Experimental Study on Head Loss Due to Cluster of Randomly Distributed Non-Uniform Roughness Elements in Supercritical Flow

Abstract

Accurate estimation of head loss introduced via randomly placed roughness elements found in natural or constructed streams (e.g., fish passages) is essential in order to estimate flow variables in mountain streams, understand formation of niches for aquatic life, and model flow structure. Owing to the complexity of the involved processes and the often missing detailed data regarding the roughness elements, the head loss in such streams is mostly approximated using empirical models. In our study, we utilize flume experiments to analyze the effects of the spatial distribution of roughness elements on water surface levels and head loss and, moreover, use the produced data to test three empirical models estimating head loss. The experiments were performed in a 15 m long, 0.9 m wide flume with a slope of 5% under large Froude numbers (2.5–2.8). Flow velocities and water levels were measured with different flow rates at 58 points within a 3.96 m test section of the flume. We could show that different randomly arranged patterns of roughness elements significantly affected head loss (differences up to 33.6%), whereas water jumps occurred when flow depths were in the same size range as the roughness elements. The roughness element position and its size influenced water surface profiles. None of the three tested empirical models were able to well reproduce the differences in head loss due to the different patterns of roughness elements, with overestimated head loss from 12 to 94.7%, R² from 41 to 73%, NSE from −21.1 to 0.09, and RRMSE from 18.4 to 93%. This generally indicates that these empirical models are conditionally suitable to consider head loss effects of random patterns of roughness elements.

1. Introduction

The spatio-temporal dynamics of the flow depths and velocities in mountainous streams are influenced by cobbles, boulders, and step pools. The cobbles, boulders, and step pools cause head loss along the stream flow pathway [1,2,3,4,5], and understanding the spatio-temporal flow dynamics is essential for modelling [6]. The highly variable flow velocities and flow depths form niches for different flora and fauna, and are important to fish lifecycles [7] and biodiversity in general [8,9]. A thorough investigation is required on the ecology and morphology of watershed and streams [10,11] in order to make any changes in this region that help mankind with hydraulic and hydrologic knowledge about mountain streams [1,12,13]. The stream flow depth and its velocity are important for assessing the morphological response of streams [14], as well as to design fish passages [15,16,17].

The Mannings and Darcy–Weisbach models are generally used for estimating head loss in open-channel flows. However, these models cannot be used when vortices formed in the wake region of macro-roughness elements [18,19]. For supercritical flows, the Darcy–Weisbach friction factor can be estimated using empirical models developed based on the field data when boulders are in the submerged condition [2,5,20]. In earlier studies, it was assumed that the roughness elements were submerged and the proposed models are valid for such flow conditions. However, the roughness elements present in mountainous streams are partially exposed to the atmosphere and few are in submerged conditions. The relative submergence significantly influences the head loss, which is a function of both stream flow depth and effective diameter of macro-roughness elements [2]. In a typical mountain stream, macro-roughness elements are non-uniformly sized and are randomly distributed. Recent experimental studies [14,15,16,18,21,22,23] have considered different characteristics of both flow and macro-roughness elements to estimate head loss (

Table 1. Experimental investigation on head loss due to macro roughness elements.

Flume Characteristics			Details of Macro Roughness Element’s				Flow Characteristics			Authors
Length	Width	Bed Slope	Shape	Diameter	Distribution	Cluster Density Range †	Flow Rate	Relative Submergence ‡	Froude Number
m	m	%	–	m	–	%	m3/s	–	–	–
3.5, 6, and 9	0.25, 0.35, and 0.5	8, 18, 25, and 40	Hemi sphere	0.029 and 0.038	Random	0–30	–	0.5–10.5	0.8–2.8	Pagliara and Chiavaccii [23]
12	2	0.001 and 0.005	Hemi sphere	0.054 and 0.116	Random	2.9–74.9	0.0004–0.0105	0.4–2.0	<1	Jordanova [19]
8.89	0.92	1.5, 3, and 5	Sphere	0.12–0.16	Structured	–	0.0250–0.1600	0.77–1.6	0.39–0.78	Baki et al. [15]
8.89	0.92	1.5	Sphere	0.14	Structured	–	0.1400–0.1980	1.56–1.90	<1	Baki et al. [21]
7 and 10	0.25 and 1	1–9	Cylinder	0.035 and 0.115	Structured	8–16	0.0100–0.0900	0.3–2.8	0.14–1.8	Cassan et al. [16]
4	0.4	1–9	Cylinder	0.035	Structured	8–19	0.0010–0.0180	>1	0.2–1.4	Cassan and Laurens [20]
7	0.92	5	Sphere	0.036–0.498	Random	0.7–16, 44	0.05–0.45	0.2–2.4	1.2–2.8	Thappeta et al. [25] *

† Cluster density is the ratio between area occupied by the roughness elements and the encompassed cluster area; ‡ relative submergence is the ratio between flow depth measured from channel bottom and size of the roughness element. * This study is based on numerical experiments.

). Different types of empirical models were proposed to estimate the head loss caused by macro-roughness elements in mountainous streams [23,24,25]. Moreover, Baki et al. [15,26] and Cassan et al. [27] also developed models to find head losses in fish passages. In these studies, empirical models were developed by considering the channel slope, flow velocity, flow depth, cluster density, size, and spatial distribution of roughness elements. However, the effect of random distribution of non-uniform sized macro-roughness elements for a given cluster density was not investigated in any of the above mentioned studies. Understanding empirically the importance of randomness of roughness elements in the head loss models is still necessary. However, nowadays, the flow resistance due to the presence of roughness elements has been studied widely using 3D computational fluid dynamics (CFD) models [25,26,28].

In this paper, flume experiments are focused on three random distribution/patterns of non-uniform size roughness elements under super critical flow conditions. An investigation was undertaken to study how random patterns of roughness elements affect head loss for a given cluster density. The proposed empirical models in the literature [23,25,27] are chosen based on the applicability of current experimental conditions and tested using the flume experimental data. As our flume experiments are limited to a single constant cluster density, this study aims (1) to experimentally analyse the effect of random distributions or patterns of non-uniform size roughness elements under supercritical flow conditions on surface water levels and head loss, and (2) test available empirical models in the literature that are proposed to estimate head loss in natural or constructed streams.

2. Methodology

2.1. Experimental Setup

Experiments were conducted in a horizontal rectangular flume with a length of 15 m and width of 0.9 m. An acrylic sheet was used to obtain a bed slope of 5% [2,15], which helps to create supercritical flow. This inclined false bottom surface has a length of 10 m, as shown in Figure 1. The flow that enters the flume is free from the flow disturbances because the upstream tank is located at a distance of 6.7 m from the flume entrance. Two centrifugal pumps with 25 HP capacity each were used to recirculate the flow. A test section of 3.96 m length starts at a distance of 3.38 m from the flume entrance (Figure 1). Spherical stainless-steel balls of size ranging from 0.0254 to 0.106 m were considered as roughness elements, placed randomly in three different patterns; planes 1, 2, 3, 4, and 5 in the flow direction were defined in the test section (Figure 2). The number of roughness elements (N = 6) and the cluster density (λ = 3%; Equations (1) and (2)) were kept constant for all of the experiments.

$$\lambda =\frac{N\frac{\pi }{4}{D}_{eff}{}^2}{l·b}=3\%$$

(1)

$$D_{eff}=\sqrt{\frac{D_1^2+D_2^2+\cdots +D_N^2}{N}}=0.067 \mathrm{m}$$

(2)

where, D₁, D₂, D₃…D_N are the individual diameters of the roughness elements R1, R2, R3…RN, respectively, and D_eff is the effective diameter. Among the six roughness elements, D₁ is the largest size; D₃ and D₆ are 72% of D₁; D₅ is the medium size (48% of D₁); and D₂ and D₄ are the smallest size (24% of D₁). These sizes were selected randomly. In the three patterns, the position of the largest size (D₁ = 0.106 m) R1 was changed along the central plane 3, in order to avoid disturbances near the boundary (Figure 2). The position of R2, R3, and R6 in the test section was changed for the three patterns, but the position of R4 and R5 was unchanged. The cluster length l (1.350 m) represents the longitudinal distance between the starting point of the first boulder and the end point of the last boulder, and the cluster width b (0.575 m) represents the lateral distance between the left-most and the right-most boulders. Grid width (0.15 m) was not varied in the flow direction until 5.73 m, whereas it was changed in the transverse direction. The roughness elements were placed away from the side walls of the flume to avoid the wall effect. Three flow rates were used in this study (0.110, 0.121, and 0.131 m³/s). Corresponding relative submergence [25] was calculated using Equation (3), resulting in 0.93, 1.00, and 1.09 (

Table 2. Details of various experimental parameters.

Pattern	Flow Rate (m 3/s)	Average Flow Depth at “Section-U” Hu (m)	Relative Submergence Hu/Deff	Froude Number at “Section-U” Fu	Cluster Density λ (%)
A, B, and C	0.110	0.0616	0.93	2.71	3
	0.121	0.0666	1.00	2.65	3
	0.131	0.0721	1.09	2.56	3

).

Relative submergence = (H_u/D_eff)

(3)

where H_u is the flow depth of the approaching flow at the upstream of the roughness elements.

2.2. Measurements

An ultrasonic flow meter (UFM 6760, Adept Fluidyne, Pune, India) with an accuracy of ±1% was used to control the flow rate delivered by the pumps. The flow velocity and water levels were measured at 58 grid points in the test section (Figure 2) using an Acoustic Doppler Velocimeter (ADV; Model N4000-72, Nortek, Boston, USA) fitted to an automatic traverse system and a digital point gage (Model SDV-24E -vertical; Mitutoyo, Kanagawa, Japan), respectively. The grid was prepared in such a way that the ADV can measure the velocity data without any difficulty owing to the presence of roughness elements. A probe check was carried out to confirm the signal strength as per the guidelines of the instrument. The digital point gage can measure water depth up to 0.6 m with an accuracy of ±0.05 × 10⁻³ m. The water level at each grid point was measured three times to avoid errors and an average value was taken. The ADV data were despiked as per Goring and Nikora [29] standards before using the data for further analysis, as discussed below. The 3D point velocities along the flow depth at each grid point were measured at an interval of 0.005 m starting from the flume bed using an ADV side looking probe with a sampling frequency of 25 Hz. The temporal mean velocity was unchanged beyond a sampling period of 180 s. The velocity data recorded by the ADV have shown a correlation coefficient (COR) and signal-to-noise ratio (SNR) between 40% and 90% and 15 and 25 dB, respectively. For highly turbulent and aerated flow, the COR and SNR values were reported as up to 30% and 5 dB, respectively [30]. Initially, the measured velocity data were passed through a filtering scheme of SNR ≥15 dB and COR ≥40%, and the spikes were eliminated [29]. The total head at any section along the flow is the summation of datum head, pressure head, and velocity head. The total head at upstream (section-U) and downstream (section-D) sections is denoted as TE_u and TE_d, respectively (Figure 1 and Figure 2). Subscripts ‘u’ and ‘d’ denote the upstream and downstream sections of the channel, respectively. The total head loss (ΔTE = TE_d − TE_u) occurs owing to the presence of both roughness elements (ΔE) and bed friction (ΔE_bed). The head loss due to bed friction (ΔE_bed) was calculated using experimental data obtained without roughness elements, and ΔE_bed was found to be 0.058 m, 0.048 m, and 0.046 m for a flume discharge of 0.110, 0.121, and 0.131 m³/s, respectively. The Mannings n is equal to 0.01, calculated from the flow data without roughness elements. Later, ΔE can be calculated (ΔE = ΔTE − ΔE_bed). The Froude number ($F_u$) of our study varied over a small range from 2.56 to 2.71. For our flume experimental conditions, it is found that velocity profiles using an ADV side looking probe can be measured accurately for a minimum flow depth of 0.0616 m and the corresponding $F_u$ is 2.71, whereas 2.56 is fixed based on the maximum allowed flow rate of 0.131 m³/s.

2.3. Proposed Empirical Models

The following empirical formulae were proposed by different authors to find the increase in Darcy–Weisbach friction factor (f_i), head loss (ΔE), and flow rate $\left(q\right)$ due to presence of roughness elements valid for supercritical flow. However, we converted them to the total head loss (ΔTE) using the well-known general Equations (13)–(15) and compared them with our observed total head loss data. The detailed comparison and corresponding error for different patterns are listed in

Table 3. Comparison of total head loss (ΔTE) with the models reported in the literature.

Present Experimental Study Observations					Pagliara and Chinvacini [23] Model fi = cλd (c = 0.6; d = 0.7)			Cassan et al. [27] Model $\mathit{q}=\sqrt{\frac{2\mathit{g}{\mathit{S}}_{\mathit{o}}\left(1-\mathit{\lambda }\right)}{{\mathit{C}}_{\mathit{D}}\mathit{f}\left({\mathit{F}}_{\mathit{c}}\right)\mathit{f}\left(\frac{{\mathit{H}}_{\mathit{u}}}{{\mathit{D}}_{\mathit{e}\mathit{f}\mathit{f}}}\right)\frac{{\mathit{C}}_{\mathit{r}}}{{\mathit{D}}_{\mathit{e}\mathit{f}\mathit{f}}}+\mathit{\beta }\frac{{\mathit{B}}_{\mathit{f}}}{{\mathit{H}}_{\mathit{u}}}}} {\mathit{H}}_{\mathit{u}}$ $\mathit{g}$$=9.81 \mathbf{m}/{\mathbf{s}}^2; {\mathit{S}}_{\mathit{o}}=0.05; {\mathit{C}}_{\mathit{r}}$$=0.006; {\mathit{C}}_{\mathit{D}}$$=0.3$$; {\mathit{D}}_{\mathbf{eff}}=0.067 \mathbf{m}; \mathit{\beta }$$=0.935; \mathit{B}=0.005$				Thappeta et al. [25] Model ΔTE = ΔE + ΔEbed $\mathbf{\Delta }\mathit{E}=\mathit{I}\left[\mathit{N}\right]\left[\mathbf{\Delta }{\mathit{E}}_{\mathit{s}\mathit{r}}\right]$ $\mathit{N}=6$
Pattern	Q	Vu	ΔEbed	ΔTE	fi	ΔE	ΔTE	Error	q	ΔTE	Error	ΔE	ΔTE	Error
	(m3/s)	(m/s)	(m)	(m)	-	(m)	(m)	(%)	(m2/s)	(m)	(%)	(m)	(m)	(%)
A	0.110	1.98	0.058	0.131	0.050	0.187	0.243	−85	0.187	0.167	−22	0.092	0.150	−12
	0.121	2.02	0.048	0.110	0.050	0.170	0.217	−97	0.232	0.153	−28	0.082	0.130	−15
	0.131	2.02	0.046	0.092	0.050	0.170	0.212	−130	0.225	0.144	−36	0.074	0.120	−23
B	0.110	1.98	0.058	0.124	0.050	0.187	0.243	−96	0.187	0.167	−26	0.092	0.150	−17
	0.121	2.02	0.048	0.127	0.050	0.170	0.217	−71	0.232	0.153	−17	0.082	0.130	−2
	0.131	2.02	0.046	0.095	0.050	0.170	0.212	−123	0.235	0.144	−34	0.074	0.120	−20
C	0.110	1.98	0.058	0.154	0.050	0.187	0.243	−58	0.187	0.167	−7.9	0.092	0.150	+2
	0.121	2.02	0.048	0.147	0.050	0.170	0.217	−48	0.232	0.153	−4	0.082	0.130	+11
	0.131	2.02	0.046	0.087	0.050	0.170	0.212	−144	0.225	0.144	−39	0.074	0.120	−27

.

Pagliara and Chiavaccini [23] model:

f_i = cλ^d

(4)

Thappeta et al. [25] models:

$$\mathsf{\Delta }E=I\left[N\right]\left[\mathsf{\Delta }{E}_{sr}\right]$$

(5)

$$I=2.0{\left(\lambda \right)}^{-0.37}{F}_u{}^{0.22}{\left(\frac{H_u}{D_{eff}}\right)}^{-0.57}{\left(\frac{H_u}{LD}\right)}^{1.00}$$

(6)

$$\mathsf{\Delta }{E}_{sr}=LD\left[0.062\times {F_u}^{1.42}{\left(\frac{H_u}{D_{eff}}\right)}^{-1.61}{\left(\frac{H_u}{LD}\right)}^{0.88}\right]$$

(7)

Cassan et al. [27] models:

$$q=\sqrt{\frac{2g{S}_o\left(1-\lambda \right)}{C_{D}f\left(F_c\right)f\left(\frac{H_u}{D_{eff}}\right)\frac{C_r}{D_{eff}}+\beta \frac{B_f}{H_u}}} H_u$$

(8)

$$B_f=\frac{2}{{\left(5.1log\left(\frac{H_u}{k_s}\right)+6\right)}^2}$$

(9)

$$f\left(F_c\right)=min{\left(\frac{1}{1-\frac{F_c^2}{4}}, \frac{1}{F_c^{0.67}}\right)}^2$$

(10)

$$F_c=\frac{F_u}{1-\sqrt{\frac{C_r l}{b}}}$$

(11)

$$f\left(\frac{H_u}{D_{eff}}\right)=1+\frac{0.4}{{\left(\frac{H_u}{D_{eff}}\right)}^2}$$

(12)

General models:

$$\mathsf{\Delta }E=\frac{ f_{i}L{V}_u^2}{8gR}$$

(13)

ΔTE = ΔE + ΔE_bed

(14)

$$\mathsf{\Delta }TE=\frac{q^2{n}^{2}L}{H_u^{10/3}}$$

(15)

where f_i = increase in Darcy–Weisbach friction factor due to roughness elements; c = 0.6; d = 0.7; I = interaction factor; $\mathsf{\Delta }{E}_{sr}$ = head loss due to a single roughness element; $F_u$ = Froude number; $F_c$= correction factor for $F_u$; LD = longitudinal distance from section-U to the nearest boulder (for the present study, LD = 0.947 m); $g$ = 9.81 m/s²; S_o = bed slope; $C_r=\frac{D_{eff}{}^2}{lb}$ = roughness elements concentration assuming $l, b$ are equal to the cluster center-center distance of in longitudinal and lateral direction, respectively; $C_D$ = drag coefficient (=0.3 for spherical shape roughness element [31]); $\beta =1-\sqrt{\frac{C_{r}b}{l}}-\frac{\left(\lambda \right)}{2}$; $\frac{H_u}{D_{eff}}$ = relative submergence; $B_f$ = bed friction factor; $f\left(F_c\right)$ = correction function to consider the influence of Froude number on flow resistance; $f\left(\frac{H_u}{D_{eff}}\right)$ = correction function to consider the interaction between flow and channel bed; q = flow rate per unit width; L = length of test section; n = Mannings roughness value; $k_s={\left(6.7n\sqrt{g}\right)}^6$ = equivalent sand grain size roughness value; $V_u=$ averaged velocity; and R = hydraulic radius. Subscripts ‘u’ and ‘d’ denote upstream and downstream sections of the channel, respectively, as mentioned earlier. Pagliara and Chiavaccini [23] model was developed based on purely empirical data. Cassan et al. [27] model was a semi-analytical one based on momentum balance and experimental data. Accurate selection of functional relation for $B_f$ is an important aspect while estimating head loss. Equation (9), used in our study, can accurately estimate bed friction for hydrodynamically rough bed with roughness elements. However, Blasius formula can be used for the hydrodynamically smooth bed where flow is smooth [16]. The study of Thappeta et al. [25] is based on complete numerical experiments using the 3D CFD ANSYS-CFX model. Thus, we considered three forms of head loss models to test the random pattern effect on head loss.

3. Results

3.1. Water Surface and Velocity Profiles

Figure 3 shows experimental images for a flow rate of 0.110 and 0.131 m³/s. For pattern A (Figure 3a,b), it can be observed that the roughness element R3 was in submerged condition when the discharge was increased from 0.110 to 0.131 m³/s. However, at the same flow rate for pattern C, a water jump has not formed at R3 owing to a change in the position of R1 (Figure 3e). Meanwhile, in pattern B, the water jump formed over roughness elements (R3 and R6), which may be because of the upstream protruding boulder R1 (Figure 3c,d). For pattern C, the water jump does not form over R3, although the discharge increased, but no protruding boulder was present upstream of the same plane (Figure 3f). It can be inferred that the position of the roughness elements (especially R1) and discharge (or Froude number) together influenced the formation of the water jump, and thus the flow structure, although the cluster density was unchanged. The water jumped over the boulder in the form of a jet and splashed onto the downstream water pool at some distance from the boulder when flow depth (y) was less than D. Such water jumps occurred noticeably along plane 3 for all the three patterns.

Figure 4 shows the water surface profiles in four planes for the three patterns of roughness elements for a constant discharge (Q = 0.110 m³/s) and cluster density. The profiles along plane 3 were shown separately in Figure 5 owing to water jump formation. A maximum water surface level of 0.064, 0.068, and 0.066 m along plane 1 was observed owing to the presence of R5 at 5.28 m longitudinal distance for patterns A, B, and C, respectively (Figure 4a). However, the water surface levels dropped before and after the roughness element. For pattern A and B, the water surface profiles along plane 1 and 2 did not vary significantly. This is because of the unchanged position of largest roughness element R1 (Figure 4a,b). Meanwhile, a significant difference was observed for pattern A and B along plane 4 and 5 due to the jump formation over R3 at 4.83 m in pattern A, and no jump in pattern B (Figure 4c,d). Although the roughness elements were not present in plane 2, the variation in the water surface profiles can be noticed in pattern C because of adjacent roughness elements R1, R2, R3, R5, and R6. However, it was observed that the change in the magnitude of water surface level depends on both position and size of the roughness element (Figure 4). For pattern A along plane 4, the water level suddenly dropped to 0.047 m at a distance of 5.13 m owing to the jump formation over R3, which leads to the discontinuity in the water depth. Meanwhile, for other patterns at 4.98 m onward, the water surface profiles are unchanged owing to no such water depth discontinuity because of the change in the position of R1 and R3 (Figure 4c). A similar observation was noticed along plane 5, whereas water level suddenly dropped for pattern A (Figure 4d). However, all water surface profiles were not identical for a given plane, discharge, and cluster density when the pattern is different (Figure 4). Therefore, as hypothesized, the three patterns/position of roughness element influenced the observed water surface levels spatially.

Water jump formation was also evident in the water surface profiles along plane 3 for Q = 0.110 m³/s (Figure 5), regardless of roughness element pattern. Jets were formed over roughness elements R1, R3, and R6 in all three patterns (Figure 5). Measurements were taken to delineate multiple free surfaces, i.e., the lower and the upper nappes of the jets, and the free surface of the pool. The jet thickness was found to vary from 3 to 5 mm. An air pocket got formed between this water jet originating from the boulder to the downstream water pool. In pattern A, the water jet created at 4.38 m by R1 jumped to a height of 0.18 m, and then splashed into the downstream side pool at 0.45 m from the boulder (Figure 5a). In pattern B, the water jet created at 4.38 m by R1 splashed onto the downstream side the roughness element R3, and the depth of the downstream water pool between these two elements was shallow (Figure 5b). Similarly for pattern C, jump was observed at R1. The three patterns (especially different positions of R1 and R3) caused jump formation for constant Q and cluster density.

Four representative grid points in the upstream and wake regions of the roughness elements were chosen to show velocity profiles. Two grid points (1st and 4th) were located upstream of R1; the other two (11th and 13th) were located downstream of R1 (Figure 2a). The observed vertical velocity profiles for pattern A at Q = 0.110 m³/s are shown in Supplementary Figure S1 for the grid points. No significant variation in the velocity magnitude near to bed was found at all the four grid points, which might result from the supercritical flow conditions. A similar observation is given by Cassan et al. [27].

Observations from our flume experiments indicate that the roughness elements interact with the water surface profile, giving rise to highly nonlinear spatial water surface variations when the roughness elements are just beneath the emergent conditions in supercritical flows. This interaction effect depends on the spatial locations of the non-uniformly sized elements within the cluster density for a given discharge.

3.2. Head Loss

The total head loss induced by the bed roughness and the turbulent flow around clustered roughness elements was measured in our experiments for all three roughness patterns (Table 3). In general, our results are in line with the finding of earlier studies [15,23,32], indicating that head loss decreases with an increasing flow rate/relative submergence. Partly, the experimental results of earlier studies relating to head loss due to macro roughness elements in steep open channels were used to derive empirical models (see Equations (4)–(12)) to estimate head loss under different conditions [23,25,27]. However, as none of these models were considered a parameter for randomness, we tested these models against our experimental data. As shown in Figure 6, the empirical models were only partly able to substantially estimate the observed head loss. The model of Pagliara and Chiavaccini [23] (Equations (4), (13) and (14)) generally leads to an overestimation of the measured ΔTE (mean overestimation 94.7 ± 32.9% (Table 3)). There is some explanatory power of the model, giving an R² of 42%, which describes the degree of collinearity between observed and estimated ΔTE, and larger values of R² can be obtained even that model estimations differ considerably in magnitude [33]; overall, however, the model performance is weak given an NSE of −21.1 and an RRMSE of 93%. The model of Cassan et al. [27] (Equations (8)–(12) and (15)) also leads to an overestimation of the measured ΔTE (mean overestimation 24 ± 12% (Table 3)). There is some explanatory power of the model giving an R² of 58%; overall, however, the model performance is also weak with an NSE of −2.6 and an RRMSE of 33.5%. The empirical model of Thappeta et al. [25] (Equations (5)–(7) and (14)) leads to an overestimation of the measured ΔTE (mean overestimation 12 ± 12% (Table 3)). There is some explanatory power of the model giving an R² of 73%; overall, however, the model performance is better than the other two models giving an NSE of 0.09 and an RRMSE of 18.4%.

4. Discussion

The results presented in the above section are discussed in the overall context of the possible reasons for overestimation of head loss by the three empirical models for our experimental work. The models do not explicitly account for patterns in random roughness elements, hence the models of Pagliaria and Chincacinin [23], Cassan et al. [27], and Thappeta et al. [25] do not show any changes in head loss with different patterns. A different general performance in estimating the head loss under supercritical flow can be recognized (Figure 6 and Table 3). This might be for several reasons. In the Pagliaria and Chincacinin [23] model: (i) f_i is a function of cluster density alone and the model does not address the effects of the approaching flow (Froude number and relative submergence); (ii) it assumes constant empirical coefficients for any random pattern of roughness elements; and (iii) hemispherical roughness elements were used in the study. In the Cassan et al. [27] model: (iv) the semi-analytical model is originally developed for the staggered pattern with uniform size roughness elements, where the present study focused on non-uniform size random patterns; and (v) both studies considered sub and superficial flows for the model development, $F_u$: 0.14–1.8 [27] and $F_u$: 0.8–2.8 [23]. (vi) $\lambda$, $F_u$ of Cassan et al. [27] study do not fall under the range of current experimental study. However, all remaining parameters are well within the range, as are those for the Pagliara and Chiavaccini [23] and Thappeta et al. [25] study (Table 1 and Table 2).

As mentioned earlier, the Thappeta et al. [25] model predicted the head loss better than the other two models, although the parameters in this model do not explicitly account for the randomness. A potential reason might be that the Thappeta et al. [25] model was developed for the Froude number ($F_u$) ranging from 1.20 to 2.80, considered purely supercritical flow, which is similar to this flume study. Nevertheless, the Thappeta et al. [25] model also overestimated by 12 ± 12%, which potentially results from a missing representation of different patterns of roughness elements.

From Table 3, it is observed that, for all three patterns at Q = 0.110 m³/s, the observed ΔTE varied from 0.124 m to 0.154 m (19.5% difference). At Q = 0.121 m³/s and Q = 0.131 m³/s, the observed ΔTE ranged from 0.110 to 0.147 m (33.6% difference) and from 0.087 to 0.095 m (9.2% difference), respectively. Thus, it is evident empirically that the tested three patterns affected ΔTE significantly for the given flow rate.

Overall, it is evident from our model testing that all proposed empirical models are not able to adequately represent the effects of randomly positioned roughness elements on head loss.

5. Conclusions

The objectives of the present study were to (i) understand changes in water surface levels and head losses due to the random patterns of non-uniform size roughness elements and (ii) test three different empirical head loss models. Therefore, we first conducted flume experiments under super critical flow with large Froude numbers (2.5–2.8) and measured the change in head loss in the case of different non-uniformly sized macro-roughness elements arranged in the test section. From these experiments and the model testing, we can draw the following conclusions.

Overall, significant changes in head loss (up to 33.6%) were observed in the case of all three random roughness element patterns. For the different patterns, water surface profiles were not identical for a given plane, discharge, and cluster density. The local water depth suddenly rose at the upstream and dropped at the downstream sides of the roughness element if the flow depth exceeded the size of the roughness element, whereas the formation of a water jump was noticed if the flow depth was slightly less than the size of the roughness element. In general, head loss decreased with an increase in the flow rate (or relative submergence).

The tested empirical models incurred large errors: 12–94.7%, considering the effect of the random patterns of roughness elements for a given cluster density and flow conditions.

The experimental data available from this study, especially water jump profiles at boulders, are rare for large Froude numbers, and are thus a valuable basis for further tests of empirical or three-dimensional mathematical models.