# Influence of Water Depth and Slope on Roughness—Experiments and Roughness Approach for Rain-on-Grid Modeling

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## Abstract

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## 1. Introduction

#### 1.1. Modeling Extreme Events

#### 1.2. Aim of the Study

## 2. Background

#### 2.1. Flow Equations

_{N}or the Strickler roughness coefficient k

_{S}. In principle, these coefficients consider the total resistance. The only dimensionless factor is Darcy Weisbach’s friction factor f, which is the most general and valid for all states of flow [17,18]. The friction factor f is part of the following Darcy–Weisbach equation:

_{H}is the hydraulic radius, S is the channel slope and A is the cross-sectional area. There are different possibilities to calculate the friction factor f. According to Prandtl [19], the friction factor depends on the Reynolds number Re for hydraulically smooth areas and laminar flow, and according to Nikuradse [20], it depends on the relative roughness k

_{N}/D

_{H}(hydraulic diameter) for the hydraulically rough flow regime. For the experiments conducted in this study, only the hydraulically rough flow regime is valid (see Reynolds numbers in Table 1). In addition to the friction factor f, another coefficient to analyze roughness is used here, the equivalent sand–grain roughness k

_{N}, by Nikuradse.

_{r}is influenced by channel shape. For surface runoff with the assumption of an infinitely wide channel bed, the parameter can be estimated to be p

_{r}= 3.05 [21].

_{N}[26,27] are available. The influence of roughness on the flow differs depending on the surface and bed conditions. There is resistance due to bed roughness, such as concrete in channels, and vegetation resistance, where vegetation is either emergent or submerged. In the literature, roughness for solid surfaces is assumed to be a constant, empirically calculated coefficient, whereas the consideration of vegetation depends on the submergence of vegetation structures.

#### 2.2. Vegetation Resistance

_{veg}). This submergence ratio depends on the water depth h and vegetation height h

_{veg}. Other studies state that conventional flow resistance equations with constant values are not valid for vegetated flow [16,26] because vegetation resistance depends on vegetation type and water depth [31]. By analyzing the performance of different models in vegetated channels, Vargas-Luna et al. [33] concluded that a separate consideration of emergent and submerged conditions offers the best results. One possibility to describe different resistance conditions is a distinction in a resistance layer (flow through vegetation) and a surface layer (flow above vegetation) [16,31]. This two-layer approach has been proposed by several studies [31,35,36].

_{N}on a shape parameter, the roughness density and the height for selective roughness. Huthoff [27] presents an equation for k

_{N}that is based on a resistance coefficient, the ratio of the frontal cylinder area and the specific volume between roughness elements (product of the specific bed surface area and the height of roughness elements). Another approach for k

_{N}was proposed by Gualtieri et al. [26]. To formulate a constant roughness coefficient k

_{N}, roughness height and density are used. Gualtieri et al.’s approach is valid for submergence ratios (water depth/obstacle height) of 5 and higher and therefore represents strongly inundated vegetation. Another approach is provided by Ferro and Guida [38]. They used a new resistance approach for the friction factor f and validated it with experimental data by Bond et al. [39] for grassland. This approach has a calibration parameter for different types and properties of grass and depends on slope, Reynolds number and Froude number. For experiments by Bond et al. [39], the calibration parameter ranges from 0.273 to 0.314 [39]. Conversely, solving the equation yields a very similar form of the Darcy–Weisbach equation (Equation (1)).

_{N}and Equation (2). f″ can be calculated with Equation (3); in the following “vegetation approach” [21,35]:

_{D}is the bulk drag coefficient. Factors d

_{veg}(diameter of vegetation) and D

_{veg}(density of vegetation or roughness elements [pieces per area]) represent the specific frontal area of the vegetation in the x-direction, the roughness density [40].

_{veg}> 5), resistance will transform to conventional equations, which are not dependent on water depth but are constant in their values [16,26,31].

#### 2.3. Conditions for Overland Flow

## 3. Materials and Methods

Reference | Artificial Grass | Wheat | Cement-Based Coating | Asphaltic Emulsion | Exposed Aggregate Concrete | Aluminum | |
---|---|---|---|---|---|---|---|

Description | Nubby blade of grass: length: 2.5 cm height: 1.5 cm predominantly rigid | Dried wheat height: 0.5 m 500 pc./m ^{2}fixed in 3 cm Styrodur and on top: 2 cm cement-based coating predominantly rigid (bending was avoided) | Mixture of masonry mortal and tile adhesive (ratio 1:2) | Grain size: 0–8 mm | Texture: gravel; grain size: 5–20 mm | Plates, 2 mm thick | |

Installation | Sticked and tightened to a coated plywood plate | 4 separate boxes | 4 separate boxes | 4 separate boxes | 4 pieces | 4 pieces sealed with silicone | |

Flow condition | Submerged vegetation Submergence: 2.1–7.5 | Emergent vegetation | Submerged Solid surface | Submerged Solid surface | Submerged Solid surface | Submerged Solid surface | |

Total number of experiments | 149 | 77 | 168 | 119 | 119 | 98 | |

Q [l/s] | 5–70 | 5–35 | 5–70 | 5–70 | 5–70 | 5–70 | |

h [cm] | 3.1–11.2 | 1.2–14.3 | 1.0–9.5 | 1.1–7.1 | 1.3–7.5 | 0.9–8.0 | |

Re | 2.48 × 10^{4}–3.31 × 10 ^{5} | 2.75 × 10^{4}–1.76 × 10 ^{5} | 2.78 × 10^{4}–3.75 × 10 ^{5} | 2.65 × 10^{4}–3.64 × 10 ^{5} | 2.63 × 10^{4}–3.60 × 10 ^{5} | 2.92 × 10^{4}–3.74 × 10 ^{5} | |

S [%] | 1 | X | X | X | X | X | X |

2 | X | X | X | ||||

3 | X | X | X | ||||

4 | X | X | X | ||||

5 | X | X | X | X | X | X | |

10 | X | X | X | X | X | X | |

15 | X | X | X | X | X | X | |

20 | X | X | X | X | X | X | |

25 | X | X | X | X | X | ||

30 | X | X | X | X | X | X | |

35 | X | X | X | X | X | ||

40 | X | X | X | X | X |

## 4. Results and Discussion

_{N}of each experiment with calculated f and Equation (2). Figure 3 summarizes the data of all types of bed roughness in this study with slopes from 1% to 20%. For this first overview, no distinction is made between solid or vegetation bed roughness, and for every experiment, coefficient k

_{N}is considered as bed roughness to show the range of values. For further analyses, the friction factor f or coefficient k

_{N}is used, depending on the surface. The plot depicts the dependence of water depth on roughness k

_{N}. With a split y-axis, the range of data points for low k

_{N}values can be better recognized. For submerged vegetation (artificial grass), roughness decreases with increasing water depth and submergence. In contrast, for emergent vegetation (wheat), roughness increases with increasing water depth. These results of change in roughness depending on submergence are plausible regarding results from the literature for vegetation in the channel [40] and for overland flow [14,17]. Solid surfaces (cement-based coating, asphaltic emulsion, exposed aggregate concrete and aluminum) show a much smoother roughness with a reasonable constant roughness coefficient k

_{N}.

#### 4.1. Consideration of Roughness for Submerged Vegetation

#### 4.1.1. Analyses and Evaluation of Experimental Results

_{veg}). The experiments of Ruiz Rodriguez and Trost [58,59] with artificial grass are comparable to the surface used in this study. Accordingly, the data points largely agree with the measurements of this study. Scheres et al. [60] observed a smoother surface for a “species-poor grass-dominated mixture” with a vegetation coverage of 82%. The friction factor for these experiments deviates widely from the results of this study. Here, friction factors are constant for all submergence ratios, and in total, values are lower. Experiments by Wilson and Horritt [61] were carried out in a flume with a bottom slope of 1% and grass blades of 7 cm height. A curve of friction factors for different submergence is given, although data points show lower friction factors for a given submergence than data from the literature. In a study by Karantounias [62], he used a grass mat with blades, which were 8 cm long and bent to the ground. Unfortunately, the height of the vegetation layer was not reported, but one photo was presented. For comparison in Figure 4, a height of 1 cm was assumed to calculate submergence. With this assumption, the results of this study also fit the results of the present study. By increasing the assumed vegetation height, the data points slide to the left. For all data, friction factors are highest for minimum submergence. Reports by Nepf [40] and Abrahams et al. [14,17] and the results of wheat in this study indicate that the curve changes with a submergence of 1 (change from submerged to emergent). Hence, the maximum roughness is reached with submerged vegetation (submergence = 1). With this knowledge, an assumption of a vegetation height of 1 cm for experiments by Karantounias seems plausible because a submergence of 1 is not exceeded.

#### 4.1.2. Existing Models

_{N}values: Gualtieri et al. [26]: 0.205 m; Huthoff [27]: 0.017 m; Schröder [37]: several meters. The k

_{N}value from the approach by Schröder [37] strongly deviates from the experimental results of this study, whereas values from the approaches by Gualtieri et al. [26] and Huthoff [27] reproduce the mean values of the experimental data (see Figure 3 and Figure 5a). However, the k

_{N}value resulting from the approach by Gualtieri et al. [26] should be valid for submergence ≥5 when hydraulic resistance reaches a constant value. In this study, a constant value of k

_{N}= 0.1 m is reached for high submergence (see Figure 5a).

_{N}(Figure 5a) and submergence against f (Figure 5b) for different approaches in comparison to the experimental data. In Figure 5a, experimental data are shown separately for each slope condition, and in Figure 5b, data from this study are shown as black point clouds for better clarity compared to other data. Both plots show a constant k

_{N}, f values obtained with a constant k

_{N}(gray, dashed line), as suggested in the literature, and k

_{N}/f values obtained with a constant k

_{S}(black, dotted line), as it is state of the art in 2D models [63]. All k

_{S}values are transformed to f with a combination of the Gauckler–Manning–Strickler equation and Darcy–Weisbach equation (Equation (5) in the following paragraph); conversion of f to k

_{N}and backward is carried out with Equation (2). Constant k

_{N}and constant k

_{S}were chosen as values for high submergence, as these values are listed in the literature for channel flow (see paragraph “Vegetation Resistance”) and therefore fit high water depth. For that, the values of k

_{N}= 100 mm and k

_{S}= 25 m

^{1/3}/s represent these conditions. Mean values for k

_{N}and k

_{S}could fit the experimental data better as a whole, but over- and underestimate specific ranges of values. The influence of water depth for k

_{S}in Figure 5a and for k

_{N}in Figure 5b arises from the factor of hydraulic radius R

_{H}in both transformation equations. The plot depicts that a constant k

_{N}as well as a constant k

_{S}merely applies with experiments with high water depths (submergence >6 to 7), although constant k

_{N}values can represent the quality of the curve progression in Figure 5b. In Figure 5b, it can be seen that data from Scheres et al. [60] fit with a constant k

_{S}and data from Wilson and Horritt [61] fit with a constant k

_{N}. Furthermore, Wilson and Horritt conducted experiments with grass blades of 7 cm. If these blades bend, submergence rises, and the curve of data points from Wilson and Horritt approximates the data curve from this study. Properties, especially the density of vegetation, deviate in each experiment, so different values for hydraulic resistance are reasonable.

_{H}in Equation (2). Nevertheless, an additional dependence of k

_{N}and water depth is shown in Figure 5a. According to these findings, a novel approach for k

_{N}depending on water depth is introduced.

#### 4.1.3. Novel Approach

_{N}is dependent on water depth. This dependence can be described with a linear approach (Figure 5, blue, solid line) in Equation (4).

_{N}consists of three parts: (a) a part for submergence ≤1, (b) a part for high submergence and (c) a part for the intermediate area. For submergence ≤1 (h ≤ h

_{veg}), the conditions change to emergent, and the hydraulic response is different. To simplify the novel approach, the roughness coefficient k

_{N}for h ≤ h

_{veg}is assumed to be constant (k

_{N-S1}). For water depths with high submergence (h >> h

_{veg}or submergence ≥5), a constant value (k

_{N}) can be assumed as well [6,16,31]. For the intermediate area, k

_{N}will be changed approximately linearly between k

_{N-S1}and k

_{N}. This valid range of values is constrained by the minimum k

_{N}for high submergence and the maximum k

_{N-S1}for low submergence. The change in roughness for 1 < h/h

_{veg}< 5 to 7 depends on the gradient Δk

_{N}/Δ submergence and therefore on the water depth.

_{N}for water depths <3 cm was assumed for all the following analyses (as shown in Figure 5a, blue line).

_{N}and submergence vs. f (Figure 5), a comparison of the predicted velocity to the measured velocity (calculated with measured discharge and water depth) for all approaches is shown in Figure 6a–c. For k

_{S}= 25 m

^{1/3}/s = constant (a) and k

_{N}= 0.1 m = constant (b), approaches approximate measured values for high velocities and therefore for high water depths. For data with high bottom slopes, the approaches fit less well. The root mean square error (RMSE) of flow velocity v is 0.463 for k

_{N}= constant and 0.604 for k

_{S}= constant. Altogether, both approaches lead to velocities that are too high. In the right column of Figure 6d–f, the plot Reynolds number Re against friction factor f indicates the same trend. Accordingly, both approaches cause friction factors that are too small for the predicted scenario (dashed lines). For the novel approach (bottom row), the velocity fits well for small values; for higher velocities, the values scatter (Figure 6c). The RMSE of velocity v is 0.138 for all measurements and 0.094 for v < 1 m/s. The friction factor f can be predicted more precisely than with constant approaches (Figure 6f). Here, an RMSE of 0.57 can be reached for all measurements.

_{N}) with water depth or submergence do not show clear dependence. From Figure 5a, it can be assumed that roughness increases with decreasing slope for a given water depth. However, some slopes deviate, e.g., 1% and 25% to 40%. The plot of Re vs. f (Figure 6f) shows that the friction factor can be predicted adequately with k

_{N}depending on the water depth, and an additional dependence on the bottom slope is not necessary.

_{S}= constant, k

_{N}= constant and the novel approach were used in a flash flood simulation to evaluate their effects in a real catchment area.

#### 4.1.4. Implementation in a 2D Model

_{S}, either as a constant value or a value depending on the water depth. The programming code of the model converts the Strickler roughness coefficient k

_{S}into the friction factor f by merging the Gauckler–Manning–Strickler equation and Darcy–Weisbach equation into the following equation [65]:

_{H}represents the hydraulic diameter, with D

_{H}= 4 × R

_{H}. According to Hydrotec [65], the factor R

_{H}can be replaced by water depth for 2D shallow water equations.

_{N}is converted to friction factor f by using Equation (2) and then to Strickler’s roughness coefficient k

_{S}with Equation (5). For k

_{N}= constant and for the novel approach (Equation (4)), the entered values of k

_{S}are a function of water depth. Within the analysis of flash flood simulations, attention is mostly focused on the simulation results of water depth and flow velocity.

_{N}and derive the friction factor f and flow velocity. Thus, appropriate discharge for water depth and velocity is not considered. However, in 2D modeling, discharge or precipitation are initial parameters for simulations. To estimate the influence of this condition, the experimental flume was rebuilt as a 2D model, approaches were applied, and discharge from experiments was used as the initial condition. In Figure 7, measured flow velocities are compared to velocities from manually calculated approaches (a–c) and to velocities from simulations (d–f). Figure 7a–c are extracts from Figure 6a–c. For this comparison, exemplary slopes of 3%, 5%, 15% and 40% were used. This selection represents the total range of values for scattering.

_{S}= constant; with k

_{N}= constant 0.47 against 0.19 and with novel approach, 0.16 against 0.06. Experimental runs with slopes of 40% and low discharge do not bring plausible results with the 2D model and are therefore excluded from Figure 7. Water depth and flow velocity seem unstable. The reason for this is the usage of shallow water equations in 2D simulations. As presented in the introduction of the HydroAS model, only the x- and y-directions are considered for simulation, and the z-direction is not calculated. For very high slopes, such as 40% in experimental runs, conditions of high slopes and low water depth could result in resilient simulation values. However, the analysis of slopes with SRTM1 data (shuttle radar topography mission) shows a share of less than 2% for slopes of 40% and higher for the area of Germany. Consequently, a combination of the mentioned conditions is rare.

^{2}, and the elevation varies from 288.13 m to 371.61 m above sea level. In the model, there is no deviation in different land uses, so grassland approaches were applied for the entire model. This method was selected to avoid interactions with different roughness values and other processes. Hence, the discharge curve only results from different grassland approaches to evaluate their effects. Precipitation was considered with an intensity of 60 mm/h for a duration of one hour, and initial losses were taken into account at 2.5 mm. For all scenarios, the total runoff volume is equal.

- The novel approach of this study (Equation (4)) with k
_{N}as a function of water depth was applied. - The approach of Ferro and Guida [38] with friction factor f as a function of slope, Reynolds number and Froude number was applied. Here, the calibration factor was 0.21, which fits best to the measured values of high water depth in this study.

_{S}= constant leads to the earliest and highest discharge peak. The approach with k

_{N}= constant results in a similar curve but shows translational motion. Due to a higher friction factor for k

_{N}= constant, which is presented in Figure 6, overland flow slows down, and a delayed discharge is plausible. By using the novel approach with even higher roughness, translation and retention effects are clearly visible in the discharge curve. For the roughness approach by Ferro and Guida [38], the 2D model has to use water depth and velocity to calculate roughness in every time step. However, these variables should be calculated in the model with existing default roughness. Consequently, the discharge curve shows strong oscillation, and the model seems unstable. Discharge approximates the shown curves and becomes more stable due to the limitation of different equation parameters, such as Re and Fr. Nevertheless, the range of limitations is difficult to define.

_{S}and k

_{N}would lead to an intensified retention and translation of the discharge curve. However, these values over- or underestimate specific areas in the catchment. The accuracy of the hydrograph does not ensure the correct determination of the hydraulics in the entire model [15]. Therefore, the correct approach for friction is essential for obtaining precise results at each point in the model.

#### 4.2. Consideration of Roughness for Emergent Vegetation

_{D}, which is an empirically determined value. In this study, the shape of a cylinder is assumed for wheat stalks, and therefore, a value for C

_{D}of 1.2 is selected [21]. The second parameter is the specific frontal area of the vegetation in the x-direction. Due to the experimental setup of this study, a mean diameter of wheat stalks of 3.7 mm and a density of vegetation of 500 pieces/m

^{2}were used for calculation.

_{N}derived from experiments with a cement-based coating. Here, the approach was simplified by an assumption of k

_{N}= 0.006 m as a uniform value for each experiment. Scattering of results from experiments with cement-based coating was not considered. However, the influence of the bottom friction f’ is almost insignificant because the values for k

_{N}vary from 2.3 mm to 10.2 mm (see section “Consideration of Roughness for Solid Surfaces”) and therefore f′ varies from 0.05 to 0.1. However, f″ varies from 0.1 to 0.8 (Figure 9). Part of the vegetational resistance f″ was calculated by subtracting f and f′.

#### 4.3. Consideration of Roughness for Solid Surfaces

_{N}to describe bed roughness. In all plots, a similar phenomenon can be perceived. For data of slopes up to 15/20%, the sand-grain roughness k

_{N}is an almost constant value for all water depths. This suggests that the roughness coefficient k

_{N}does not depend on the water depth. Compared to a condition of constant k

_{N}for h >> h

_{veg}for submerged vegetation, in this case, water depth is much higher than roughness height k

_{N}and therefore, it could be seen as a very high submergence. Against this background, constant k

_{N}values seem plausible. For higher slopes, k

_{N}scatters or increases with increasing water depth. Similar to the experimental data of the wheat, scattering increases with both bottom slope and water depth. The range of values for the derived k

_{N}are listed in Table 2. Here, the mean values for different slope conditions are presented.

_{N}values of this study are higher than the orders of magnitude listed in the literature. By comparing measured data with data from the literature, it is noticeable that values from the literature often fit within the range of data with slope = 1%. Two explanations are possible: (a) despite the assumption of 90% discharge (see paragraph “Outline”), some data with 1% slope scatter and deviate from the expected curve; perhaps even more discharge was lost, or (b) data from the literature fit with data of 1% slope because literature values are derived from channel conditions with low slopes. Overall, it could be seen that for a given water depth, k

_{N}rises with slope. However, this conclusion is weakened by considering scatter due to measurement inaccuracies. By changing the measured water depth by ± 3 mm, k

_{N}varies widely by −39 to −60% or rather +53 to +100% (for slopes up to 20%). For comparison of plots shown for submerged vegetation, here, a plot of Re and friction factor (logarithmic display) shows negatively sloping functions with low slopes. This result seems reasonable to what has been reported in the literature [14,17].

## 5. Conclusions

_{N}. However, by calculating the friction factor f with the Prandtl–Colebrook equation and a constant k

_{N}, the water depth is an influencing factor. Investigations regarding the influence of the bottom slope on roughness show uniform results. In a plot of water depth and roughness k

_{N}, the influence of slope could be seen for some data points. Due to scattering, this trend is vague for high slopes and high water depths. However, when considering the friction factor f, the influence disappears or is only slightly visible.

_{N}values are reasonable, although the measurements of this study lead, in total, to a slightly higher roughness than presented in the literature. The results of emergent vegetation show that the existing vegetation approach depicts the mean curve for measured values. Scattering rises with increasing water depth and slope. In contrast, the consideration of submerged vegetation is not fully applicable to existing approaches. Consequently, a novel approach appropriate for measured data has been introduced to consider water depth-related roughness. Simulations of an RoG model with different approaches clearly show visible changes in discharge curves. The decision of the roughness approach to correctly calculate water depth and flow velocity has a very high impact on the catchment response. Thus, this response depends on the consideration of the correct roughness value at each point in the entire catchment. To evaluate the quality and decision about the most reasonable approach, real measurements of water level gauges at several points in the catchment area are necessary. For the best model performance, a simple approach is recommended. Calibrated RoG models can be used to simulate specific precipitations and investigate precautions for real flash floods.

_{N}/Δ submergence should be derived by vegetation properties with data from this study and data from the literature. Thus, further research is needed.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Limitations

## Notation

A | Cross-sectional area (m^{2}) |

C_{D} | Bulk drag coefficient (-) |

CORR | Corrected |

D_{H} | Hydraulic diameter (m) |

d_{veg} | Diameter of the vegetation (m) |

D_{veg} | Density of the vegetation (pcs/m^{2}) |

DEM | Digital elevation model |

f | Friction factor (Darcy-Weisbach) (-) |

f′ | Bottom friction factor (-) |

f″ | Vegetation friction factor (-) |

Fr | Froude number (-) |

g | Gravitational acceleration (m/s^{2}) |

h | Water depth (m) |

h_{veg} | Vegetation height (m) |

k_{N} | Equivalent sand-grain roughness (Nikuradse) (m) |

k_{S} | Strickler roughness coefficient (m^{1/3}/s) |

p_{r} | Shape coefficient (-) |

Q | discharge (m³/s) |

R_{H} | Hydraulic radius (m) |

Re | Reynolds number (-) |

RMSE | Root mean square error |

RoG | Rain-on-Grid |

S | Channel slope (-) |

SRTM | Shuttle radar topography mission |

x | Longitudinal direction along the flume |

y | Transverse direction of the flume |

v | Flow velocity (m/s) |

2D | Two-dimensional |

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**Figure 2.**Photos of the experimental flume with different surfaces: (

**a**) artificial grass, (

**b**) wheat, (

**c**) cement-based coating, (

**d**) asphaltic emulsion, (

**e**) exposed aggregate concrete, and (

**f**) aluminum.

**Figure 3.**Relationship between water depth h and equivalent sand–grain roughness k

_{N}for different bed roughnesses. Ordinate was split to better show data points with low k

_{N}.

**Figure 5.**Relationship between water depth or submergence and roughness parameters: (

**a**) water depth h against equivalent sand–grain roughness k

_{N}with courses of the novel approach (Equation (4)), a constant k

_{N}and a constant k

_{S}in comparison to measured values of this study (dots, separately for each slope condition) and (

**b**) submergence against friction factor f with courses of the novel approach (Equation (4)), a constant k

_{N}and a constant k

_{S}in comparison to measured values of this study (black dots) and data from the literature [58,59,60,61,62].

**Figure 6.**Comparison of approaches k

_{S}= constant, k

_{N}= constant and novel approach for quality assurance. (

**a**–

**c**): Comparison of predicted velocity and measured velocity. (

**d**–

**f**): Comparison of Reynolds number Re and friction factor f (logarithmic) for measurements (continuous lines) and predictions (dashed lines).

**Figure 7.**Comparison of approaches k

_{S}= constant, k

_{N}= constant and novel approach, manually calculated and implemented in 2D model. (

**a**–

**c**): Comparison of predicted velocity and measured velocity for 3%, 5%, 15% and 40%. (

**d**–

**f**): Comparison of velocity from the 2D model and measured velocity for 3%, 5%, 15% and 40%.

**Figure 8.**Model area and 2D results: (

**a**) DEM of the model and location of the main outlet, (

**b**) hydrograph for different scenarios at the main outlet [38].

**Figure 9.**Relationship between submergence (h/h

_{veg}) and friction factor f″ for measured values of experiments with wheat (dots) in comparison to the vegetation approach (blue, solid line, Equation (3)).

**Figure 10.**Relationship between water depth h and equivalent sand–grain roughness k

_{N}for measured values of (

**a**) cement-based coating, (

**b**) asphalt emulsion, (

**c**) exposed aggregate concrete, and (

**d**) aluminum.

Surface | Range of Mean k_{N} Values(For S = 2%–S = 20%) | Literature [66] |
---|---|---|

Cement-based coating (Figure 10a) | 2.3–10.3 mm (for slope = 1%: k _{N} = 1.4 mm) | Concrete, smooth: k _{N} = 1–6 mm |

Asphaltic emulsion (Figure 10b) | 8.3–9.7 mm (for slope = 1%: k _{N} = 2.3 mm) | Asphaltic concrete or mastic asphalt: k_{N} = 1.5–2.2 mm |

Exposed aggregate concrete (Figure 10c) | 12.4–18.4 mm (for slope = 1%: k _{N} = 8.3 mm) | Concrete smooth—rough: k _{N} = 1–20 mm |

Aluminum (Figure 10d) | 1.3–4.6 mm (for slope = 1%: k _{N} = 0.4 mm) | Steel: k_{N} = 0.04–0.1 mm |

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**MDPI and ACS Style**

Hinsberger, R.; Biehler, A.; Yörük, A.
Influence of Water Depth and Slope on Roughness—Experiments and Roughness Approach for Rain-on-Grid Modeling. *Water* **2022**, *14*, 4017.
https://doi.org/10.3390/w14244017

**AMA Style**

Hinsberger R, Biehler A, Yörük A.
Influence of Water Depth and Slope on Roughness—Experiments and Roughness Approach for Rain-on-Grid Modeling. *Water*. 2022; 14(24):4017.
https://doi.org/10.3390/w14244017

**Chicago/Turabian Style**

Hinsberger, Rebecca, Andreas Biehler, and Alpaslan Yörük.
2022. "Influence of Water Depth and Slope on Roughness—Experiments and Roughness Approach for Rain-on-Grid Modeling" *Water* 14, no. 24: 4017.
https://doi.org/10.3390/w14244017