Hydraulic Scale Modeling of Pressurized Sediment Laden Flow

Abstract

In hydropower tunnel systems, unlined pressurized tunnels in competent rock are commonly used for cost-effective construction. Incorporating pressurized sand traps at the downstream end of these tunnels can increase plant capacity and improve energy efficiency. The present work focuses on optimizing the performance of existing pressurized sand traps. Hydraulic scale models were developed and tested at the Hydraulic Laboratory of NTNU, Within the 960 MW Tonstad Hydropower Plant in southern Norway as a case study. This study compares 1:1 velocity/sediment scaling with Froude scaling through physical experiments, analyzing velocity profiles via Particle Image Velocimetry (PIV) and sediment trap efficiency. Results show that Froude scaling, combined with geometric sediment scaling, provides superior accuracy in trap efficiency scaling across varying factors. However, in many practical hydropower applications, the large scaling factor required for laboratory models results in very small model sediments, leading to cohesion limitations. In such cases, Froude scaling may not be feasible. The 1:1 scaling method provides a conservative alternative. Hence, for practical applications, 1:1 scaling may be more cost-effective and sufficient for designing pressurized sand traps. This study emphasizes the importance of accounting for unscaled parameters and flow phenomena in hydraulic model design.

1. Introduction

Hydraulic scale modeling is a widely used method for researching, developing, and testing hydraulic structures. It involves creating smaller, manageable versions of real-world hydraulic systems to study their behavior in a controlled laboratory setting. Various approaches can be employed to study hydraulic systems, including experimental and physical modeling, as well as theoretical, conceptual, and numerical modeling. Each approach has its own advantages and limitations. While numerical models offer flexibility and can simulate complex scenarios, physical models were selected for this study to minimize potential sources of error. In physical models, water, sand, and gravity are real and directly represent the prototype conditions, eliminating the need for theoretical approximations [1]. Moreover, physical models avoid the discretization of time and space inherent in numerical simulations, which can introduce inaccuracies [2]. This direct representation of physical phenomena is particularly crucial when investigating complex sediment transport process in pressurized flows, where subtle interactions can significantly impact the results. Despite its widespread use, recent studies have highlighted ongoing uncertainties and challenges in accurately scaling pressurized, sediment-laden flows [3,4,5,6]. These challenges are especially pronounced when attempting to predict sediment transport behavior in such systems.

The fundamental principles of hydraulic scale modeling rely on achieving geometric, kinematic, and dynamic similarity between the model and prototype [5,7,8]. Among these, dynamic similarity is the most critical for ensuring a consistent ratio of forces acting on the system (i.e., the derived dimensionless parameters are equal in both model and prototype), typically presented by dimensionless parameters such as the Froude, Reynolds, and Euler numbers [7,8,9]. However, the complex flow dynamics of pressurized sediment-laden flows, governed by inertia, gravity, pressure, and viscosity, pose significant challenges. When scaling between a model and its prototype, no single fluid can simultaneously meet all force ratio criteria. It’s crucial to identify the dominant forces to determine an appropriate scaling method. Once identified, this scaling law can be employed to estimate prototype forces by scaling up those observed in the model, even acknowledging the inevitable occurrence of scale effect [5]. The scaling laws utilized in the design of physical models are established through dimensional analysis [10]. In typical hydraulic modeling, the most critical dimensionless parameters are those formed with respect to inertia. The outcome of a basic dimensional analysis is a set of independent dimensionless parameters, or groupings, that are essential for understanding flow and sediment transport based on [8,11,12,13,14]: the basic dimensional analysis with Buckingham π-theorem yielded to pi group of ${\pi }_A=f_A\left({\displaystyle \frac{P}{{\rho U^{*}}^2}},{\displaystyle \frac{{\tau }_0}{{\gamma }^{\prime }{d}_s}},{\displaystyle \frac{{\rho U}^{*}{d}_s}{\mu }},{\displaystyle \frac{{\rho }_s}{\rho }},{\displaystyle \frac{B}{d_s}},sin\theta ,{\displaystyle \frac{{U^{*}}^2}{{gd}_s}},{\displaystyle \frac{k_s}{d_s}},{\displaystyle \frac{w_s}{U^{*}}},{\displaystyle \frac{L}{d_s}}\right)$ where A in π_A might represent dependent variables such as flow resistance (energy gradient), sediment transport, or other hydraulic channel variables, P pressure, ρ water density, ${\rho }_s$ sediment density, U* shear velocity, τ_o bed shear stress, ${\gamma }^{\prime }$ submerged sediment specific weight, d_s sediment size, $\mu$ dynamic viscosity of water, B Tunnel width, L rib spacing and sin θ slope, k_s surface roughness, g gravitational acceleration and $w_s$ sediment settling velocity. Variables and all possible π terms (dimensionless parameters) together with their independent variables are defined in Section 2.3 and Table 3 respectively.

Traditionally, Froude scaling, which balances inertial and gravitational forces, has been widely used for open-channel flows [7,15]. However, its applicability to pressurized flows remains a subject of debate. Recent studies have used an alternative approach, such as Euler scaling, which focuses on accurately replicating pressure losses and turbulence [3,16,17]. Additionally, the 1:1 velocity/sediment size scaling approach, incorporating Euler scaling and Shields parameters, has gained a potential advantages over Froude scaling in pressurized systems [3]. This approach aims to maintain equal head loss and local losses in both the model and the prototype, thereby improving model accuracy.

The complicated nature of sediment transport, involving both bed load (rolling or sliding along the bottom) and suspended load carried within the water column) mechanisms, further complicates the scaling process [18]. Factors such as sediment particle size, flow turbulence, and fluid properties significantly influence sediment movement [19,20]. Despite advancements in two-phase flow models, their application in pipeline sediment transport remains challenging [21].

The selection of scaling criteria for sediment laden flows remains an active area of debate, with various researchers proposing different sets of dimensionless parameters [8,22,23,24]. Traditional scaling approaches present notable limitations, such as the inability to simultaneously achieve similarity in both the Shields parameter and sediment transport intensity [9,23], as well as challenges in maintaining similarity in relative fall velocity [22,25]. Previous researches [3,26] have raised important questions regarding the accuracy of physical modelling techniques, highlighting the need for further investigation.

This research contributes to identifying the most appropriate scaling method for modeling pressurized hydraulic sediment laden flows in hydropower sand traps and similar applications. Specifically, it aims to assess how accurately hydraulic scale models can represent pressurized sediment laden flows, particularly in the context of sand traps used in hydropower plants. However, the findings are expected to have broader implications for hydraulic modeling involving similar flow conditions.

A “Model family” of four geometrically scaled M1 (1:1), M2 (1:2), M3 (1:3.5 and M4 (1:7) was constructed to investigate how 1:1 scaling (where velocity and sediment size is kept constant between the models) and Froude and geometrical sediment scaling influences on the scale models sediment trapping accuracy. The study experimentally compares the two different scaling methods by comparing results from the four different physical scale models and using the largest of the four as the reference prototype, and the three smaller models to provide results with the two different scaling methods. To ensure the relevance of the research to real-world applications, the sand trap of the 960 MW Tonstad hydropower plant, located in southern Norway, was selected as a basis for the modelling, however, the geometry has been simplified to ensure broader applicability. The purpose of the modelling is not to perfectly model the real-world Tonstad sand trap, but to compare the results of scaling between the four different models. This approach aims to identify the most accurate scaling method for pressurized sediment laden flows and to contribute to improved design and analysis of hydraulic structures.

2. Methods and Materials

2.1. Reference Prototype and Geometrical Scaling Factor

The reference prototype for the models is the pressurized sand trap of Tonstad Hydropower Plant (960 MW), Southern Norway. The sand trap is an open sand trap of 200 m long, maximum flow of 80 m³/s and cross-section of 120 m². The sand trap geometry is described by [3,26]. Recently, the sand trap was upgraded with Rib-and-Ramp solution where a concrete ramp with 8^° inclination and eight ribs have been installed in the downstream end of the sand trap to increase trap efficiency, and this was based on 1:1 scaling methods. For modelling purposes, geometry is divided into three sections as shown in

Table 1. Tonstad sand trap geometry.

Geometry	Tunnel Section	Ramp Section	Rib Section
Length (m)	200	23	9
Width (m)	11	11	11
Height (m)	11	11–9.2	9.2

and Figure 1 presenting geometry in prototype measures.

2.2. Model Development

Four geometrically similar models were developed and tested at scales of 1:28.65, 1:57.3, 1:100 and 1:200 in reference to Tonstad sand trap III in Norway. For the remainder of this study, the largest model was considered as the prototype, and the scaling references were made relative to this model as illustrated in

Table 2. Model geometric scale between models and real-world reference (Tonstad).

Model	Scale	Scale (Tonstad)
M1	1:1	1:28.65
M2	1:2	1:57.3
M3	1:3.5	1:100
M4	1:7	1:200

.

2.3. Dimensional Analysis

To determine the physical scale modelling parameters, a dimensional analysis was conducted with Buckingham’s π-theorem [10]. In this work an effort has been made to investigate all potential outcomes of a dimensional analysis. The resulting dimensionless parameter from a dimensional analysis depends on which variables are selected as independent variables. There exists no consensus to which are the most correct and different researchers recommend different choices. Hence, to get a full overview of all possible dimensionless numbers that can characterize physical behavior, all possible variables that can have an effect are included, and all possible combinations of independent variables have been tested

Table 3. Dimensionless π-terms derived for the hydraulic and sediment transport system based on all relevant combinations of independent variables.

PI–Terms	Possible Independent Parameters
	ρ, v, B	ρ, v, ds	ρ, ${\mathit{\tau }}_0$ , ds	ρ, ${\mathit{U}}^{\mathit{*}}$, ds	ρ , ${\mathit{\gamma }}^{\prime }$, ds
π1	${\displaystyle \frac{P}{{\rho \mathrm{U}}^2}}$	${\displaystyle \frac{P}{\rho {\mathrm{U}}^2}}$	${\displaystyle \frac{P}{{\rho U^{*}}^2}}$	${\displaystyle \frac{P}{{\rho U^{*}}^2}}$	${\displaystyle \frac{P}{{\gamma }^{\prime }{d}_s}}$
π2	${\displaystyle \frac{\rho {\mathrm{U}}^2}{B{\gamma }^{\prime }}}$	${\displaystyle \frac{\rho {\mathrm{U}}^2}{{\gamma }^{\prime }{d}_s}}$	${\displaystyle \frac{{\tau }_0}{{\gamma }^{\prime }{d}_s}}$	${\displaystyle \frac{\rho {U^{*}}^2}{{{\gamma }^{\prime }d}_s}}$	${\displaystyle \frac{{\tau }_o}{{\gamma }^{\prime }{d}_s}}$
π3	${\displaystyle \frac{\rho \mathrm{U}B}{\mu }}$	${\displaystyle \frac{{\rho \mathrm{U}d}_s}{\mu }}$	${\displaystyle \frac{\rho U^{*}{d}_s}{\mu }}$	${\displaystyle \frac{\rho U^{*}{d}_s}{\mu }}$	${\displaystyle \frac{{\rho U}^{*}{d}_s}{\mu }}$
π4	${\displaystyle \frac{{\rho }_s}{\rho }}$	${\displaystyle \frac{{\rho }_s}{\rho }}$	${\displaystyle \frac{{\rho }_s}{\rho }}$	${\displaystyle \frac{{\rho }_s}{\rho }}$	${\displaystyle \frac{{\rho }_s}{\rho }}$
π5	${\displaystyle \frac{B}{d_s}}$	${\displaystyle \frac{B}{d_s}}$	${\displaystyle \frac{B}{d_s}}$	${\displaystyle \frac{B}{d_s}}$	${\displaystyle \frac{B}{d_s}}$
π6	${\displaystyle \frac{L}{B}}$	${\displaystyle \frac{L}{d_s}}$	${\displaystyle \frac{L}{d_s}}$	${\displaystyle \frac{L}{d_s}}$	${\displaystyle \frac{L}{d_s}}$
π7	$sin\theta$	$sin\theta$	$sin\theta$	$sin\theta$	$sin\theta$
π8	${\displaystyle \frac{U^2}{gB}}$	${\displaystyle \frac{U^2}{{gd}_s}}$	${\displaystyle \frac{{U^{*}}^2}{{gd}_s}}$	${\displaystyle \frac{{U^{*}}^2}{g{d}_s}}$	${\displaystyle \frac{{\gamma }^{\prime }}{g\rho }}$
π9	${\displaystyle \frac{k_s}{d_s}}$	${\displaystyle \frac{k_s}{d_s}}$	${\displaystyle \frac{k_s}{d_s}}$	${\displaystyle \frac{k_s}{d_s}}$	${\displaystyle \frac{k_s}{d_s}}$
π10	${\displaystyle \frac{w_s}{\mathrm{U}}}$	${\displaystyle \frac{w_s}{U}}$	${\displaystyle \frac{w_s}{U^{*}}}$	${\displaystyle \frac{w_s}{U^{*}}}$	${\displaystyle \frac{w_s}{U^{*}}}$

.

According to [3,26], Julien [11], Yen [13] and Yalin [8] the sediment load q_s is influenced by up to thirteen parameters: mean approach flow velocity U, or bed shear stress τ_o, densities ρ, ρ_s, pressure p, Tunnel width B, rib spacing L and slope sin θ, dynamic viscosity of water μ, gravitational acceleration g, submerged sediment specific weight γ’, surface roughness k_s, sediment size d_s, sediment settling velocity Ws. The sediment discharge rate q_s can be represented as: flow parameters (ρ, µ, U*, d, g, P), sediment parameters (d_s, ρ_s, σ_g, ${\gamma }^{\prime }$, W_S), tunnel geometry (B, $sin\theta ,L$)

$$q_s=f_1\left(\rho ,U^{*}, B, L,sin\theta ,P,\mu ,{\rho }_{s, }g,k_s,{\gamma }^{\prime },d_s,w_s\right)$$

(1)

By selecting the most suitable independent variables ρ, U* and d_s from Table 3 that are resulting from all possible π-terms for all other combinations of independent variables considering a two-phase flow. And thus, the selected independent variables one can derive the π-terms are presented in the fourth column of the table (Table 3).

Table 3 shows that for all possible combinations of independent variables, the system is characterized with the same forces as: Shield parameters, Particle Reynolds number, relative density, relative width, ratio of rib spacing to sediment size, relative roughness, and slope. However, variations in the representation of forces such as Euler number, Froude number, and relative settling velocity are evident across different combinations, as presented in Table 3. These variations indicate that the representation of these forces depends on the selected independent variables.

Reynolds scaling is challenging, as it would require impractically high velocities for large geometrical scaling factors. Additionally, the Reynolds number cannot be simultaneously matched with the Euler and Froude numbers. Typically, the Reynolds number is not strictly scaled in hydraulic modeling, as its effects are considered limited when the flow is fully turbulent [7]. Therefore, scaling factors should be selected to ensure that the flow remains in the turbulent regime and that it is hydraulically smooth.

The Euler and Froude model laws result in the same scaling factors as shown in

Table 4. Scale factors applied for Euler and Froude similarity laws used in the hydraulic model analysis.

Parameters	Dimensions
Length	Lr
Area	Lr2
Volume	Lr3
Time	Lr1/2
Velocity	Lr1/2
Discharge	Lr5/2
Pressure	Lr

. Furthermore, water density (1000 kg/m³), gravity (9.81m/s²), viscosity (8.93 × 10⁻⁷ m²/s and temperature must be kept equal in both the model and the prototype.

2.4. Experimental Setup and Instrumentation

Four hydraulic scale models M1, M2, M3 and M4, were built at scales of 1:1, 1:2, 1:3.5 and 1:7, respectively. While M1 and M2 were built in a hydraulic flume of 0.6 m in width, 0.74 m in depth, and 12 m in length, M3 and M4 were constructed in a mini flume of 0.18 m in width, 0.38 m in depth, and 2.4 m in length at the hydraulic laboratory of NTNU.

The models were designed as square closed conduits, built from transparent acrylic glass (plexiglass) to ensure visibility and prevent corrosion contamination that could affect sediment transport. To optimize time and material costs, models M1 and M2 were built side by side within the larger flume. In contrast, models M3 and M4 were built separately to fit within the mini flume, ensuring economic feasibility. Model dimensions are illustrated in

Table 5. Overall dimensions of the physical scale models.

Geometry	M1	M2	M3	M4
Length (m)	6.98	3.49	2	1
Width (m)	0.384	0.192	0.11	0.055
Height (m)	0.384	0.192	0.11	0.055

.

The sand trap box was positioned at 6.67 m, 3.33 m, 1.91 m, and 0.96 m downstream of the conduit inlet and 0.768 m, 0.384 m, 0.22 m, and 0.11 m before the tunnel exit of M1, M2, M3, and M4 respectively considering geometric similarity of the prototype and the models. Each model was fully flooded and submerged within the flume, with the roof secured using with pressure weight of water and a locking mechanism placed on top.

Water circulation was maintained through two interconnected reservoir tanks operating in a recirculating closed-loop system. Both the inlet and outlet of the conduit were connected to the hydraulic reservoirs via circular pipes with an outer diameter of 0.2 m (Figure 2a). The mini flume was also integrated into a recirculating closed-loop system (Figure 2b), utilizing a lower reservoir tank equipped with three submersible pumps and a flow control valve.

This setup maintained a uniform hydraulic grade, minimizing large-scale turbulence formation and ensuring even water distribution throughout the flow. To further enhance flow uniformity, mesh flow straighteners were installed 3 m and 4 m before the inlets of M1 and M2, respectively, and 0.2 m before the inlets of M3 and M4. Each model maintained a uniform cross-section along its length, except in the final section, where the sand trap consisting of a ramp and ribs was installed. The dimensions of the trap section are detailed in

Table 6. Cross-sectional dimensions of the model components, including ramp, sand trap, and outer upstream tunnel.

Geometry	Ramp				Sand Trap				Outer Upstream Tunnel Cross Section
	M1	M2	M3	M4	M1	M2	M3	M4	M1	M2	M3	M4
Length (mm)	800	400	229	115	314	157	90	45	-	-	-	-
Width (mm)	384	192	110	55	384	192	110	55	400	200	120	65
Height (mm)	62.8	31.4	18	9	62.8	31.4	18	9	400	200	120	65
Ramp angle (°)	4.5	4.5	4.5	4.5	-	-	-	-	-	-	-	-
Rib spacing and rib width (mm)	-	-	-	-	34.9	17.45	10	5	-	-	-	-
Rib Thickness (mm)	-	-	-	-	10	5	2.87	1.5	-	-	-	-

.

The experimental setup includes a 0.7 bar differential pressure sensor (PS) for pressure measurements, an electromagnetic flow meter (FM), as illustrated in Figure 2. Differential pressure measurements were obtained using Aplisens APR-2000ALW sensors with an accuracy of ±0.025%, corresponding to a maximum measurement error of ±1.0 mm at the model scale. This error translates to approximately 0.0029 m at the prototype scale, which is considered negligible. The flow meters have an uncertainty of ±0.4%, resulting in a maximum measurement error of 0.007 L/s at the model scale.

2.4.1. PIV System and Data Processing

Particle Image Velocimetry (PIV) is an image-based velocity measurement technique widely used to analyze turbulent flow fields with high spatial resolution. In this study, a classical two-dimensional (2D) PIV system from TSI was employed. The water was seeded with neutrally buoyant tracer particles (~1000 kg/m³), which were illuminated by a laser sheet generated using a Nd:YAG (neodymium-doped yttrium aluminum garnet) double-pulsed laser. The laser sheet was aligned vertically and parallel to the flow direction.

One 4-megapixel CCD (charge-coupled device) cameras captured particle images at a frequency of 20 Hz for 3 min per discharge condition, yielding a total of 3600 image pairs. These images were processed using TSI’s Insight 4G software in three stages: pre-processing (enhancing particle clarity), processing (defining interrogation window size and overlap), and post-processing (interpolating and correcting bad vectors). Instantaneous velocity fields were obtained through statistical cross-correlation of paired images, separated by a time delay Δt (1–10 ms). Figure 3 illustrates the PIV setup, and Figure 4 presents a sample raw image and resulting vector field for model M3.

2.4.2. Seeding Particle Selection and Validation

To evaluate the suitability of the seeding particles used for PIV measurements, the ratio of the Stokes number to the Froude number (St/Fr), was assessed, following the recommendation by Mathai, et al. [27]. This ratio represents the relative influence of particle inertia to gravitational acceleration in turbulent flows and should ideally be much less than one (<<1) to ensure that particles accurately follow the flow dynamics.

In this study, polyamide particles with a diameter of 50 μm and density close to that of water (~1000 kg/m³) were used as seeding material. The mean flow velocity in all models was 0.661 m/s, with characteristic lengths of 6.98 m, 3.49 m, and 2 m for M1, M2, and M3 respectively. Based on the estimated characteristic time scales (T_f) and particle relaxation times (T_p) the corresponding Stokes numbers were calculated as 1.48 × 10⁻⁵, 2.96 × 10⁻⁵ and 5.16 × 10⁻⁵. The Froude numbers based on the flow velocity and characteristic length scale were 0.341, 0.482, and 0.637 for the three models. Consequently, the calculated St/Fr ratio for all the models showed values well below 0.001, This analysis ensures that the particles can be considered accurate Lagrangian tracers of the flow, justifying the use of PIV measurements for detailed velocity field characterization in our pressurized system.

2.5. Sediment Materials

The sediments used in the tests are obtained from the real-world prototype sand trap. The sand was dried in an oven at 115 °C prior to testing and a sieve curve of Figure 5 is developed from the sieve analysis. The sediment sizes were selected based on the scaling methods applied, as detailed in

Table 7. Sediment particle sizes used in the experiments.

Scaling Method	Sediment Size (mm) (Natural Sand)
	M1	M2	M3	M4
1:1 velocity and sediments	0.1–0.25	0.1–0.25	0.1–0.25	0.1–0.25
	0.25–0.5	0.25–0.5	0.25–0.5	0.25–0.5
	0.5–1	0.5–0.1	0.5–0.1	0.5–0.1
	0.5–2	0.5–2	0.5–2	0.5–2
	1–2	1–2	1–2	1–2
	2–4	2-4	2–4	2–4
Froud scaling/geometric sediment	0.5–1	0.25-0.5	0.15–0.3	0.075–0.15
	0.5–2	0.25-1	0.15–0.6	0.075–0.3
	1–2	0.5-1	0.3–0.6	0.15–0.3
	2–4	1–2	0.6–1.16	0.3–0.6

.

The selected sand particles, within the specified grain size range, have a specific gravity of 2.65. Based on sieve analysis, sand grains in the 0.075–2 mm range (Figure 5) were specifically targeted for sediment distribution, based on sediments scaling though approximately 80% of the trapped sediment mixture falls within 0.5–2 mm ranges. This grain size distribution closely aligns with a previous study [3]. The amount of sand used in each test was determined based on the volume available beneath and between the ribs in the sand trap section. To ensure consistent sediment transport throughout the study, each test was conducted with a sand volume corresponding to 74% of the total capacity of the sand trap. Sediments were introduced into the model using a funnel at the inlet of each model experiment and prototype, with sand quantities of 1024 g, 128 g, 24 g, and 3 g placed into six individual cups for models M1, M2, M3, and M4, respectively.

To prevent excessively high sand concentrations in the water, a controlled time interval was established between consecutive cup releases. This interval was determined based on the time required for water to travel from the model inlet to the outlet, allowing the timing to be scaled proportionally across different models. Consequently, the time intervals for table per cup under the 1:1 scaling method were set to 10.56 s for M1, 5.28 s for M2, 3.03 s for M3, and 1.51 s for M4. Under Froude and geometrical sediment scaling, the feeding intervals were adjusted to 10.56 s, 7.47 s, 5.65 s, and 4.00 s for M1, M2, M3, and M4, respectively

Under the 1:1 scaling method, the same sediment size as the prototype was used in the model. However, for models using the Froude and geometrical sediment scaling approach, the sediment size was scaled proportionally to the model geometry. The corresponding d₅₀ was chosen for each sediment size range which are presented in Table 7.

2.6. Experimental Procedure

The experiments were conducted under steady flow conditions with sediment inflow across a range of parameter values in the models (

Table 8. Experimental properties of the physical models.

	1:1 Velocity/Sediment Scaling				Froude/Geometrical Sediment Scaling
	M1	M2	M3	M4	M1	M2	M3	M4
Discharge, L/s	97.4,	24.4	8	2	97.4	17.2	4.28	0.76
Average flow velocity, m/s	0.661	0.661	0.661	0.661	0.661	0.467	0.354	0.250
Particle diameter (d50), mm	0.375 0.75 1.25 1.5 3	0.375 0.75 1.25 1.5 3	0.375 0.75 1.25 1.5 3	0.375 0.75 1.25 1.5 3	- 0.75 1.25 1.5 3	- 0.375 0.625 0.7 1.5	- 0.215 0.358 0.45 0.859	- 0.11 0.179 0.225 0.429
Plexiglass roughness (ks), mm	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.015
f (Moody) (-) f (measured) (-)	0.014 0.0135	0.017 0.0173	0.019 0.022	0.023 0.027	0.014 0.0135	0.018 0.019	0.023 0.025	0.028 0.030
Re (-)	2.84 × 105	1.42 × 105	8.14 × 104	4.07 × 104	2.84 × 105	1.00 × 105	4.36 × 104	1.54 × 104
Eu (-)	0.127	0.150	0.173	0.209	0.127	0.164	0.209	0.255
Fr (-)	0.341	0.482	0.636	0.900	0.341	0.341	0.341	0.341
${\mathsf{\tau }}_{\mathbf{*}}$(-)	0.126 0.063 0.03 0.031 0.016	0.148 0.074 0.045 0.037 0.019	0.17 0.085 0.051 0.043 0.021	0.207 0.103 0.062 0.052 0.026	- 0.0630 0.0378 0.0315 0.0157	- 0.0810 0.0486 0.0405 0.0202	- 0.0988 0.0593 0.0494 0.0253	- 0.1203 0.0722 0.0601 0.0301
${\mathbf{R}}_{\mathbf{*}}$(-)	11.61 23.22 38.71 46.45 92.89	12.61 25.21 42.02 50.42 100.85	13.53 27.05 45.09 54.11 108.22	14.88 29.77 49.61 59.53 119.07	- 23.22 38.71 46.45 92.90	- 9.31 15.52 18.62 37.24	- 4.78 7.97 9.56 18.69	- 1.87 3.11 3.73 7.46
Relative Fall velocity, (-)	22.89 30.77 34.22 35.03 36.84	21.08 28.34 31.52 32.27 33.93	19.65 26.41 29.37 30.07 31.62	17.86 24.01 26.70 27.33 28.74	- 3.39 4.87 5.46 8.12	- 2.22 3.64 4.23 6.81	- 1.38 2.60 3.15 5.58	- 0.61 1.37 1.77 4.03

). The width of both the flume and the models remained constant throughout all tests. At the start of each experiment, the flume gradually flooded with water until the entire model was fully submerged, maintaining a water depth twice the model’s depth (h) to ensure pressurization. The pressure sensor was calibrated for every test needed.

To maintain a consistently pressurized system and verify flow conditions, a tail gate was installed downstream of the pressurized tunnel within the flume. Head loss measurements at the tunnel inlet and before sand trap were obtained using a differential pressure sensors. Sediment was introduced to the models as explained in Section 2.4. The experiment continued until equilibrium was reached but each sediment exiting the system was collected with syphoning technique within a given time interval, until sediment inflow equaled sediment outflow. At this stage, the trap efficiency of the models was assessed, and the maximum sediment-holding capacity of the sand trap (at equilibrium) was determined.

Additionally, the incipient motion of larger sediment particles of 0.5–2mm was analyzed at varying flow velocities over specific time intervals to evaluate particle movement in each model. The flow velocity of particles in the mid-section of the tunnel was measured using Particle Image Velocimetry (PIV) to analyze velocity profiles, and the resulting shear forces. The duration of each experiment was determined based on the time required for sediment transport to reach equilibrium.

Upon completion of each experiment, the tail gate was opened carefully to allow controlled flushing of water from the tunnel. Once the water was fully drained, measurements of the trapped sediment were taken to complete the analysis. The range of model properties used in the experiments are listed in Table 8 below.

2.7. Model Accuracy

This section presents a method to assess hydraulic scale modelling accuracy. The accuracy was evaluated by comparing the results from the four hydraulic scale models. It was determined using the equation presented in Equation (2), where P_m is the scale model parameter, and P_p is the prototype parameter that the accuracy is calculated for.

$$1-{\displaystyle \frac{\left|P_m-P_p\right|}{P_p}}$$

(2)

3. Results

3.1. PIV Analysis

Figure 6a illustrates a typical turbulent flow profile within a pressurized pipe. The mean flow velocity, derived from the double-averaged velocity profile, resulted to 0.67 m/s for M1, 0.67 m/s for M2, and 0.69 m/s for M3, closely aligning with the mean flow velocity of 0.66 m/s recorded by the flow meter. The PIV results further revealed maximum flow velocities of 0.72 m/s in M1, 0.73 m/s in M2, and 0.79 m/s in M3, corresponding to increases of 8.6 %, 10 %, and 16 %, respectively, relative to the mean flow velocity. The results from vertical flow velocity Figure 6b shows the ratio of maximum velocity in the horizontal to maximum in the vertical velocity and the minimum velocity in the horizontal flow to the maximum velocity in the vertical flow is very small in all the models its effect is low and insignificant on the horizontal velocities as the horizontal flow velocities are significantly larger than the vertical velocities The contour plots in Figure 6c depicts the flow field under steady state condition in the x-direction, results showed that maximum and minimum flow velocities in the models match close to the maximum and minimum flow velocities in Figure 6a.

Figure 7 presents the Reynolds shear stress ($R_{xy}=-\rho \overline{u^{\prime }{v}^{\prime }}$) profiles obtained from time averaged PIV measurements across the tunnel height for the three scaled models (M₁, M₂, and M₃). The raw data obtained was post-processed to remove outliers and local spikes. Noisy vectors exceeding a threshold of 0.01 in the u′ and v′ were replaced using moving average filter with linear interpolation. The results indicate a consistent trend: as model scale decreases, the magnitude of R_xy increases near the bed. The prototype model (M1) exhibits a relatively balanced profile with less intense shear stress gradients, while the smaller-scale models (M₂ and M₃) exhibit sharper, more pronounced stress gradients concentrated near the wall. This behavior suggests enhanced near-wall turbulence and momentum transfer in M₂ and M₃, corresponding with the increased sediment mobilization observed during experiments.

3.2. Dimensional Analysis and Comparison of Dimensionless Numbers

Figure 8 illustrates the influence of dimensionless parameters on trap efficiency in the hydraulic models under 1:1 velocity and sediment scaling (U_avag = 0.661 m/s, d₅₀ = 1.25 mm) and Froude and sediment geometric sediment scaling. The results of Figure 8a indicate that as the model size decreases, an increase in the Euler number (Eu), Shields parameter (τ*), Particle Reynolds number (Re)*, and Froude number (Fr) leads to a reduction in trap efficiency. Conversely, an increase in the Reynolds number (Re), the ratio of rib spacing to particle size (L/d₅₀), and the ratio of sediment settling velocity to shear velocity (Ws/U*) corresponds to an improvement in trap efficiency as the model size increases. The results from Froude scaling in Figure 8b shows that the effect of Euler number (Eu), Shields parameter (τ*), relative roughness (ks/d₅₀) on trap efficiency between the models is low and the trap efficiency is somewhat even closer to each other in M2, M3 and M4 compared to 1:1 scaling results. The Particle Reynolds number (Re)*, relative settling velocity (Ws/U*), and Reynold number (Re) with those the trap efficiency increases with increasing model size from M4 to M1and their effect on model trap efficiency similarity is low in comparison with 1:1 scaling. Unlike to 1:1 scaling the Froude number, relative diameter (d₅₀/B) and the ratio of rib spacing to particle size (L/d₅₀) are similar in all the models with a better and close trap efficiency among the scale models. In general, these opposing trends underscore the significant influence of dimensionless parameters on achieving similarity in trap efficiency across the model series.

3.3. Incipient Motion

Incipient motion tests were conducted along the centerline of each model, using a scaled sediment bed volume at the conduit invert. Incipient motion was defined as the point at which more than five sediment particles were observed to move according to [28]. In this study, the incipient motion was achieved at an average threshold flow velocities of 0.31 m/s in M1 and M2, 0.30 m/s in M3, and 0.29 m/s in M4 with corresponding head loss measurements of 0.89 mm, 0.99 mm, 1.09 mm, and 1.11 mm, respectively. However, M1 exhibited bed shear stress approximately 12% lower than the other models, indicating a slight deviation from perfect similarity as shown in Figure 9 where results from M2, M3 and M4 almost closely overlapped.

3.4. Maximum Capacity of Sand Trap

The maximum sediment capacity of the sand traps was measured in terms of the accumulated sediment within the sand traps and the sediment exiting the traps. The measurements were recorded every minute in all the models in the 1:1 velocity and sediment scaling. But in Froude and geometrical sediment scaling measurements were taken every 5-min intervals for model M2 and 10-min intervals for models M3 and M4, using a siphon tube. Dynamic equilibrium was achieved when sediment inflow and outflow were balanced, as Figure 10 illustrates the temporal progression towards equilibrium of the four models. These combination scaling was applied to ensure appropriate scaling of both flow and sediment characteristics within the models.

3.5. Trap Eefficiency for a Range of Particle Size

Hydraulic model tests employing 1:1 velocity and sediment scaling and Froude and geometric sediment scaling were conducted across a range of particle sizes (d₅₀). While maintaining a constant flow velocity of 0.661 m/s with 1:1 velocity and sediment scaling. The results, presented in Figure 11a, indicates that trap efficiency in M1 increased with sediment size (0.75 to 3 mm), primarily due to its relatively low bed shear stress and minimal scaling effects on rib spacing. In contrast, M2 and M3 exhibited an increase in trap efficiency within the d₅₀ range of 0.75 to 1.5 mm, followed by a decline at larger sediment sizes, likely attributed to the scaling influence on rib spacing relative to sediment size. For d₅₀ values below 0.375 mm, significant flocculation was observed, leading to suspended particle transport and, consequently, low trap efficiency in M1, M2, and M3. M4 consistently exhibited very low trap efficiency, influenced by both scaling effects and flocculation. Notably, with the absence of a trap at the model exits resulted in a complete loss of trap efficiency across all models, emphasizing the critical role of model boundary conditions in sediment retention assessments.

On the other hand, employing Froude scaling in conjunction with geometrical scaling of sediment size, were performed across a matrix of four distinct flow velocities and four varying particle sizes. The results in Figure 11b revealed a consistent trend: trap efficiency increased across all models with increasing sediment size. However, significant flocculation was observed with sediment sizes below 0.375 mm, resulting in suspended particle transport and a corresponding reduction in trap efficiency in M3 and M4. Furthermore, the absence of a sediment trap at the model exits led to a complete loss of trap efficiency in all experimental runs.

3.6. Trap Efficiency at Variable Flows

To evaluate the convergence of trap efficiency across different model scales, tests were conducted using 1:1 velocity and sediment with d₅₀ = 1.25 mm while varying the flow velocity. A 1:1 scaling approach was applied for both velocity and sediment size. Figure 12 presents the resulting trap efficiency for all three models, demonstrating a clear trend toward 100% trap efficiency as the flow velocity approaches the incipient motion velocity, on the contrary M4 will have pronounced low trap efficiency even close to the incipient motion velocity as indicated in Figure 12 below. However, M4 is highly affected by scaling, even closed to the incipient motion velocity the trap efficiency will not the same or close to the other models.

3.7. Sensitivity Tests

To assess the sensitivity of trap efficiency results, reverse engineering tests were conducted on the models with four different testing ways. The first test involved maintaining 1:1 sediment scaling while adjusting flow velocity to achieve comparable trap efficiencies and accuracy across different scales. The second test involved rib spacing (L) which was modified by 67% to assess its impact on the consistency of trap efficiency between models with a flow of 0.661 m/s and sediment size of 0.5–2 mm, results indicate that trap efficiency is affected in both models M1 and M2 by 0.4% but the effect is very high in smaller models M3 and M4. The results of these sensitivity analyses are presented in

Table 9. Sensitivity test results for variable flow velocity.

Model	ds (mm)	Original Test		Sensitivity Test		Accuracy (%)
		Uavg (m/s)	η (%)	Uavg (m/s)	η (%)
M1	0.5–2	0.66	96	0.83	92	100.0
M2	0.5–2	0.66	89	0.60	92	99.9
M3	0.5–2	0.66	73	0.37	91	99.2
M4	0.5–2	0.66	26	0.35	75	82.3

and

Table 10. Sensitivity test results for changes (modified) in rib spacing.

Model	Original Test		Sensitivity Test		Accuracy (%)
	L/d50 (-)	η (%)	L/d50 (-)	η (%)
M1	28	96	47	96	99.6
M2	14	89	23	89	99.6
M3	8	73	13	79	94
M4	4	26	7	61	64

, as well as in Figure 13a and Figure 13b, respectively

A third sensitivity test was conducted using Froude scaling for the flow, adjusting sediment size to achieve similar trap efficiencies across the models. As presented in

Table 11. Sensitivity test results for Froude scaling flow with sediment size tuning.

Model	Original Test			Sensitivity Test			Accuracy (%)
	Uavg (m/s)	ds (mm)	η (%)	Uavg (m/s)	ds (mm)	η (%)
M1	0.661	0.5–2	96	0.661	0.5–2	96	100
M2	0.467	0.25–1	94	0.467	1–2	95	98.96
M3	0.354	0.15–0.6	91	0.354	0.6–1.16	95	98.96
M4	0.25	0.075–0.3	87	0.25	0.3–0.6	88	91.7

and Figure 14a, this approach yielded trap efficiency with an accuracy of 98.96%, 98.96%, and 91.7% for M2, M3, and M4, respectively, relative to the prototype.

Furthermore, the fourth test was based up on geometrical scaling of sediment size with tuned flow rates was investigated. This method demonstrated improved accuracy, resulting in trap efficiency accuracy of 99%, 99%, and 97% for M2, M3, and M4, respectively, compared to the prototype, as shown in

Table 12. Sensitivity analysis results using geometric sediment scaling by tuning flow velocity to achieve comparable trap efficiency.

Model	ds (mm)	Original Test		Sensitivity Test		Accuracy (%)
		Uavg (m/s)	η (%)	Uavg (m/s)	η (%)
M1	0.5–2	0.661	96	0.827	92	100
M2	0.25–1	0.467	94	0.467	91	99
M3	0.15–0.6	0.354	91	0.354	91	99
M4	0.075–0.3	0.25	87	0.284	89	97

and Figure 14b.

3.8. Comparison of the Scaling Methods

A comparative analysis of trap efficiency reveals distinct differences between scaling approaches applied in hydraulic models. In this study, trap efficiency served as the primary metric to evaluate the performance of 1:1 velocity scaling versus Froude scaling. Figure 15 demonstrates that when Froude scaling is combined with geometric sediment scaling, trap efficiency is improved across models M1 to M4. Conversely, the 1:1 velocity and sediment scaling approach resulted in lower trap efficiencies, representing a more conservative method especially in smaller models.

4. Discussion

4.1. Flow Dynamics Based on PIV Results

4.1.1. Velocity Profiles

A comparative analysis of the double-averaged velocity profiles across the different model scales reveals significant variations (Figure 6a) given with 1:1 velocity scale. In particular, the horizontal velocity profile (U_avg) in M3 exhibits a distinct parabolic shape, indicating a strong influence of wall effects at this scale. As the model scale decreases, wall effects become increasingly dominant, altering the velocity distribution. With a decreasing scale factor (Lr), the flow profile transitions toward a characteristic logarithmic distribution, typical of pressurized turbulent flow.

However, the effect of the vertical flow velocity in all the models is low and considered insignificant (Figure 6b), as the horizontal velocities are significantly larger than the vertical velocities. Considering the ratio of minimum horizontal velocity to maximum vertical velocity, and maximum horizontal velocity to maximum vertical velocity, it is evident that vertical velocities have a relatively minor impact on the overall flow pattern. Nevertheless, their potential influence on boundary layer effects, turbulence generation near the model walls, and the development of secondary flow patterns warrants further investigation.

The contour plots in Figure 6c confirm that wall roughness effects are substantial, as velocities near the boundaries are lower than those at the center of the flow in each model, demonstrating that wall effects become more pronounced as the model size decreases.

Additionally, the reflection issues observed in M1, caused by the diffusion of the laser beam over the plexiglass surface from M2, appear to be visualized as part of the boundary layer effect. However, this phenomenon is not related to flow behavior; it results from optical reflections from M2 and the bed surface, as both models were constructed side by side within a single plexiglass flume.

4.1.2. Reynolds Shear Stress Profile

The observed increase in near-bed Reynolds shear stress (Figure 7) in M₂ and M₃ is primarily attributed to their lower Reynolds numbers, which result in higher relative wall friction and intensified turbulence near boundaries. These findings are consistent with numerical and experimental studies in turbulent ducts and channels, where reduced scale or Reynolds number leads to sharper stress gradients near walls [30,31,32]. Furthermore, experimental studies in constrained and open-channel ducts report similar stress distribution patterns under varying Reynolds conditions [33]. The steeper stress gradients in M₂ and M₃ explain the increased sediment mobilization and reduced trap efficiency observed experimentally. In contrast, the prototype model’s higher Reynolds number supports more distributed shear and reduced near-wall turbulence, favoring sediment deposition and higher trap efficiency as observed in the experiments for sediment trapping and incipient motion tests. These results reinforce the importance of Reynolds’ shear stress behavior in understanding scale effects in pressurized tunnel flows and sediment transport.

4.2. Dimensional Analysis

The principle of hydraulic scale modeling relies on establishing geometric, kinematic, and dynamic similarity between the model and prototype [5,7,8]. Among these, geometric and kinematic similarity were largely achieved across the models, dynamic similarity, which is essential for preserving proportional forces, is represented by dimensionless parameters such as the Froude, Reynolds, and Euler numbers [7,8,9].

To determine the key similarity criteria necessary for designing the scale models, a dimensional analysis was performed, and the most important parameters are presented in Table 3. Using 1:1 velocity and sediment scaling, these similarity criteria were not fully satisfied (except for ρ/ρs, ks/d₅₀, and slope θ) due to variations in head loss across the models. These differences stemmed from variations in head loss among the model scales, leading to different friction levels, despite all models being constructed from the same plexiglass material. This discrepancy also resulted in different bed shear stresses across the models.

Consequently, the dimensionless parameters Eu, Fr, τ*, Re*, Ws/U*, and Re exhibited dissimilarity between the models. Similarly, Froude scaling also failed to achieve complete similarity, except for the Froude number itself and a close similarity in trap efficiency compared to the 1:1 scaling. Furthermore, dimensionless numbers such as d₅₀/B and L/d₅₀ were influenced by sediment size and rib spacing, contributing to lower trap efficiency in the smaller-scale models and general dissimilarity in trap efficiency among them.

With Froude and geometric sediment scaling, the relative roughness ks/d₅₀ contributed to dissimilarity, due to sediment scaling. The primary factor influencing the dissimilarity of dimensionless parameters between the models in both scaling approaches is the fact that friction does not scaled proportionally between the models, and the effects of the Reynolds number are significant and cannot be neglected, as the flow is not in a fully rough, turbulent regime but rather smooth and turbulent [7].

4.3. Incipient Motion

Experiments were conducted in four scale models (M1–M4) using 0.5–2 mm naturally rounded quartz sediment, aiming to observe the threshold of sediment motion and compare it to the Shields diagram (Figure 9). The tests utilized controlled sandboxes located at distances of 3.106m, 1.553m, 0.889m, and 0.445m from the inlet, respectively, with corresponding dimensions of 0.384 m × 0.384 m and 0.0025 m height in model M1; 0.192 m × 0.192 m and 0.0025 m height in M2; 0.11 m × 0.11 m and 0.0025 m height in M3; and 0.055 m × 0.055 m and 0.0025 m height in M4. The corresponding sediment masses were 602.32 g, 152.3 g, 50 g, and 12.5 g, respectively.

Shields (1936c) discussed both bed load extrapolation and visual methods for determining critical shear stress, but the exact method used was not explicitly stated [34]. In this study, the Kramer visual observation method (weak transport, no bedforms) was employed [35]. However, all experimental data points fell below the Shields curve, indicating motion should not have occurred. Recent studies have shown similar results [36,37].

This discrepancy is likely attributable to several factors. First, Buffington and Montgomery [38] demonstrated that the bed load extrapolation method, generally accepted as Shields’ technique [39,40,41], overestimates critical Shields values compared to the visual observation method. Our results align more closely with visual observation-based experiments and are lower than those based on bed-load extrapolation.

Second, Shields’ experimental setup, which utilized open flumes with distinct boundary conditions, differed significantly from the pressurized hydraulic model used in this study. Pressurized systems are more susceptible to boundary effects, which can influence sediment motion.

Third, sediment characteristics, including shape, size, density, and sorting, significantly impact incipient motion [34,42,43]. Shields’ experiments used angular sediments with varying densities, which inherently resist motion due to higher particle friction [44], unlike the rounded quartz particles employed in this study.

Fourth, the location of the test section where fully developed flow was established and the reduction in the cross-section due to the 2.5 mm sediment bed might have facilitated the earlier initiation of sediment motion.

Therefore, the observed deviation from the Shields curve is likely the combined result of the chosen observation method, differences in the experimental setup, variations in sediment characteristics, and the configuration of the test section. Nevertheless, similarity at incipient motion was closely achieved across all models, except for M1.

4.4. Maximum Capacity of Cand Trap

The experiment aimed to determine the sediment capture efficiency of each scale model over time, until dynamic equilibrium was achieved, using both 1:1 scaling (velocity and sediment) and Froude scaling combined with geometric sediment scaling. Under 1:1 scaling (Figure 10a), larger models captured significantly more sediment than smaller ones, despite identical velocity and sediment size. This difference is attributed to: (1) increasing bed shear stress from M1 to M4; (2) unscaled rib spacing and sediment size within the sand trap; and (3) the proximity of the inlet to the mini flume in the smaller models (M3 and M4), resulting in faster flow, as evidenced by Particle Image Velocimetry (PIV) measurements.

However, Froude scaling combined with geometric sediment scaling improved sediment capture in smaller models compared to 1:1 scaling, owing to the appropriately scaled flow conditions and sediment size. Equilibrium time was longer in the smaller models, as the slower flow facilitated sediment capture. Nevertheless, sediment accumulation in the center of the sand trap led to some sediment escaping before full equilibrium was achieved, slightly affecting the capacity values compared to M1 and M2.

Overall, Froude scaling with geometric sediment scaling resulted in more comparable sand trap capacities across the models and produced longer equilibrium times compared to 1:1 scaling.

4.5. Trap Efficiency at Variable Sediment Sizes

In the present study, 1:1 velocity and sediment size scaling, as well as Froude scaling combined with a geometrical scaling approach, were employed for multiphase hydraulic model tests. Hydraulic model tests using 1:1 velocity and sediment scaling, conducted at Graz University in collaboration with NTNU on a 1:36.67 geometrical scale model [3], yielded a trap efficiency of 87% for sediment sizes between 0.3–1 mm. This result aligns with our findings for models M2 (88.6% trap efficiency) and M3 (69.2% trap efficiency), which operated within a similar sediment size range.

Additionally, tests performed at NTNU using Froude scaling with a 1:20 geometrical scale model [26] achieved a trap efficiency of 89% for sediment with a d₅₀ of 3 mm, with sediment deposition evenly distributed along the sand trap’s length. The trap efficiencies obtained with Froude scaling in the present study, for sediment sizes d₅₀ ranging from 0.1125 mm to 1.5 mm, ranged from 58.9% in M4 to 95% in M2, and were generally higher than those reported in the previous studies.

Across our scale models, sediment transport analyses using both scaling methods indicated that trap efficiency generally increased with sediment size, although this trend was less pronounced in model M4 under 1:1 scaling due to scaling effects. Experiments conducted without the sand trap resulted in near-zero trap efficiency across all models and sediment size ranges for both scaling methods, a finding consistent with the results reported by [3].

Overall, a comparative analysis of sediment trapping for varying sediment sizes indicates that Froude scaling combined with geometric sediment scaling demonstrates a higher sediment trapping capability compared to 1:1 scaling. This suggests that 1:1 scaling provides a more conservative estimation of sediment trapping efficiency than Froude scaling.

4.6. Trap Efficiency at Variable Flows

To examine the impact of flow velocity on trap efficiency, tests were conducted using 1:1 scaling for both velocity and sediment size, employing a consistent sediment size of 0.5–2 mm, while varying the average flow velocity across the models. The results revealed that the accuracy of trap efficiency comparisons between models improved as the average flow velocity decreased. Specifically, when sediment transport occurred mainly as bed load, trap efficiency increased with decreasing average flow velocity.

Conversely, at higher flow velocities, sediment transport transitioned to a combined bed load and suspension regime, with suspension becoming dominant in the smaller models (M3 and M4). This pronounced effect in the smallest model (M4) is likely attributed to boundary shear stress effect and scaling effects, which significantly limit potential improvements in trapping efficiency compared to the larger models (M1, M2, and M3).

4.7. Sensitivity Tests

A reverse engineering approach was applied to conduct sensitivity tests aimed at investigating ways to narrow the trap efficiency discrepancies between the hydraulic scale models. In the first test, sediment size (0.5–2 mm) was kept constant while flow velocity was varied, with M1 having the highest flow and M4 the lowest (M1 > M2 > M3 > M4). This adjustment was intended to compensate for bed shear stress differences and mitigate boundary effects in the smaller models, effectively reducing the differences in trap efficiency (Table 9, Figure 13a).

The second involved modifying the rib spacing within the sand trap to address scale effects related to sediment size and rib spacing [45], which were more pronounced at higher scale factors (Lr). This modification further reduced the trap efficiency variations between models (Table 10, Figure 13b). Both tests were performed under 1:1 scaling (velocity and sediment) to assess the sensitivity of trap efficiency to these parameters.

Sensitivity tests were also conducted using Froude scaling, with adjustments made to sediment size. These tests showed improved trap efficiency accuracy compared to the original experiments. This improvement is likely due to the larger sediment sizes used, which resulted in slower sediment transport. The original tests’ discrepancies were possibly influenced by sand resistance, critical Shields parameter variations, turbulence, and scale effects (Table 11, Figure 14a).

Finally, sensitivity tests incorporating geometric sediment scaling and flow tuning were performed. In this case, flow adjustments were made to counteract bed shear stress and turbulence effects in M1 and M2, and boundary effects in M3 and M4. These adjustments resulted in comparable trap efficiency and improved accuracy across all scale models (Figure 14b, Table 12).

4.8. Comparison of Scaling Laws

To evaluate the accuracy of 1:1 scaling (velocity and sediment) and Froude scaling (flow) combined with geometric sediment scaling, hydraulic scale model tests were conducted, focusing on quantifying the accuracy of these methods based on sand trap efficiency (Figure 15). The 1:1 scaling results revealed a decrease in trap efficiency accuracy from M1 to M4. This decline is attributed to several factors: variations in bed shear stress due to friction; pronounced boundary and scale effects in M3 and M4 (due to sediment size and rib spacing), and the short inlet distance (20cm) in M3 and M4, which imparted additional momentum to sediment transport.

Additionally, the limitations of 1:1 scaling, as suggested by Yalin [8], where minimum model scale is constrained by the prototype Reynolds number (U*ks/ν), may have contributed to the observed discrepancies, even though Yalin’s study focused on open river models.

Conversely, Froude scaling combined with geometric sediment scaling significantly improved trap efficiency consistency across all models, particularly in M3 and M4. While complete similarity is unattainable with either scaling method, geometric and kinematic similarity were achieved in both approaches. However, dynamic similarity concerning Eu, τ*, and Re* was not achieved in 1:1 scaling, primarily due to friction and Reynolds number effects [8], leading to lower trap efficiency accuracy. Froude scaling partially achieved dynamic similarity through the Froude number, but not for τ* or Re*, again due to friction-related issues. Despite this, Froude scaling demonstrated superior trap efficiency accuracy compared to 1:1 scaling.

These findings suggest that 1:1 scaling yields more conservative results relative to Froude scaling with geometric sediment scaling. Although [34,42,43] argued for the conservatism and accuracy of 1:1 scaling over Froude scaling, our results indicates that Froude scaling combined with geometric sediment scaling provides more consistent results across different scaling factors. However, for many practical applications, especially in hydropower, the necessary scaling factor to obtain models that can be constructed in a laboratory environment becomes large, and the resulting model sediments become smaller which can lead to the limit of cohesion. In such cases the Froude scaling may not be practically possible. For such cases, the 1:1 scaling provides an alternative, which is found in this research to provide conservative results. Hence, for practical application, the 1:1 scaling methods may still be preferred, less costly, and sufficient to find conservative designs in the case of pressurized hydropower sand traps. But to achieve a more comprehensive comparison and to further mitigate trap efficiency differences between models with both scaling methods, accurately scaling friction is recommended as a potential solution.

4.9. Limitations and Scope of the Study

While this study provides valuable insights into pressurized sediment-laden flows using both 1:1 and Froude-scaled models, certain limitations must be acknowledged: in light of these factors, we consider our findings on Froude and geometrical sediment scaling versus 1:1 scaling to be valid within the scope of the tested conditions and configurations. However, care should be taken in generalizing these conclusions to other systems without further validation, particularly where flow geometry, sediment type, or scale differs substantially.

5. Conclusions

This study compared 1:1 scaling (velocity and sediment) and Froude scaling combined with geometric sediment scaling under steady flow conditions. PIV analysis of double-averaged velocity profiles revealed significant scale-dependent variations, with smaller models exhibiting pronounced wall effects and a transition toward logarithmic velocity distributions which lead to low trap efficiency. However, vertical flow velocity was found to have a negligible effect on the overall flow structure.

Dimensional analysis indicated that while geometric and kinematic similarity were largely achieved, dynamic similarity—critical for accurate force representation- was compromised. This was particularly evident in the 1:1 scaling approach due to unscaled friction and Reynolds number effects, and to a lesser extent in Froude scaling.

Incipient motion experiments, conducted using the Kramer visual observation method, revealed deviations from the Shields curve. These deviations are attributed to differences in experimental setup, sediment properties, and the inherent overestimation of critical Shields values associated with bed-load extrapolation methods.

Regarding sand trap performance, 1:1 scaling exhibited reduced sediment capture efficiency as model size decreased, primarily due to variations in bed shear stress, unscaled rib spacing, and the relative proximity of the inlet. In contrast, Froude scaling with geometric sediment scaling significantly improved the consistency of trap efficiency across different model scales, particularly in smaller models.

Furthermore, sediment capture efficiency was observed to increase with sediment size and to decrease with increasing flow velocity. The1:1 scaling approach provided more conservative estimates in comparison to Froude scaling. Sensitivity analyses, employing reverse engineering techniques, confirmed the critical influence of flow velocity, rib spacing, and sediment size on trap performance–highlighting the necessity of accurately scaling these parameters. Overall, while 1:1 scaling offers conservative and practical estimates Froude scaling combined with geometric sediment scaling demonstrated superior accuracy and consistency across different model scales in replicating the trap efficiency of pressurized sand traps. However, in many practical applications-particularly in hydropower the large scaling factor required for laboratory implementation results in extremely small model sediments, where cohesion effects become significant. Under such constraints, Froude scaling may not be practically feasible. In these cases, the 1:1 scaling remains a viable and cost-effective alternative for developing conservative designs of pressurized hydropower sand traps. To further enhance similarity and reduce discrepancies between scaling approaches, accurate representation of frictional effects is strongly recommended.