2.1. Drainage Length (L)
In China, various highway grades typically feature a single lane width of 3.75 m. For multi-lane roads, like a unidirectional four-lane expressway, the total width for lanes in one direction can reach 15 m. This implies that, during rainfall, the runoff’s drainage length on a unidirectional road could extend up to 15 m. On extensive slopes, this increased drainage length might prompt distinctions between laminar and turbulent flow along the slope, necessitating differentiation in theoretical analyses.
Fluid dynamics recognizes two primary flow states: laminar and turbulent flow. Laminar flow occurs when fluid particles move in a parallel fashion without mixing, and are characterized by smooth paths. Turbulent flow, however, involves chaotic paths where fluid particles mix, collide, and create disorder. These distinct states not only vary in particle trajectories but also exhibit entirely different internal flow structures, including velocity distribution and pressure characteristics. Consequently, their head loss and diffusion patterns differ, resulting in entirely distinct head losses along the flow. Hydraulic studies extensively investigate these associated head losses.
When evaluating hydraulic aspects of runoff on sloping surfaces, understanding the disparities between laminar and turbulent flows holds significant importance. Theoretically establishing the critical point where these flow states transition is pivotal, enabling distinct discussions on runoff characteristics within each flow state range. During the theoretical analysis of these critical transition conditions, it becomes essential to examine the criterion distinguishing laminar and turbulent flows—the critical Reynolds number. In industry standards, for a circular pipe, the industry has experimentally determined
= 2320 and
= 12,000–50,000. Due to the instability of the upper critical Reynolds number, engineers typically use the lower critical Reynolds number in practical applications. Therefore, for a circular pipe:
When < = 2320, the flow state is laminar; otherwise, it is turbulent.
The Reynolds number can be understood as the ratio between the inertial forces and the viscous forces in a fluid flow.
Inertia force
, and its dimension is
; viscous force
, and its dimension is
. The ratio of inertial force and viscous force can be expressed as
The above formula is the dimensional composition of the Reynolds number, where is the characteristic flow velocity, L is the characteristic length, and υ is the kinematic viscosity coefficient.
The above criteria are for liquid flow in circular tubes. For other flow boundaries, they also include laminar flow and turbulent flow, as well as corresponding Reynolds numbers and critical Reynolds numbers. The characteristic length of an open channel can be characterized by its hydraulic radius.
Hydraulic radius refers to the ratio of the water-passing cross-sectional area
A to the wetted perimeter
χ, represented by
R, and has the length dimension. Therefore, for a pipe flow with diameter
d, the hydraulic radius is
Then, the corresponding Reynolds number and critical Reynolds number are
The critical Reynolds number relative to an open channel is roughly 500.
In this article, the characteristic length used when examining the Reynolds number is the depth
h of the water film on the road surface; that is, the Reynolds number with slope runoff is:
In analyzing road runoff, determining the hydraulic gradient J is crucial. It is essential to discern whether the flow pattern of slope runoff is laminar or turbulent during the analysis. Differentiating between the runoff characteristics of these flow patterns is imperative as they exhibit distinct behaviors. Hence, examining the Reynolds number of slope runoff becomes necessary to identify the critical threshold that distinguishes these flow states. This critical position allows for a separate discussion of the runoff characteristics on either side of this threshold.
The formula for calculating the Reynolds number of slope runoff () highlights two key parameters for examination on the right side of the equation: v and h. Here, v denotes the runoff velocity at a specific point along the slope, while h represents the slope’s gradient at that same point.
The viscosity of the flowing liquid, denoted as υ in the formula, determines the depth of the water film. It is found that at 15 °C, the dynamic viscosity (υ) of water is 1.139 × 10−6 m2/s. When evaluating v and h, directly determining the relationship between the Reynolds number of slope runoff and 500 from these two quantities becomes challenging. On one hand, the water film depth (h) on the slope, which this article aims to calculate, remains an unknown quantity. Additionally, the runoff velocity (v) is difficult to examine in isolation. On the other hand, the Reynolds number (Re) is currently a variable dependent on two variables, making direct analysis more difficult.
Hence, finding a simpler method to distinguish the flow pattern of slope runoff becomes necessary. As per the research model outlined in this article, the relationship between rainfall recharge and cross-section runoff flow is established as:
In the formula, q represents the flow value of a certain section along the runoff direction; I represents rainfall intensity; s represents the distance between the section and the center line of the road surface, that is, the drainage length of slope runoff; α represents the slope angle of the slope surface and the slope of the road i = sinα ≈ tanα.
In this way, the Reynolds number of slope runoff can be expressed as:
In the formula above, the Reynolds number’s most pertinent variables are rainfall intensity (I), drainage length (L), and cosα associated with the slope. Among these factors, for a specific slope under consideration, rainfall intensity and cosα remain constant. This simplifies the Reynolds number of slope runoff into a singular variable that solely varies with drainage length, significantly minimizing the complexity of the discussion regarding the magnitude of the Reynolds number.
Moreover, the determination of the transition between laminar and turbulent flow hinges upon the Reynolds number. This critical point is situated at a specific location along the slope in the runoff direction, represented by a particular drainage length “L”. By equating the equation with a constant and comparing it to 500, a critical value of “L”, referred to as “critical s”, can be derived. This value precisely indicates the critical juncture between laminar and turbulent flow in surface runoff. Such an approach enables a clear-cut discussion of surface runoff on both sides of the critical value, “critical L”.
The Reynolds number calculation formula, , shows that at a fixed drainage length, the determined Reynolds number varies with different rainfall intensities (I) and slope gradients (i). When considering a rainfall intensity of I = 3.0 mm/min = 0.05 × 10−3 m/s and a slope gradient of i = 5%, . When Re is 500, s equals 8.78 m, which indicates that the critical s is 8.78 m.
To streamline the discussion, this study concentrates solely on investigating laminar flow conditions while examining surface runoff characteristics. The analysis of turbulent flow scenarios remains a subject for future studies. For simplicity, and based on the previously calculated critical value of L, an approximation assumes that within a drainage length of 9 m, surface runoff predominantly maintains laminar flow. Consequently, this research delves into scrutinizing surface runoff within a drainage length of 9 m for the calculation of road surface water film depths during rainfall conditions. This research scope serves adequately for typical single-lane, two-way roads, and these simplified research findings possess reference value even for shorter slope lengths. The drainage length, denoted as “L”, ranges from 0 to 9 m, encompassing values at 0 m, 1 m, 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, and 9 m, totaling ten values. At s = 0, it signifies the initial depth of the water film on the road surface, referred to as h0.
2.2. Rainfall Intensity (I)
Rainfall intensity, denoted as I, stands as a crucial factor impacting water film depth on road surfaces during rainy conditions. Empirical evidence underscores that different levels of rainfall intensity wield a substantial influence on the water film’s depth over road surfaces. When maintaining a consistent drainage length, road surface gradient, and raindrop velocity, there is a clear trend indicating escalated water film depth with increased rainfall intensity. This observation aligns with historical experimental data, emphasizing the pronounced impact of rainfall intensity, especially during heavy downpours or stormy weather, on road surface water film depth. Hence, delving into the effects of diverse rainfall intensities on road surface water film depth assumes significance, contributing to a comprehensive comprehension of water film behavior during rainfall. This paper recognizes the significant influence of rainfall intensity among the factors affecting road surface water film depth. It seeks to explore the correlation between varying levels of rainfall intensity and road surface water film depth.
To study the relationship between road surface water film depth and rainfall intensity, the primary concern is determining the magnitude of rainfall intensity under different rainy conditions. Regarding the selection of rainfall intensity values in the design calculation process, the “Highway Drainage Design Specification” (JTG/T D33—2012) provides explicit regulations, as follows.
Using standard rainfall intensity contour maps and relevant conversion coefficients, calculate the rainfall intensity according to the equation q = cpctq5,10, where q5,10 represents the standard rainfall intensity (mm/min) for a 5-year return period and 10 min duration; cp signifies the return period conversion coefficient, which is the ratio of design return period rainfall intensity qp to the standard return period rainfall intensity q5; ct stands for the duration conversion coefficient, which is the ratio of rainfall intensity qt for duration t to rainfall intensity q10 for a 10 min duration
By consulting the “China 5-year return period 10-min rainfall intensity contour map”, it can be observed that the 5-year return period 10 min rainfall intensity, q5,10, in the southeastern coastal regions of China generally ranges from 2.0 mm/min to 2.5 mm/min, with the maximum national value of q5,10 being 3.0. Upon converting the standard rainfall intensity to values corresponding to durations below 10 min according to the table above, the converted rainfall intensity surpasses 3.0 mm/min, reaching approximately 4 to 5 mm/min.
Considering the aforementioned factors, this paper outlines the range of rainfall intensity utilized in the calculations, set between 0.5 to 5 mm/min. Specifically focusing on examining the correlation between road surface water film depth and rainfall intensity, the analysis involves comparing various rainfall intensities: 0.5 mm/min, 1.0 mm/min, 1.5 mm/min, 2.0 mm/min, 2.5 mm/min, 3.0 mm/min, 3.5 mm/min, 4.0 mm/min, 4.5 mm/min, and 5.0 mm/min. These ten values were utilized to juxtapose the curves of rainfall intensity versus water film depth, intending to establish the general trend of road surface water film depth (h) concerning the variations in rainfall intensity (I).
2.3. Road Surface Gradient (i)
In road surface runoff, the road gradient is another factor affecting the water film depth during rainfall. While studies indicate that the influence of the gradient (i) on road surface water film depth ranks relatively low among rainfall factors, its impact becomes substantial in cases of substantial gradient variations. Consequently, various research reports on road surface water film depth under rainfall consider the effect of road surface gradient (i). The gradient encompasses the longitudinal slope along the road’s profile and the cross-slope across the road, including combinations of both longitudinal and cross-slopes. Under rainfall, the actual water film depth on the road surface results from the combined effects of these diverse gradient scenarios.
The typical cross-slope of road surfaces, denoted as i, commonly ranges between 1.0% and 3.5%. For wider road surfaces aiming to enhance drainage efficiency, this percentage can be appropriately increased to approximately 4.0% or even higher. Considering these factors comprehensively, and aiming to qualitatively explore the impact of road surface gradient on water film depth concerning rainfall intensity, the gradient range (i) was set between 1.0% and 6.0%.
In particular, for a specific analysis centered on the effect of increased road surface gradient while assuming consistency in the other three factors, gradient values (i) were set as 0.5%, 1.0%, 1.5%, 2.0%, 2.5%, 3.0%, 3.5%, 4.0%, 4.5%, 5.0%, 5.5%, and 6.0%. These twelve values were employed to compute the corresponding water film depths for each gradient, establishing a quantitative relationship between water film depth and road surface gradient.
2.5. Physical Parameters of Raindrops
The physical attributes of raindrops during rainfall play a significant role in shaping the water film on road surfaces. These key parameters typically encompass the angle between raindrops influenced by wind and the vertical line (referred to as the rainfall angle,
β in
Figure 1), along with the terminal velocity of the raindrops (
u0 in
Figure 1).
It is commonly understood that as atmospheric water vapor cools and condenses, it transforms into minute liquid or solid water droplets, occasionally developing into ice crystals. These minuscule droplets or crystals merge and accumulate over time, gradually growing in size and weight, forming larger water droplets. When these droplets’ gravitational force surpasses the upward atmospheric movement, they descend, initiating rainfall. The terminal velocity (u0) of raindrops signifies the immediate speed at which a raindrop reaches the ground following its descent.
The descent of raindrops involves a process marked by variable acceleration. As a raindrop descends, both the forces influencing it and its speed undergo continuous changes. The forces acting on a falling raindrop typically include gravity, atmospheric buoyancy, and air resistance, the latter fluctuating as the raindrop’s speed changes during its descent. Air resistance is approximated to be proportional to the square of the falling speed for larger raindrops, while for smaller ones, it is considered proportional to the falling speed itself.
Research indicates that the terminal velocity of raindrops increases with their size. Raindrops with a diameter of 6 mm (commonly maintaining a diameter of less than 6 mm during descent) can achieve a terminal velocity of around 10 m/s upon reaching the ground. This substantial speed carries significant momentum upon impact, highlighting the importance of considering raindrop terminal velocity in discussions regarding surface runoff processes.
Numerous studies, both domestically and internationally, have delved into the terminal velocity of falling raindrops. Theoretical formulas for the terminal velocity (
u0) have been derived through theoretical analysis, yet actual measured data remain limited. Notably, related research [
35,
36] has conducted dedicated experiments exploring raindrop terminal velocity, compiling the statistical data presented in
Table 2. These experiments utilized pulse induction and recorded droplet fall durations via computerized means. By adjusting the height difference of droplet falls, approximate terminal velocity values were derived. The methodology employed in the experiment was comprehensive, carrying significant importance for research focused on rainfall phenomena.
In this study regarding the calculation of road surface water film depth under rainfall conditions, consideration was given to the impact of raindrop terminal velocity (
u0). In several existing reports [
24,
27,
37] on the calculation of road surface water film depth, both domestically and internationally, there has been little discussion about the influence of raindrop terminal velocity on road surface water film depth. However, based on the aforementioned analysis, during heavy rainfall, raindrop terminal velocities can reach speeds of nearly 10 m/s, resulting in a substantial impact on the ground. Regarding the value of
β, previous research has indicated that under typical wind conditions (up to level 6 wind), the maximum angle of raindrop descent does not exceed 50° [
38,
39,
40,
41]. In the study of road surface runoff, considering the relatively shallow and lightweight nature of the water film on the road surface, the impact caused by a higher momentum due to raindrop terminal velocity cannot be disregarded when acting on the water film on the road surface, as it significantly influences the entire road surface runoff.