample points were located between the laminar boundary and the turbulent boundary. It could be considered that laminar flow and transitional flow all occurred. There was no turbulent flow, and with increasing rainfall intensity, these points moved towards the transition zone. Splashing raindrops could be a main cause of this phenomenon as this made the streamline unstable. Raindrops also accelerated the flow velocity of overland flow and decreased the water depth, resulting in a lower flow resistance and more turbulent water flows. However, since there was still a transitional flow zone when the rainfall intensity was 0 mm/h, it cannot be simply considered that the rainfall was the only cause of transitional flow. During this test, the water surface was destabilized and formed waves when the hydraulic parameters reached a critical value. They had steep wave fronts and gentle wave rears that move along the slope intermittently. As they moved in the manner of a snowball, it is called ‘rolling-wave flow’ in hydraulics. Therefore, the reason why it was difficult to maintain laminar flow could be explained by the view of Wang et al. (2012), that is, due to the early transition of rolling-wave flow to transitional flow.

Zhang et al. (2011) proposed a new method for defining the flow regime of sheet flow by using a critical parameter δ/h, which defines the ratio of the thickness of the viscous sublayer to the water depth. The laminar instability zone (rolling-wave flow zone) is defined as δ/h ≤ 0.12. Next, experimental data were analyzed from the perspective of roll waves, as shown in Figure 3.

Figure 3

The relationship between δ/h and resistance coefficient.

Figure 3

The relationship between δ/h and resistance coefficient.

As shown in Figure 3, all curves for various conditions fluctuated on the right side of the critical parameter δ/h = 0.12, which defines the laminar instability zone (rolling-wave flow zone). There were differences in results when using different judgments to determine the flow regime of overland flow on this slope. These two methods could be physically compared as follows:

The inertial force F can be expressed as m multiplied by acceleration a: 
formula
(9)
From a dimensional standpoint: 
formula
(10)
where the viscous force T can be defined using Newton's law of friction: 
formula
(11)
From a dimensional standpoint: 
formula
(12)
The dimension of the ratio between inertia force and viscous force was as follows: 
formula
(13)
where v is the characteristic velocity; L is the characteristic length; and ν is expressed as the coefficient of kinematic viscosity.

The Reynolds number was shown in Equation (2) and its dimension was shown in Equation (13). That is, the physical meaning of the Reynolds number is the ratio of inertial force to viscous force.

The ratio of viscosity to depth is calculated as: 
formula
(14)

Its dimension is the reciprocal of the Reynolds number dimension, thus the physical significance of the viscosity-depth ratio can be expressed as the ratio of viscous force to inertial force. When the Reynolds number is significantly small or δ/h is relatively large, viscous force has a larger effect whereas inertial force has a smaller effect, and viscous force has an inhibitory effect on the motion water particles in sheet flow. In contrast, when the Reynolds number is relatively large or δ/h is significantly small, inertial force has a larger effect and viscous force has a smaller effect. Inertial force drives particles in the thin-layer flow to move downward and intermix. It can therefore be seen that the Reynolds number and viscosity–depth ratio have the same physical essence.

However, experimental points of sheet flow were all above the laminar flow line and the smooth transition line, which can be called an ‘increasing resistance’ phenomenon of overland flow. This phenomenon is not the same as that in open-channel flow which indicates that the resistance coefficient λ in the laminar flow and transitional areas has nothing to do with bed roughness ks, but is only determined by the Reynolds number. It is similar to the resistance characteristics of the turbulent zone in open-channel flow, meaning that sheet flow exhibits turbulent characteristics within the laminar flow zone, as well as in the transitional flow zone. Therefore, determining the flow regime using the hydraulic method remains controversial. In this study, it is suggested that the method based on the viscosity-depth ratio is superior to the traditional Reynolds number discriminant method. The main reason for this is that the former can reflect movements of overland flow more directly, and it is considered that roll waves can easily develop in flume experiments (Yoon & Wenzel 1971; Emmett 1978), especially when flow is barely impacted by roughness elements (Rouse 1938). Rolling waves are forms of energy expenditure and add further resistance to the flow (Li 2009). Hence, it is reasonable to analyze flow patterns of sheet flow and ‘increasing resistance’ phenomenon from the prospective of roll waves. Future research should focus on roll waves, such as the critical conditions for the generation and disappearance of roll waves, wave height, wave length, and wave frequency, to improve the roll wave discriminant method and to better explain the hydrodynamic characteristics of sheet flow.

Flow patterns

With respect to the different flow patterns of sheet flow, previous studies have often distinguished these using Fr = 1 as a critical value. However, it is noteworthy that the method using the Froude number is only applicable to the condition of ‘micro-waves’, but in the process of this experiment, it often appeared as ‘rolling-wave flow’. Moreover, overland flow is easily affected by a series of factors, such as surface tension and velocity distribution. Therefore, judging the flow patterns of sheet flow from the perspective of a Froude number is not appropriate. In this study, critical water depth was used to identify patterns of sheet flow. Because the deduction process of the critical depth formula in open-channel flow is not suitable for slope flow, a formula related to critical depth of sheet flow was deduced as follows: 
formula
(15)
where h = average water depth (m); θ= gradient; α= correction coefficient of kinetic energy; and g = acceleration of gravity (=9.81 m/s2). In open-channel flow, cos θ ≈ 1 is often used as the downslope has a shallow gradient. However, for sheet flow, cos θ ≈ 1 cannot be used because of a steeper slope gradient.
Critical water depth (hk) represents the water depth corresponding to the minimum specific energy of cross sections when flow charge, cross sections' shape and size are determined. The derivation of h is as follows: 
formula
(16)
where Q= flow charge; and A= the wetted cross-sectional area. Because the width of the water surface B = dA/dh, Equation (16) can be simplified to: 
formula
(17)

When dEs/dh < 0, h<hk, cos θFr2 < 0, that is, water flow can be defined as torrent flow. Conversely, when dEs/dh > 0, h>hk, cos θFr2 > 0, that is water flow can be defined as subcritical flow. When dEs/dh = 0, h=hk, that is , water flow can be defined as critical flow.

To determine critical depth hk, it is assumed that dEs/dh = 0, therefore: 
formula
(18)
Thus: 
formula
(19)
Because the cross section of the experimental fume was rectangular, the width of the bottom b = Bk. Ak = bhk, which can be obtained by combining Equations (18) and (19) as follows: 
formula
(20)
where q= unit charge and q= Q/b. Generally, the correction coefficient for kinetic energy ranges from 1.4 to 1.6. Considering that the velocity distribution was uniform in this experiment, it can be simplified as wide-shallow flow, therefore α is equal to 1.5. The results of this experiment are shown in Figure 4 according to Equation (20).
Figure 4

The relationship between water depth and critical depth.

Figure 4

The relationship between water depth and critical depth.

It can be seen from Figure 4 that overland flow rarely existed in the form of subcritical flow, and most experimental points were located in the torrent zone. Figure 4(a) shows that experimental points move downward to the right with increasing roughness. That is, points move towards the subcritical flow direction of hk/h < 1, hk<h. Figure 4(b) shows that experimental points tended to move upward to the left with an increase in rainfall intensity. That is, experimental points develop towards the torrent direction of hk/h > 1, hk>h, but this trend is not obvious in Figure 4(b), indicating that roughness played a more important role in this process compared to rainfall intensity. Compared with the open-channel formula, this critical depth formula considers the effect of slope and is more rigorous than methods using Fr. Moreover, the critical depth formula can more intuitively depict hydraulic resistance laws of overland sheet flow under different conditions. Therefore, for overland flow, the critical depth method is clearly superior to the Froude number method. In this experiment, flow patterns were studied only when the gradient was 15°. However, hydraulic characteristics also vary with the slope. Therefore, it is necessary to focus on resistance laws of sheet flow under gentle slope conditions, so as to enrich and improve current understanding.

Influence of roughness on overland flow resistance

Figure 3 shows that when the rainfall intensity was certain, the resistance coefficient was negatively correlated with the Reynolds number, so the regression parameter b in obtained by previous quantitative simulation is negative (Roels 1984; Abrahams & Parsons 1994; Ogunlela & Makanjuola 2000). Because these experimental data were essentially parallel to the open-channel laminar streamline, the value b approximated as −1. The resistance coefficient increased with increasing bed roughness, which shows that roughness had an ‘increasing-resistance’ effect. Without considering the flow patterns of the test water, it can be found that if rainfall intensity and flow discharge are certain, the resistance at a roughness of ks = 3.680 mm was not necessarily greater than that at a roughness of ks = 1.770 mm. The reason for this may be that when the roughness was 3.680 mm, the average height of sand particles on the bed was greater than water depth. This would mean that the submergence ratio is less than 1, and the water flow was submerged flow. Under this condition, the shape, as well as size of sand particles had more complex effects on the flow. Some researchers classified flow resistance on rough beds into particle resistance and form resistance, and these may change in the process of increasing roughness, so that the positive correlation between roughness and resistance coefficient may not be established indefinitely (Abrahams & Parsons 1994; Hu & Abrahams 2006).

The mechanism of increasing resistance is related to frictional resistance, the thickness of the viscous sublayer, pressure drag, and roll waves. As for roll waves, the occurrence of waves leads to an increase in the resistance coefficient, which is similar to the hydraulic jump in open channel flow, as seen in Figure 5. Savat (1980) also captured rolling waves in the laboratory and found that when rolling waves occurred, the cross section of water suddenly increased and the height of rolling waves was twice the average water depth. In the process of rolling wave migration, liquid particles on the surface and at the bottom were mixed with each other, resulting in high turbulence intensity. It also causes uneven cross section velocity distribution and large enegy loss, leading to a larger coefficient. In this paper, the mechanism of increasing resistance was mainly analyzed from three aspects: (1) frictional resistance, (2) the thickness of viscous sublayer, and (3) pressure drag.

Figure 5

Roll waves under a rough surface.

Figure 5

Roll waves under a rough surface.

Frictional resistance

Table 1 lists the mean water depth under different roughness conditions at a given rainfall intensity to illustrate the ‘drag-increasing’ effect of frictional resistance on the side walls. It can be seen that water depth increased with increasing roughness, so that the contact area between water flow and the side walls increased. For example, the water depth h increased from 1.96 mm to 3.24 mm when the roughness rose from 0.009 mm to 3.680 mm. That means that the energy consumed in the working of water flow against resistance tended to increase, which directly led to the increasing flow resistance.

Table 1

The changing law of parameter h under different conditions of roughness and rainfall intensity

Roughness (mm) Rainfall intensity (mm/h)
 
60 90 120 150 
0.009 1.96 1.67 1.51 1.46 
0.061 2.04 1.86 1.90 1.92 1.82 
0.12 2.23 2.24 2.14 2.13 2.10 
0.7 2.92 2.93 2.96 2.96 2.77 
1.77 3.26 3.17 3.15 3.11 3.08 
3.68 3.24 3.19 3.11 3.04 3.21 
Roughness (mm) Rainfall intensity (mm/h)
 
60 90 120 150 
0.009 1.96 1.67 1.51 1.46 
0.061 2.04 1.86 1.90 1.92 1.82 
0.12 2.23 2.24 2.14 2.13 2.10 
0.7 2.92 2.93 2.96 2.96 2.77 
1.77 3.26 3.17 3.15 3.11 3.08 
3.68 3.24 3.19 3.11 3.04 3.21 

Thickness of the viscous sublayer

Figure 6 illustrates the relationship between the resistance coefficient and the thickness of the viscous sublayer under certain rainfall intensities. The aim here was to explore the effect of the thickness of the viscous sublayer on the ‘drag-increasing’ mechanism.

Figure 6

Relationship between underlying thickness of the viscous sublayer and the resistance coefficient under different conditions of roughness.

Figure 6

Relationship between underlying thickness of the viscous sublayer and the resistance coefficient under different conditions of roughness.

As can be seen from Figure 6, when the rainfall intensity was constant, curves tended to move upwards towards the left with increasing roughness. That is, the greater the roughness, the smaller the thickness of the viscous sublayer, leading to a greater resistance coefficient. The ‘drag-increasing’ phenomenon in this experiment may be due to changes in the thickness of the viscous sublayer; when the sublayer is thick enough to cover the roughness of the bed (making it similar to the smooth surface), roughness will have no effects on the resistance coefficient. Conversely, when it becomes too thin to cover the roughness of the bed, rough elements will emerge and extend into the mainstream, which is equivalent to the flow of water on uneven surfaces; streamlines become displaced, making the flow path increase; water flow is more chaotic causing energy loss, meaning the resistance coefficient increases.

Pressure drag

Pressure drag is closely related to the Reynolds number. In order to study the effect of pressure drag on the mechanism of ‘drag-increasing’, the double logarithmic relationship between drag coefficient and Reynolds number was plotted for different rainfall intensities and different roughness conditions, as shown in Figure 7.

Figure 7

Relationship between turbulent flow Reynolds number and resistance coefficient under different conditions of roughness.

Figure 7

Relationship between turbulent flow Reynolds number and resistance coefficient under different conditions of roughness.

With an increase in turbulence intensity, a trailing vortex appears and the pressure on the upstream and downstream surfaces of particles are different, resulting in pressure drag, also known as form resistance. It can be seen from Figure 7 that the flow around Reynolds number increased with increasing bed roughness except for the bed condition ks = 3.680 mm, which showed a decline in the viscous resistance. Therefore, the turbulence intensity increased, meaning the pressure differences between the upstream and downstream particles increase. Pressure drag (a component of total flow resistance) accordingly rises, resulting in an increase in the resistance coefficient. When ks = 3.680, this study argues that the height of roughness elements was sufficiently high so as not to be fully inundated by the flow. Lawrence (1997) said that the degree to which the roughness elements are inundated plays a dominant role in determining the resistance offered to the flow by the surface. Therefore, the condition ks = 3.680 had little effect on the ‘drag-increasing’ mechanism.

Influence of rainfall intensity on overland flow resistance

Relationship between resistance coefficient and rainfall intensity

It can be seen from Figure 8 that the relation is similar to that of Figure 2. When the roughness is constant, the resistance coefficient is negatively correlated with the Reynolds number, and it decreases with an increasing Reynolds number. With an increase of rainfall intensity, the distance between each point and laminar flow line, as well as smooth transition line decreases, indicating that the resistance coefficient of water diminished. That is, rainfall had a ‘reducing-resistance’ effect, but compared with smooth surface conditions, this effect was not obvious on artificially roughened bed surfaces. This indicates that bed roughness had a more important role in flow resistance than rainfall intensity.

Figure 8

Relationship between resistance coefficient and Reynolds number under different conditions of rainfall intensity (results for ks = 0.170, 0.700, 1.770, and 3.680 are not shown as the results were almost the same as for ks = 0.061).

Figure 8

Relationship between resistance coefficient and Reynolds number under different conditions of rainfall intensity (results for ks = 0.170, 0.700, 1.770, and 3.680 are not shown as the results were almost the same as for ks = 0.061).

Mechanism of ‘reducing resistance’ by rainfall intensity

It can be seen from Figure 8 that the effect of roughness on the resistance coefficient was much greater than that of rainfall intensity. Therefore, the mechanism for the ‘reducing-resistance’ effect of rainfall was examined by comparing simulated rainfall tests with a smooth surface and roughened bed. Figure 9 shows the relationship between water depth and the resistance coefficient under different rainfall intensities when the bed roughness is constant.

Figure 9

Relationship between water depth and the resistance coefficient under different conditions of rainfall intensity.

Figure 9

Relationship between water depth and the resistance coefficient under different conditions of rainfall intensity.

From Figure 9, there was a considerable drop in the drag coefficient with increasing rainfall intensity when roughness is constant. However, under the condition of artificial roughness, the decline of the drag coefficient is obviously less than that of smooth surface. Under smooth conditions, water depth had a decreasing trend with increasing rainfall intensity, that is, the curves move to the left on the graph. However, under the condition of artificial roughness, it is difficult to see that water depth decreased with increasing rainfall intensity.

For the interpretation of the relationship between rainfall intensity and water depth, this was due to the rainfall intensity which promotes and speeds up downslope water flow. According to the continuity equation, when the flow charge is constant, there is a decline in water depth with increasing flow velocity. This speculation is consistent with the view of Pan et al. (2010), that when a slope is greater than six degrees (the slope of this test was 15 degrees), rainfall increases the surface velocity to varying degrees. This is due to the larger raindrop momentum in the downslope direction. According to the momentum theorem, when the velocity of raindrops falling on the bed surface is larger than the surface velocity, the surface velocity of water will increase and the velocity of raindrop will decrease until the speed of the two drops is consistent. The formula of resistance coefficient can be expressed by: 
formula
(21)

It can be seen that when R (approximately = h) decreases and u increases with rainfall intensity, there is an apparent decline in λ. That is, rainfall can reduce the drag coefficient theoretically, which is consistent with the experimental results. The reason for a significant decline in drag coefficient on the smooth bed is that roughness has a more obvious influence on the drag coefficient than rainfall intensity. When a bed is smooth, the ‘reducing-resistance’ effect of rainfall intensity has a greater influence on the drag coefficient. When the bed is rough, the ‘increasing resistance’ effect of roughness is dominant, and rainfall intensity becomes secondary. In addition, as shown in Figure 9, under certain conditions of slope angle, flow charges, and rainfall intensity, water depth of the smooth bed surface was smaller than that of the rough bed surface. Therefore, water flow on the smooth surface was more sensitive to the ‘reducing-resistance’ effect of rainfall intensity, leading to a smaller drag coefficient than the rough bed.

In addition, analyzing the effect of rainfall intensity on the drag coefficient considering the thickness of the viscous sublayer is helpful to understand the ‘reducing-resistance’ mechanism of rainfall. Figure 10 depicts the relationship between the thickness of viscous sublayer and the resistance coefficient under different rainfall intensities and with certain roughnesses.

Figure 10

Relationship between underlying thickness of the viscosity and the resistance coefficient under different conditions of rainfall intensity.

Figure 10

Relationship between underlying thickness of the viscosity and the resistance coefficient under different conditions of rainfall intensity.

As can be seen from Figure 10, the thickness of the viscous sublayer increased and drag coefficient decreased with increasing rainfall intensity with smooth conditions. That is, curves move downward to the right. However, under artificially roughed conditions, this phenomenon was less obvious than for under smooth surface conditions due to the effect of roughness on the thickness of the viscous sublayer (as described above). The thickness of the viscous sublayer was mainly affected by the bed roughness, which decreased with increasing roughness. That the above-mentioned rainfall increased the thickness of the viscous sublayer corresponds to the effect of reducing the bed roughness, so that rainfall had a ‘reducing-resistance’ effect. This is consistent with Gilley & Finkner (1991), who found that the addition of rainfall may serve to reduce random roughness.

Calculating the resistance of overland flow

Composition of drag coefficient

Based on the notion of segmentation, resistance to overland flow could be partitioned into several categories of grain resistance, form resistance, wave resistance, and rain resistance. Because all points were obtained on fixed beds and there were no changes in the bed shape during the tests, wave resistance was ignored. Thus, the resistance of overland flow could be expressed as: 
formula
(22)

Derivation of resistance formula

Grain resistance is derived from the energy loss caused by overcoming the fixed boundary. It can also be seen as the friction resistance between water flow and the fixed bed, which is mainly affected by particle size and the Reynolds number. Emmett (1978) indicated that there is a relationship between grain resistance and Reynolds number on a smooth bed with only particle resistance. In addition, the index b of laminar flow was −1. Dunne & Dietrich (1980) attributed variation of parameters a and b to the surface roughness in the power function relationship between drag coefficient and the Reynolds number. However, Abrahams et al. (1992) thought that these changes were related to the variation of inundation of rough elements. In summary, granular resistance produced by rough elements is a substantial source of total resistance. This experimental flow belongs to the laminar transition zone, and the Reh index is about −1.19 (Table 2), which is consistent with Abrahams et al. (1992) and Nearing et al. (1997) (b= − 1.1). Therefore, it is considered that grain resistance λg can be expressed as: 
formula
(23)
Table 2

Parameter b under different conditions of roughness and rainfall intensity

Roughness (mm) Rainfall intensity (mm/h)
 
60 90 120 150 
Smooth surface 1.188 1.180 1.114 0.849 
0.061 1.403 1.340 1.261 1.136 1.064 
0.120 1.273 1.236 1.129 1.121 0.859 
0.700 1.463 1.222 1.032 1.028 0.776 
1.770 1.283 1.191 1.174 1.156 1.087 
3.680 1.486 1.391 1.379 1.189 1.106 
Roughness (mm) Rainfall intensity (mm/h)
 
60 90 120 150 
Smooth surface 1.188 1.180 1.114 0.849 
0.061 1.403 1.340 1.261 1.136 1.064 
0.120 1.273 1.236 1.129 1.121 0.859 
0.700 1.463 1.222 1.032 1.028 0.776 
1.770 1.283 1.191 1.174 1.156 1.087 
3.680 1.486 1.391 1.379 1.189 1.106 

This is consistent with the resistance coefficient (Re < 900) of Shen & Li's (1973) study without rainfall.

Form resistance can be expressed as: 
formula
(24)
where ρ is the density of water; u is the mean velocity; A is the projection area of rough elements in the u direction; and Cd is the coefficient of drag, which is related to the shape of particles and the Reynolds number Red.
Hirsch (1996) integrated the upper formula and Darcy–Weisbach drag coefficients, and obtained the formula for calculating form resistance as follows: 
formula
(25)
where Ai is the projected area of the first rough element in the direction of the current. When Λ ≥ 1, Ai = Dr·Dr, when Λ < 1, Ai = Dr · h, where Dr is the particle size of the first rough element, which can be replaced by ks in this experiment, and Ab is the bed area equals to B*L, where B is the width of the flume and L is the length of the flume.
From the above analysis, it can be seen that form resistance is mainly related to Reynolds number Red and degree of submergence. On the dual logarithmic coordinate diagram, the Reynolds number Red and submergence degree have a good linear relationship with the additional resistance. In fact, Reynolds number Red and degree of submergence have a power function with the additional resistance after deducting the particle resistance as shown in Figures 11 and 12. The formula for calculating the form resistance is obtained through multiple regression analysis: 
formula
(26)
Figure 11

The relationship between inundation ratio and additional resistance.

Figure 11

The relationship between inundation ratio and additional resistance.

Figure 12

Relationship between turbulent flow Reynolds number and additional resistance.

Figure 12

Relationship between turbulent flow Reynolds number and additional resistance.

It is found that rainfall resistance is not only related to the rainfall intensity Ri, as well as Reynolds number Reh, but is also affected by the bed roughness ks, that is, the greater the ks, the smaller the contribution of rainfall to the resistance coefficient. The formula for rainfall resistance through multiple regression analysis can be expressed by: 
formula
(27)
In summary, the formula for calculating the resistance coefficient of overland flow under rainfall conditions is: 
formula
(28)

In order to discuss the rationality of this formula, the drag coefficient is calculated by Equation (28) and verified with the measured value, as shown in Figure 13.

Figure 13

Comparison between the calculated and measured resistance coefficient.

Figure 13

Comparison between the calculated and measured resistance coefficient.

It can be seen from Figure 13 that the calculated and measured drag coefficients are all around the 1:1 line, (R2 = 0.8244), but that there is some scatter around the correlation line. This is due to the interaction of different roughness elements under the same rainfall intensity, and the rolling wave phenomenon in this test is known to affect measurement accuracy to a certain extent.

CONCLUSIONS

Indoor artificial simulated rainfall experiments were performed with six kinds of bed roughness, five flow discharges, and five rainfall intensities. The experimental results showed that:

  1. Overland flow can be classified as laminar flow and transitional flow using a binary Reynolds number. If the method of judging viscosity–depth ratio is adopted, overland flow belongs to the rolling wave zone. Considering the gradient, a critical water depth formula for overland flow was re-derived, and it was concluded that this water flow belonged to supercritical flow in most cases, while it was subcritical flow in only a few cases.

  2. Roughness has an ‘increasing resistance’ effect. Frictional resistance, the thickness of viscous sublayer, pressure drag, and roll waves all result in an increasing drag coefficient. Rainfall has a ‘reducing-resistance’ effect. Under smooth surface conditions, water depth decreased with increasing intensity. Conversely, under artificially roughened conditions, it was almost impossible to see any changes in water depth with rainfall intensity. In addition, it was also found that rainfall can increase the thickness of the viscous sublayer, which is equivalent to reducing the effect of bed roughness, thus causing ‘drag reduction’.

  3. According to the notion of resistance segmentation, the resistance coefficient can be divided into grain resistance λg, form resistance λf, rainfall resistance λr, and wave resistance λw. Based on this, a formula for calculating the resistance coefficient of overland flow under rainfall conditions was obtained by multiple linear regression.

ACKNOWLEDGEMENTS

This research was supported financially by the National Natural Science Foundation of China (Grant No. 51579214, 41877076), Fundamental Research Business Expenses of Central Universities (2452017321), Science and Technology Project of Yangling Demonstration Zone (2017NY-03), and Post-doctoral Supporting Fund of Shaanxi Province.

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