High-velocity air–water flows in rough-walled tunnels Open Access

Air–water flows are typically found in natural and human-engineered systems, where they play a crucial role across a broad spectrum of applications, spacing from hydraulic and aerospace engineering to industrial and chemical processes. These flows are characterized by complex interactions between air and water phases, leading to a variety of flow regimes. A thorough understanding of these interactions is necessary for optimizing design, ensuring safety, and improving efficiency across these diverse applications. Examples include gas–liquid reactors (e.g., Kantarci , 2005; Ramezani , 2015), cooling systems (e.g., Kröger, 2004; Makarov , 2017), and urban drainage networks (e.g., Hager, 2010; Butler , 2018).

High-velocity air–water flows frequently occur at hydraulic structures, e.g., spillways, and in tunnel chutes downstream of gates, e.g., low-level outlets (LLO) of high-head dams. These flows are typically characterized by significant aeration and turbulence levels (Fig. 1). Aeration is a key feature that must be considered for several reasons, including the mitigation of cavitation effects, drag reduction, and flow bulking (Valero , 2024). After the pioneering investigation of air entrainment in spillway flows by Straub and Anderson (1958), many studies advanced the understanding of air entrainment processes (e.g., Chanson, 1996), air–water flow properties (e.g., Chanson and Toombes, 2002; Kramer and Valero, 2023), turbulence (e.g., Kramer , 2020), and scale effects (e.g., Hohermuth , 2021a). Additionally, important insights on flow properties and design recommendations were developed for smooth-invert spillways (e.g., Kramer , 2006), micro-rough invert spillways (Felder , 2023), macro-rough stepped spillways (e.g., Boes and Hager, 2003a; 2003b), and tunnel chutes (e.g., Hohermuth , 2020). In spillway flows, air entrainment occurs when the turbulent boundary layer reaches the free-surface (Wood, 1991), at which point the free-surface perturbations and instabilities overcome surface tension and buoyancy forces (Valero and Bung, 2018). Downstream of this location, the flows are a complex mixture of air–water interfaces of different sizes and shapes (Pfister and Hager, 2011; Valero and Bung, 2016). Flow aeration significantly affects the properties of the flow, i.e., by increasing flow depth (Falvey, 1980), reducing drag (Kramer , 2021a), enhancing air–water mass transfer (Gulliver and Rindels, 1993), and mitigating cavitation risk (Falvey, 1990).

In tunnel chutes without upstream gate, the air entrainment process closely resembles that observed in spillways. However, the presence of an inflow-controlling gate is a common feature in LLOs. The transition from pressurized flow to free-surface flow at the gate results in the formation of a high-energy water jet, with velocities reaching up to approximately 60 m/s at prototype scale, depending on the available head. Downstream of the gate, the flow is conveyed through the tunnel chute with typical longitudinal slopes ranging from 0% to 15%. The flow along the tunnel typically resembles a high-velocity open-channel flow with strong aeration (Felder , 2019; Hohermuth , 2020; 2021a). With increasing head and gate opening, the air–water mixture may fill the entire cross section, becoming a high-velocity, pressurized, and closed conduit flow. Previous research has mostly focused on the air demand of LLOs (e.g., USACE, 1964; Sharma, 1976; Hohermuth , 2020; Pagliara , 2024b), with limited attention given to the properties of such flows (e.g., Speerli and Hager, 2000; Felder , 2019). In addition, recent prototype (Hohermuth , 2020) and large-scale laboratory (Pagliara , 2024b) studies suggested that tunnel roughness significantly influences air demand, but little information has been reported on the effects on the key flow properties. Table I summarizes important prototype and laboratory studies that investigated the effects of wall roughness in high-velocity air–water flows. Straub and Anderson (1958) and Anderson (1965) were the first to present a broad dataset of void fractions characterizing high-velocity flows over spillways with smooth and micro-rough inverts. Several studies complemented this research, reporting properties of high-velocity flows over spillways with smooth and micro-rough inverts (e.g., Aivazyan, 1986; Chanson, 1996; and Felder , 2023) and macro-rough inverts (e.g., Chanson, 1994; Boes, 2000; and Felder and Pfister, 2017).

Conversely, the literature on the properties of high-velocity air–water flows in tunnels downstream of a high-head gate is limited. Although these flows exhibit Reynolds numbers comparable to those of spillways (⁠ $R$ ≥ 10⁵ at laboratory scale), they are distinguished by significantly higher Froude numbers (⁠ $F$ ≫ 6), which limits the applicability of findings from spillway studies. For such applications, the Reynolds number, representing the ratio of inertial to viscous forces in a fluid, has been typically defined as $R$ = D_h· $U¯$/ν, with D_h = hydraulic diameter for clear-water flow, Ū = mean flow velocity, and ν = kinematic viscosity. Similarly, the Froude number, which quantifies the ratio of inertial to gravitational forces, has been commonly expressed as $F$ = Ū/(g·h_w)^0.5, with g = gravitational acceleration and h_w = clear-water flow depth.

First contributions to this field were made by Speerli (1999) and Speerli and Hager (2000), who investigated fundamental air–water flow properties in a smooth tunnel, detailing flow patterns, mixture flow depths, and air concentrations. Subsequent studies by Felder (2019) and Hohermuth (2019) observed high-velocity flow patterns and a range of air–water flow properties in a smooth tunnel chute. Hohermuth (2021a) further extended this work by conducting prototype air–water flow measurements, providing important validation of previous laboratory campaigns.

To date, no study has systematically examined the effects of wall roughness on high-velocity air–water flows in tunnel chutes. The present study addresses this gap by investigating the influence of several wall roughness configurations on these flows. The experiments replicated typical flow conditions in LLOs of high-head dams, complementing past research (Table I). First, the experimental setup and methodology are outlined, including an assessment of scale effects. Next, the air–water flow patterns observed in the tunnel chute are qualitatively described to characterize the flow conditions. Subsequently, key air–water flow properties measured at fixed locations along the tunnel centerline are presented, alongside empirical scaling relationships. The effects of wall roughness on the air–water flow properties are then summarized through a comparative analysis of measurements taken under identical inflow conditions at corresponding non-dimensional locations along the tunnel, using the position of maximum air concentration as scaling reference.

This study advances the physical understanding of high-velocity air–water flows in tunnel chutes by highlighting the critical role of wall roughness and the associated hydraulic regime. Additionally, several empirical equations are proposed for scaling key flow properties, offering valuable design and verification tools for hydraulic engineering applications.

The experiments were conducted in a large-scale physical model at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH Zürich [Fig. 2(a)]. The hydraulic model consisted of (i) a rectangular pressurized inflow tunnel (width = 0.20 m, height = 0.25 m), (ii) a gate chamber with a sluice gate (maximum opening a_m = 0.25 m, width W_g = 0.2 m), (iii) an air vent system with a pipe (diameter = 0.15 m) connected to the gate chamber soffit, and (iv) a rectangular acrylic tunnel chute (height h_t = 0.30 m, width W_t = 0.2 m, slope S = 4%, and length L_t = 20.6 m) with varying wall roughness.

Two high-head pumps provided a water discharge 0.062 ≤ Q_w ≤ 0.256 m³/s, resulting in energy heads at the gate of 5 ≤ H ≤ 30 m. The sluice gate was operated by a motor and relative gate openings A = a/a_m = 0.2 and 0.4 were investigated. The air vent supplied air to the flow downstream of the gate; it had a loss coefficient ζ ≈ 2.1, resulting in an air vent parameter A^* ≈ 0.17 (see Pagliara , 2024b). The base configuration R1 consisted of a smooth rectangular acrylic tunnel, with equivalent sand roughness k_s = 0.05 mm [Fig. 2(b)]. The wall roughness was modified by mounting stretched aluminum plates on the acrylic as described by Pagliara (2024b). Two types of roughness typologies were adopted: (i) configuration R2, resulting in k_s = 3.9 mm and (ii) configuration R3, resulting in k_s = 13.2 mm [Fig. 2(b)], with k_s estimated from pressure measurements (see Pagliara , 2024b). For the roughness typologies R2 and R3, two wall lining setups were tested, i.e., full-lining (labeled with “f”), where the plates covered all walls, and invert-lining (labeled with “i”), where the plates covered only the invert [Fig. 2(b)]. The energy head at the gate was computed as H = (p_w,av)/(ρ_w·g) + [Q_w/(W_t·a_m)]²/(2·g), with p_w,av = average water pressure at the gate, and ρ_w = water density. The contraction flow depth downstream of the gate was defined as h_c = a·C_c, with C_c = contraction coefficient. Key non-dimensional quantities were computed at the vena contracta (subscript c) comprising the Froude number $F$_c = Q_w/[(W_g·h_c)·(g·h_c)^1/2], Reynolds number $R$_c = (4·Q_w)/[(W_g+2·h_c)·ν_w], and Weber number $W$_c = [(Q_w/(W_g·h_c))²·ρ_w·h_c/σ]^0.5, with ν_w = water kinematic viscosity at 20 °C, and σ = surface tension coefficient between air and water at 20 °C. A total of 188 flow conditions were investigated by varying the tunnel roughness (R1, R2f, R2i, R3f, and R3i), the gate opening (A = 0.2 and 0.4), and the energy head at the gate (H = 5, 10, 20, 25, and 30 m). These corresponded to Froude 18 ≤  $F$_c ≤ 47, Reynolds 9 × 10⁵ ≤ $R$_c ≤ 3 × 10⁶, and Weber 212 ≤  $W$_c ≤ 594 numbers at the vena contracta.

Air-water flow properties were measured with a dual-tip conductivity phase-detection probe (CP), currently representing the most reliable instrument to measure the internal properties of highly aerated flows at laboratory and prototype scale (Pagliara 2024a; Felder , 2024). The CP was manufactured at VAW with the design of those manufactured at the Water Research Laboratory (WRL), UNSW Sydney, which have been successfully used in past research (e.g., Felder and Pfister, 2017; Felder , 2019; and Hohermuth , 2021b). The probe tips were made of hypodermic needles (diameter = 0.5 mm) surrounding an inner platinum-wire electrode (diameter = 0.125 mm). The two tips were positioned side-by-side, separated in streamwise direction Δx = 3.80 mm and transverse direction Δy = 0.97 mm, with a negligible vertical offset (Δz ≈ 0 mm). The CP was carefully positioned in the flume centerline through openings in the tunnel soffit (alignment precision ≈ 1°) and shifted in vertical direction by a robotic arm (precision = 1 mm). The raw voltage signals from both probe tips were amplified with a WRL electronic unit to a voltage range of approximately 0.5–4.5 V and sampled with a LabVIEW high-speed data acquisition system from WRL.

Finally, data collected from Hohermuth (2019) and Felder (2019) in the same smooth tunnel chute (configuration R1) were re-analyzed and compared with the results of the present study. These were collected using a CP analogous to the one adopted in the current study, with a sampling duration of 45 s at a frequency of 500 kHz.

The scale of the physical model was 1:5–1:20 compared to high-head LLOs in the European Alps (reference length = tunnel width W_t), and the tests were scaled with Froude similitude. This leads to mismatches in Reynolds and Weber numbers between model and prototype; thus, scale effects must be considered (Heller, 2011). To minimize scale effects, $W$_c > 170 for all present tests, fulfilling the most stringent criterion for air entrainment of Skripalle (1994). Other limiting criteria were implicitly satisfied, i.e., $R$_c > 2 × 10⁵ (Pfister and Chanson, 2012), as the Morton number M is invariant for air–water flows, i.e., $M$ =  $Wc3$· $Rc−4$· $Fc−2$ = 3.89 × 10^–11. Previous studies have shown that scale effects for void fraction and interface velocity can be minimized when the above-mentioned limiting criteria are met (e.g., Boes and Hager, 2003a; 2003b). Nevertheless, within the present experimental flow conditions, significant scale effects may be expected as previously observed in terms of interface frequency, chord sizes, and particle cluster properties (Felder and Chanson, 2017). Recent prototype measurements in a tunnel chute (Hohermuth , 2021a) confirmed these laboratory observations, showing that interface frequencies and chord sizes were not scaled correctly under Froude similitude for Reynolds numbers up to $R$_c ≈ 3 × 10⁷. The present experimental setup achieved Reynolds numbers of the order of 10⁶, reaching values close to those observed in some prototype spillways (Table I).

For the smooth tunnel (R1), scale effects were previously investigated by Hohermuth (2019) comparing dimensionless experimental results of void fraction C, interface frequency F·h_crit/U_crit, and interface velocity U/U_crit for same values of $F$_c and differing $R$_c over the dimensionless flow depth z/h_crit (Fig. 3), with h_crit = [(Q_w/W_t)²/g)]^1/3 = critical depth, and U_crit = Q_w/(W_t·h_crit) = critical flow velocity. The comparison was made at the same non-dimensional streamwise distance downstream of the gate, with the location of occurrence of the maximum mean air concentration used as reference for scaling (see also Sec. IV A). The profiles of void fraction and interface velocity showed very similar distributions regardless of the Reynolds number with only a slight increase in C and U/U_crit for increasing $R$_c [Figs. 2(a) and 2(d); and 2(c) and 2(f)], despite the uncertainties in the scaling location and data scatter. Conversely, significant scale effects were found for the interface frequency, with a significant increase in the number of detected particles with increasing $R$_c [Figs. 2(b) and 2(e)], despite a consistent shape of the profiles. This suggests an increase in bubble breakup processes with increasing $R$_c, resulting in a larger number of detected air interfaces.

These observations confirm previous results in high-velocity air–water flows and provide guidance on the interpretation of the effects of wall roughness on the flow properties presented in Secs. IV and V.

This section describes the main characteristics of the flows observed during the experimental campaign, focusing on the most common flow regimes in tunnel chutes downstream of high-head gates. These include the free-surface flow regime, which is identified by a distinct air layer above the flow mixture, and the foamy flow regime, characterized by a dense foam occupying the entire cross section. In the free-surface flow regime, rapidly varied flows (RVF) occurred close to the gate, while gradually varied flows (GVF) were observed in the second half of the tunnel [Fig. 4(a)]. A nearly uniform flow was achieved at the very end of the tunnel chute (x/L_t > 0.8), limited to the rough wall configurations (R2, R3) and low heads, while uniform flow was never observed.

In RVF, shockwaves were initiated approximately at the vena contracta and propagated along the tunnel. These are classified with the following numbering in Fig. 4(a): (1) sidewall shockwaves induced by gate corner vortices (see Pagliara , 2023); (2) a central shockwave developing along the tunnel centerline due to secondary currents directed inward from the walls (Auel , 2014); (3) additional sidewall shockwaves forming approximately from the point of collapse of the central one; and (4) an additional central shockwave developing along the tunnel centerline from the point of collapse of the two sidewall ones. Shockwaves strongly contributed to the aeration of the water jet, as they promoted the formation and ejection of droplets, as well as re-impacting on the clear water core. Their formation and growth were linked to a decrease in the mean air concentration C_m along the centerline, followed by the point of collapse of the two sidewall shockwaves and formation of the central one which corresponded to the point with maximum C_m (see also Sec. IV A). Due to the peculiar flow conditions in the tunnel, the shockwaves were characterized by significant oscillations in time and space in both longitudinal and vertical directions.

The GVF in the second half of the tunnel chute was characterized by complex air–water flow features, and flow interactions with the tunnel soffit were observed for most of the flow conditions. For small Q_w, the air–water mixture resembled open-channel flow. For moderate Q_w, increasing flow aeration was observed together with increasing droplet formation with trajectories intercepting the tunnel soffit, despite the presence of a distinct air layer above the flow. For large Q_w, the air–water mixture occupied the entire tunnel cross section, limiting the air circulation above the flow and resulting in complex interactions with the tunnel soffit.

The vertical profiles of the high-velocity air–water flows could be classified into several distinct flow regions [Fig. 4(b)], resembling the visual observations of Killen (1968). Close to the chute invert, a clear-water region was found for large gate openings A and/or low heads H (i.e., for low C_m). Above the clear-water region, a bubbly flow region was observed with bubbles advected by the water flows, while the free-surface flow (or intermediate) region above was characterized by pronounced surface perturbations. At the top of the flow column, a droplet (or spray) region occurred, characterized by significant spray formation and re-circulation. The clear-water region disappeared for C_m ≥ 0.25, as a consequence of the bubbly flow region protruding into the clear-water region (Straub and Anderson, 1958). A very fine spray in the droplet region has been commonly observed in prototypes (e.g., Hohermuth , 2021a) and in large-scale physical models (e.g., present study).

The wall roughness had a pronounced effect on the flow patterns. An increase in roughness resulted in an increase in the mixture flow depth and a decrease in flow velocity. Visual observations of RVF and GVF are presented in Fig. 5, for two exemplary inflow conditions (i.e., A = 0.2 and H = 10 m, and A = 0.2 and H = 30 m) and the roughness configurations R1, R2i, and R3i. With increasing roughness, the start of GVF shifted upstream, the velocity of the flow mixture decreased, the flow bulking increased, and the foamy flow regime occurred at smaller H for a given A, due to the larger mixture flow depths. The comparison between the invert-lined and the full-lined setups for the same roughness typology (not presented) indicated larger flow depths associated with the full-lined setup due to the sidewall influence, particularly pronounced in RVF. For the roughness typologies R2 and R3, an additional increase in A or H from the foamy flow condition triggered the formation of a hydraulic jump, followed by pressurized flow (not investigated).

Typical void fraction C profiles at two longitudinal positions along the tunnel chute are shown in Fig. 6 as a function of z/h₉₀, for different combinations of A and H, and for all roughness configurations. The longitudinal position x = 4 m was in RVF, while the flows at x = 12 m were in GVF. For the smooth tunnel (R1), the C profiles had a typical S-shape with a steep gradient for 0.3 < C < 0.7 in RVF [Fig. 6(a)], whereas they tended to a straight line with increasing distance from the gate in GVF [Fig. 6(d)]. At a given position x and for a fixed A, aeration increased with H (thus, with $F$_c and $R$_c). Similar features were observed for the intermediate (R2) and large (R3) roughness configurations. In RVF (x = 4 m), the profiles tended to a straight line, with higher bottom air concentrations. In this region, there was little visual difference between full- and invert-lined setups, while h₉₀ was larger for the full-lined setup [Figs. 6(b) and 6(c)]. For R3, in RVF, large H led to double-S shaped profiles resembling those observed in hydraulic jumps, with significantly greater amount of entrained air and bottom air concentration [Fig. 6(c)]. In GVF (x = 12 m), the difference in h₉₀ between full- and invert-lined setups became more pronounced, irrespective of the wall roughness. The bottom air concentration and flow bulking decreased due to flow de-aeration [Figs. 6(e) and 6(f)]. Overall, the effect of the wall roughness on the void fraction profiles got more pronounced as H increased, resulting in a higher quantity of entrained air, a more uniform vertical air distribution, and profile shapes that reflected greater turbulence intensities and velocity fluctuations with increasing invert roughness.

Further comparisons suggested that the C distributions exhibited self-similarity in terms of C_m (see the supplementary material, Sec. S1). The C profiles approached a straight line for increasing C_m and formed a double-S shape for C_m > 0.50, confirming the strong flow aeration (visually identified in Fig. 5). The milder gradients of the C profiles close to the chute invert for C_m > 0.30 further suggested enhanced bottom aeration and a reduced cavitation risk. For C_m ≤ 0.30, the profiles were in reasonable agreement with the solution of the advection-diffusion equation of Chanson (1996), irrespective of the roughness configuration, whereas for C_m > 0.50 deviations were observed near the chute invert (see the supplementary material, Sec. S1). In this region (0 < z/h₉₀ < 0.5), high turbulence levels enhanced mixing, which was not accounted for by the advection-diffusion equation which assumes a homogeneous bubbly layer [Fig. 4(b)]. Overall, regardless of the flow conditions and roughness configurations, the C distributions of high-velocity flows in tunnel chutes can be reasonably predicted based solely on the value of C_m.

The development of the mean air concentration C_m along the tunnel is presented in Fig. 7 for all experiments, where C_m represents an important parameter to characterize the development of key flow properties along the tunnel. Irrespective of the roughness, C_m increased with increasing H and decreasing A. The development of C_m was strongly linked to the shockwave development [Fig. 4(a)]. In all cases, when captured, a first maximum C_m was observed just downstream of the gate, resulting from primary aeration in proximity of the location where sidewall shockwaves collapsed and formed a central shockwave [Figs. 4(a) and 7(c)]. Downstream of this maximum, C_m decreased due to the development of new sidewall shockwaves. A second maximum in the C_m profiles, labeled as C_m,max, consistently occurred at position x_m,max where shockwaves merged in the tunnel centerline [Fig. 7(c)], leading to additional air entrainment. The trend described was further influenced by the development of the “inception point.” Downstream of x_m,max, a consistent decrease in C_m was observed due to air detrainment in the decelerating flow along the tunnel chute. For all roughness configurations, the detrainment rate tended to decrease with increasing H and significantly decreased with increasing A (Fig. 7). Furthermore, the detrainment rate was lower for the smooth tunnel [Fig. 7(a)] compared to the rough tunnels [Figs. 7(b) and 7(e)], and greater for the full-lined configurations [Figs. 7(b) and 7(c)] compared to the invert-lined configurations [Fig. 7(d) and 7(e)]. This is a direct result of the flows being aerated beyond capacity as velocity decreased, with rough tunnels associated with stronger flow decelerations. Furthermore, the influence of the sidewalls on aeration and de-aeration processes was due to the more uniform distribution of shear stresses in the cross section. Overall, the two roughness typologies R2 and R3 showed similar detrainment rates, despite larger C_m values associated with the largest roughness.

Typical distributions of interface frequency F are presented in Fig. 10. Most of the F distributions associated with the smooth tunnel showed a maximum number of entrained bubbles F_max in the upper part of the bubbly region [C ≤ 0.3; Fig. 4(b)] or the lower part of the intermediate region [0.3 < C < 0.7; Fig. 4(b)]. The location of F_max was lower compared to smooth spillways, resulting from differences in flow velocity magnitude and distribution, which caused a more abrupt aeration. For the rough tunnels, F_max typically occurred in the intermediate region. At a given x and for the same inflow condition, the F distributions for invert- and full-lined setups were similar [Figs. 10(a) and 10(c); and 10(d) and 10(f)], suggesting that the aeration and bubble breakup processes were mainly driven by the invert roughness. For each respective roughness configuration, the number of entrained bubbles increased with H (thus, with $F$_c and $R$_c), due to the larger flow velocities. The increase in H further led to an increase in F_max and a shift in the location of F_max toward smaller z/h₉₀. For a fixed inflow condition, F decreased along the tunnel chute and the location of F_max shifted upward [Figs. 10(a) and 10(d); 10(b) and 10(e); and 10(c) and 10(f)]. These observations were linked with air detrainment, which was more pronounced for the rough tunnels due to the higher mixture flow depths and lower flow velocities.

The development of F_max along the tunnel chute (see the supplementary material, Sec. S2) resembled the trend in C_m, despite the absence of the characteristic double-peak (Fig. 8). Values of F_max were similar irrespective of the roughness configurations, and the location of the maximum value of the longitudinal profile approximately matched the corresponding location x_m,max of C_m,max.

The relationships between F/F_max and C and between F_max and C_m were analyzed to identify links between flow aeration and number of entrained bubbles (Fig. 11). For a fixed roughness typology, the F-C profiles were in close agreement when F was normalized with F_max [Fig. 11(a)], and an increase in wall roughness resulted in a shift toward larger C. The increase in wall roughness resulted in F_max to occur at larger C, on average, i.e., C(F_max) = 0.38 for R1, C(F_max) = 0.46 for R2, and C(F_max) = 0.50 for R3. The present data resembled the shape of the self-similar parabolic profiles proposed by Chanson and Toombes (2002), i.e., F/F_max = 4·C·(1 – C), despite F_max occurring for smaller C than predicted for most of the tests. Such disagreement was expected, since the parabolic profile is characterized by C(F_max) = 0.5 under the assumption that velocities and chord sizes are constant within a profile (not satisfied for our test conditions). The theoretical model derived by Toombes and Chanson (2008), which introduced correction factors depending on the local void fraction and flow conditions, gave a good agreement with the present data irrespective of tunnel roughness [Fig. 11(a)]. Therefore, a good approximation of the F profiles can be inferred based upon F_max only.

The time-averaged interface velocity U profiles resembled those of wall jets in the vicinity of the gate (Launder and Rodi, 1979) and of high-velocity open-channel flow with strong aeration and small aspect ratio further downstream (see the supplementary material, Sec. S3). In the present study, maximum interface velocities of up to U_max ≈ 24 m/s were measured for the largest head and close to the gate. Irrespective of the roughness configuration, the flows decelerated along the tunnel, linked with an increase in flow aeration and flow depth (Fig. 5), with the strongest deceleration observed for the largest roughness typology and full-lining (R3f). Typical profiles of non-dimensional interface velocity U/u^* vs z/z₀ are presented in Fig. 12, alongside the log-law [Eq. (4)], the wake-modified log-law [Eq. (5a)], and the dip-modified log-law [Eq. (5b)]. The shear velocity u^* and zero-velocity plane z₀ were obtained by fitting the log-law to the velocity profiles (see Sec. II B). The resulting roughness Reynolds numbers $ks+$ classified the flow conditions as transitional (4 < $ks+$ < 70, limited to the R1 configuration) or fully rough (⁠ $ks+$≥ 70), and high-velocity air–water flows in prototype tunnel chutes are expected to exhibit similar regimes. In all cases, the log-laws aligned well with the velocity profiles in the wall region (i.e., z/h₉₀ < 0.2), with deviations observed in the outer region, depending on the flow regime and roughness configuration (Fig. 12). Moreover, the measured profiles typically fell between the dip-modified log-law (Yang , 2004) and the wake-modified log-law (Coles, 1956). In RVF [Figs. 12(a)–12(c)], all the profiles displayed a significant velocity dip due to the wall-jet characteristics of the flow, with measurements close to the dip-modified log-law, particularly for roughness configurations R1, R2f, and R3f. The milder velocity dip observed for the invert-lined configurations (R2i, R3i) was attributed to the weaker secondary currents generated by the smooth walls. In GVF [Figs. 12(d)–12(f)], the profiles for the roughness typology R1 continued to exhibit a pronounced velocity-dip phenomenon [Figs. 12(d)], while a velocity wake was observed in the rough tunnels (R2 and R3), resembling high-velocity open-channel flow with a small aspect ratio [Figs. 12(e) and 12(f)].

This relationship is valid for z/h_Umax ≤ 1, where h_Umax is the elevation of U_max and K₁₁ is a power-law exponent (often labeled as “n”). The value of the power-law exponent for the smooth tunnel was K₁₁ = 8.1 (R² = 0.91), while the value for the rough tunnels was much smaller (K₁₁ = 3.2 with R² = 0.83), reflecting the strong velocity gradients close to the invert. For z/h_Umax > 1, data were scattered, mostly due to the interactions between the air–water flow and the tunnel soffit, and no distinct velocity profile trend was found in this region.

As emphasized in Secs. IV A–IV C, turbulence is a key driver of the observed air–water flow properties. Typical distributions of streamwise turbulence intensity are presented in Fig. 14 at position x = 12 m, expressed as u_rms/U [Figs. 14(a)–14(c)] and u_rms/u^* [Figs. 14(d)–14(f)]. The comparison includes the universal function of Monin and Yaglom (1971), i.e., u_rms/u^* = 2.3·exp(−z/h₉₀), originally proposed for smooth, uniform, two-dimensional, open-channel flows.

When expressed as u_rms/U, significantly larger turbulence values were observed in the rough tunnels [Figs. 14(b) and 14(c)] compared to the smooth tunnel [Fig. 14(a)]. Additionally, for the same roughness typology, full-lined configurations exhibited higher turbulence values than invert-lined configurations. These results align with increasing void fraction and mean air concentration (Sec. IV A) and decreasing interface frequency (Sec. IV B) and velocity (Sec. IV C) as roughness increases. In all cases, the profiles showed steep gradients close to the invert (z/h₉₀ < 0.4), and an increase in turbulence in the region z/h₉₀ > 0.8 limited to the rough tunnels and large heads (H ≥ 20 m). This was attributed to the more pronounced spray formation in the upper part of the flow column at large H and in RVF, resulting in the ejection of fast water droplets and recirculation above the flow. Overall, the profiles were consistent with findings from previous relevant studies on hydraulic jumps (Zhang , 2014), ship hulls (Perret and Carrica, 2015), stepped spillways (Kramer , 2020), and tunnel chutes (Hohermuth , 2021b).

When expressed as u_rms/u^*, the profiles for the smooth tunnel [Fig. 14(d)] and the rough tunnels [Figs. 14(e) and 14(f)] exhibited distinct shapes, reflecting the different hydraulic regimes, i.e., transitional for R1, and fully rough for R2 and R3. The u_rms/u^* profiles for the smooth tunnel followed the pattern of the corresponding u_rms/U profiles, and they deviated from the universal function of Monin and Yaglom (1971), primarily due to the effects of secondary currents and the small aspect ratio (Auel , 2014). In contrast, the u_rms/u^* profiles for the rough tunnels showed more complex shapes compared to the corresponding u_rms/U profiles. For rough tunnels (R2 and R3), similar values of u_rms/u^* were observed near the invert, with key differences emerging for z/h₉₀ ≥ 0.2. These findings are consistent with observations on the F profiles, which showed similar values of F_max for R2 and R3, but larger z/h₉₀(F_max) associated with R3 (Fig. 10).

The mean Sauter diameter d₃₂ = 1.5·C·U/F was calculated in regions where 0 < C < 0.5, assuming spherical dispersed air bubbles in water (Kowalczuk and Drzymala, 2016; Fig. 15). The d₃₂ profiles for all tests are presented in dimensional form in Fig. 15(a), revealing minimal differences across the roughness configurations. In the inner region (z/h₉₀ < 0.2), slightly larger bubbles were observed in the rough tunnels (R2, R3), reflecting higher C (∝ d₃₂) and lower F (∝ 1/d₃₂), outweighing the effect of lower U (∝ d₃₂). In the rough tunnels, the larger C and smaller U facilitated bubble clustering and coalescence processes more effectively than in the smooth tunnel, ultimately prevailing over bubble breakup processes induced by wall roughness. Overall, the observed d₃₂ values were in the lower range of reported bubble sizes for spillway models and prototypes (e.g., Chanson, 1993; Kramer, 2004) and in the mid-upper range for tunnel chute prototypes (e.g., Hohermuth , 2021a; 2022).

For the conditions investigated in this study, bubble breakup and coalescence processes were primarily governed by shear forces, with a strong correlation previously identified between shear and aeration [e.g., Fig. 13(d)]. The mean Sauter diameter d₃₂ was normalized with the theoretical maximum bubble diameter d_b,Hinze before shear-induced splitting of bubbles occurs, as defined by Hinze (1955), and plotted against the mean air concentration C_m in Fig. 15(b). While no definitive empirical equation could be established, the plots suggest a direct correlation between relative bubble size and flow aeration, confirming that shear is a critical influencing factor. Data points from the smooth tunnel (R1) exhibited larger values of d₃₂/d_b,Hinze for a given C_m [Fig. 15(b)]. This observation implies that larger relative bubble sizes are associated with the smooth tunnel compared to the rough tunnels, likely due to the less intense shearing in the smooth tunnel. Nevertheless, the Hinze diameter is purely shear driven, and thus, it neglects coalescence effects.

The analysis is further supplemented by the chord size (CS) distributions for air bubbles and water droplets (Fig. 16). These were obtained at vertical locations characterized by similar void fraction, i.e., C ≈ 0.30 (bubbly region) and C ≈ 0.95 (droplet region), for all roughness configurations, along with the main parameters characterizing the distributions. All CS distributions showed consistently the largest frequency of small particles, as well as a decreasing probability with increasing CS. In general, the distributions exhibited strong consistency across all roughness configurations [Figs. 16(a) and 16(d), solid lines]. Additionally, a flattening of the distributions toward larger CS was observed in GVF [Figs. 16(c) and 16(d), solid lines] compared to RVF [Figs. 16(a) and 16(b), solid lines], attributed to flow deceleration. In RVF, an increase in head from H = 10 m [Fig. 16(a), solid lines] to H = 20 m [Fig. 16(b), solid lines], which corresponded to increases in both $F$_c and $R$_c, resulted in mild differences of the distributions. In contrast, in GVF differences were more pronounced: the lower head (H = 10 m) resulted in flattened distributions [Fig. 16(c), solid lines], while an increase in H [Fig. 16(d), solid lines] produced distributions similar to those observed in RVF.

In contrast, the droplet CS distributions exhibited no significant variations with respect to roughness configurations, positions along the tunnel, or inflow conditions [Figs. 16(a)-16(d), dashed lines].

To summarize the effects of roughness on the air–water flow properties, available data with identical Froude and Reynolds numbers were compared at similar non-dimensional locations along the tunnel. The location of the inception point has been typically used as scaling parameter for spillways (e.g., Boes and Hager, 2003a; 2003b). Nevertheless, aeration processes are different in tunnel chutes downstream of high-head gates, and an inception point cannot be clearly identified (Figs. 4 and 5). Instead, the location x_m,max with C_m,max was used as scaling parameter. This choice was supported by the relationships found for C_max and for the development of C_m along the tunnel chute (Sec. IV A). Non-dimensional distributions of void fraction (C), interface frequency (F·h_crit/U_crit), interface velocity (U/U_crit), and turbulence intensity (u_rms/U) are presented in Fig. 17 for all roughness configurations using identical inflow conditions (i.e., A = 0.2, H = 20, $F$_c = 38.0, and $R$_c = 2.0 × 10⁶) and the same three non-dimensional locations, i.e., (x – x_m,max)/h_crit ≈ 6, 12, and 35. The normalizations of the flow properties were conducted with critical depth h_crit and critical velocity U_crit to emphasize the effects of roughness.

The comparison of the C profiles [Figs. 17(a), 17(e), and 17(i)] confirmed a significant increase in h₉₀ and C_m with increasing roughness, including greater values for the full-lined configurations compared to the invert-lined. This is a direct consequence of the increasing turbulence promoted by increasing wall roughness, which is reflected by the significantly larger amount of entrained air for the rough tunnels compared to the smooth tunnel at the same non-dimensional location. The full-lined rough tunnels (R2f and R3f) had similar C distribution shapes compared to the invert-lined counterparts (R2i and R3i) despite the different mixture flow depths, highlighting the importance of the invert lining on the overall air entrainment process. Furthermore, similar invert roughness resulted in similar bottom air concentration C_b at the same non-dimensional location [Figs. 17(a), 17(e), and17(i)]. A sufficiently large C_b is crucial for mitigating cavitation potential. For all roughness configurations, C_m decreased with increasing (x – x_m,max)/h_crit and the void fraction profiles shifted toward larger values of z/h_crit. In this context, the rough tunnels showed stronger de-aeration processes, as a direct consequence of the larger turbulence levels, ultimately leading to greater energy dissipation and air detrainment.

The F profiles [Figs. 17(b), 17(f), and 17(l)] showed a decrease in F_max with increasing roughness, and larger values of F_max associated with the invert-lined tunnels compared to the full-lined counterpart. The profiles resulting from rough tunnels presented a shift of the point of occurrence of F_max toward larger values of z/h_crit and C, thus moving from the upper end of the bubbly flow region to the intermediate region with increasing roughness [see Figs. 4(b) and 11(a)]. The significantly larger U associated with the smooth tunnel can explain the overall larger values of F compared to the rough tunnels, while the stronger velocity gradients and larger turbulence intensities in the rough tunnels caused the shift of F_max toward larger values z/h_crit and C.

The U profiles [Figs. 17(c), 17(g), and 17(m)] resembled wall-jet profiles for (x – x_m,max)/h_crit = 6, and tended to those characterizing open channel flows with increasing non-dimensional distance (in particular for the full-lined rough tunnels). The smooth tunnel led to the largest U, with the profiles shifting toward smaller values with increasing wall roughness. The significantly larger U associated with the invert-lined rough tunnels compared to the full-lined evidenced the strong influence of the sidewall roughness, especially for the most upstream position presented [Fig. 17(c)]; with increasing non-dimensional distance, the influence of the rough invert on U over the sidewalls increased [Figs. 17(g) and 17(m)].

The u_rms/U profiles [Figs. 17(d), 18(h), and 18(n)] showed a significant increase in turbulence with increasing roughness close to the invert (bubbly region), and comparable values in the intermediate and droplet regions. These features corroborate observations of larger h₉₀ and C_m, greater energy dissipation, and smaller U for increasing roughness.

Bubble and droplet chord size CS distributions evaluated at similar non-dimensional locations (see the supplementary material, Sec. S5) confirmed the findings presented in Sec. IV E.

The same analysis was performed for identical flow conditions and roughness configurations by normalizing the distances with the uniform flow depth h_u and by plotting the flow properties against z/h_u, to incorporate the effects of wall roughness (Fig. 18). The non-dimensional distances presented are (x − x_m,max)/h_u ≈ 0, 12, 34. The C profiles converged to one profile for all non-dimensional locations when normalized with h_u, effectively capturing the influence of wall roughness [Figs. 18(a), 18(e), and 18(i)]. For the other flow properties, the profiles corresponding to the rough tunnels collapsed across all non-dimensional distances, whereas those associated with the smooth tunnel typically exhibited a different pattern [Figs. 18(b)-18(d); 18(f)–18(h); and 18(l)–18(n)]. This finding supports the distinction between smooth and rough tunnels in Secs. IV A–IV E of the manuscript, as it reflects the contrasting hydraulic regimes in the smooth tunnel (transitional, $ks+$ < 70 for most flow conditions) and in the rough tunnels (fully rough, $ks+$ ≫ 70 for all flow conditions). The scatters between the profiles are primarily attributed to uncertainties in determining C_m,max and to the limited number of measurement locations along the tunnel.

Overall, the results presented in this section provide a comprehensive summary of the key effects of wall roughness on the flow properties of high-velocity air–water flows in tunnel chutes downstream of a high-head gate. They demonstrate the universality of these findings when uniform flow parameters are used for scaling, emphasizing the importance of considering both wall roughness and the associated hydraulic regime in the analysis of such flows.

Air–water flows play a crucial role in natural and human-engineered systems, occurring in a broad spectrum of applications. High-velocity air–water flows frequently occur in tunnel chutes downstream of high-head gates, such as low-level outlets, representing crucial infrastructure of large dams. Previous laboratory studies on high-velocity air–water flows in tunnel chutes have not considered wall roughness, which has been shown to have a significant influence on the air demand and flow patterns. In the present study, the hydraulics of high-velocity air–water flows was systematically investigated in a tunnel of variable wall roughness across a range of flow conditions, comprising unit discharges 0.3 ≤ q ≤ 1.3 m²/s, relative gate openings 0.2 ≤ A ≤ 0.4, energy heads at the gate 5 ≤ H ≤ 30 m, contraction Reynolds numbers 9 × 10⁵ ≤ $R$_c ≤ 3 × 10⁶, contraction Weber numbers 212 ≤  $W$_c ≤ 594, and contraction Froude numbers 18 ≤  $F$_c ≤ 47. The resulting highly aerated high-velocity water jets were characterized by maximum measured interface velocities of up to approximately 24 m/s.

In longitudinal direction, a rapidly varied flow (RVF) region was identified, characterized by strong spray formation and re-circulation, shockwave formation, followed by a gradually varied flow (GVF) region presenting flow bulking, energy dissipation, and de-aeration processes. Visual observations highlighted that the flow characteristics were significantly affected by the wall roughness, leading to an increase in the mixture flow depth and of the amount of entrained air, and a decrease in flow velocity.

The void fraction profiles showed an increase in entrained air, mixture flow depth, and a more uniform vertical air distribution for increasing roughness, and self-similarity of the profiles in terms of the mean air concentration C_m only was identified. A correlation between the development of C_m along the tunnel chute with the shockwave patterns and the wall roughness was observed, and empirical equations to predict its maximum value (C_m,max), location (x_m,max), and development downstream of the maximum (C_m/C_m,max) were proposed. The interface frequency F distributions and the development of the maximum interface frequency F_max along the tunnel showed strong analogies with the considerations made on void fractions. The interface velocity U profiles resembled those of wall jets in proximity of the gate, becoming more uniform in GVF and resembling open-channel flows with small aspect ratios. A significant increase in shear velocities u^* was observed for increasing roughness, and the log-law well represented the velocity profiles in the inner region (z/h₉₀ < 0.2), with deviations observed in the outer region; overall, the measured velocity profiles fell between the dip-modified and the wake-modified log-laws. The streamwise turbulence intensity u_rms/U profiles showed a significant increase with increasing roughness in the bubbly region, and comparable values in the intermediate and droplet regions. The streamwise turbulence expressed as u_rms/u^* showed key differences in the profile shapes between smooth and rough tunnels, leading to differences in key flow properties (e.g., interface frequency and velocity). Turbulence was identified as the key driver behind the increase in the mixture flow depth, increase in the mean air concentration, increase in energy dissipation, decrease in interface velocities, and variations in chord size distributions and in air entrainment and detrainment processes.

The findings of the present study highlight the importance of considering wall roughness for flow properties of high-velocity air–water flows, and future studies should take this into consideration. The comparison of fundamental air–water flow properties at similar non-dimensional locations along the tunnel under identical flow conditions highlighted key differences when scaled using the critical flow depth. Conversely, when the uniform flow depth was used, the distributions collapsed according to the hydraulic regime, incorporating the effects of tunnel roughness. This result emphasizes the need to account for both wall roughness and the associated hydraulic regime when dealing with high-velocity air–water flows in closed conduits, for both research and design purposes. The present study ultimately leads to a better understanding of the complex interactions between air and water phases in high-velocity air–water flows. Finally, in the context of hydraulic engineering, the empirical equations presented serve as important design and verification tools for tunnel chutes downstream of high-head gates, typically found in large dams.

See the supplementary material for extended results and insights into air–water flow properties that further support the findings of this study. While these supplementary details may be of particular interest to a more specialized subset of readers, they are presented to enhance the depth of the analysis. The supplementary material includes a discussion on self-similarity of void fraction profiles, the development of the maximum interface frequencies and of the shear velocities along the tunnel chute, representative interface velocity vertical profiles, as well as the influence of wall roughness on chord size distributions.

The support of the technical staff of the Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH Zürich, is gratefully acknowledged: Patrick Egli, Stefan Gribi, Daniel Gubser, Raphael Heini, Dorde Masovic, Mario Moser, Andreas Schlumpf, and Henry Zoller (in the alphabetical order). The project was supported by the Swiss National Science Foundation (SNSF, Grant No. 197208); https://data.snf.ch/grants/grant/197208.

The authors have no conflicts to disclose.

Simone Pagliara: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Visualization (equal); Writing – original draft (equal). Benjamin Hohermuth: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Robert Michael Boes: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Stefan Felder: Conceptualization (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

All data that support the findings of this study are available from the corresponding author upon reasonable request.

a

Gate opening (m)

A = a/a_m

Relative gate opening

A^*

Air vent parameter

a_m

Maximum gate opening (m)

C

Time-averaged void fraction

C_c

Contraction coefficient

C_m

Mean air concentration

C_m,max

Maximum mean air concentration

CS

Chord size (m)

CT

Chord time (s)

d₃₂

Mean Sauter diameter (m)

d_b,Hinze

Maximum bubble diameter before shear-induced splitting occurs (m)

$F$

Froude number

F

Interface frequency (1/s)

F_max

Maximum interface frequency in a profile (1/s)

g

Gravitational acceleration (m/s²)

H

Energy head at the gate (mWC)

h₉₀

Characteristic mixture flow depth, i.e., depth at which C(z) = 0.90 (m)

h_c

Contraction flow depth (m)

h_crit

Critical flow depth (m)

h_t

Tunnel height (m)

h_u

Uniform flow depth (m)

h_Umax

Flow depth at which U_max occurs (m)

h_w

Clear-water flow depth (m)

K_i

Empirical coefficients (with 1 ≤ i ≤ 17)

k_s

Equivalent sand roughness (m)

$ks+$

Roughness Reynolds number

L_t

Tunnel length (m)

$M$

Morton number

n

Power-law exponent

p_w,av

Average water pressure in the inflow tunnel (Pa)

q

Specific water discharge (m²/s)

Q_w

Water discharge (m³/s)

$R$

Reynolds number

R²

Coefficient of determination

R_h

Hydraulic radius for clear-water flow (m)

R_h,u

Hydraulic radius at uniform flow conditions (m)

S

Tunnel slope

U

Time-averaged streamwise interface velocity (m/s)

Ū

Mean flow velocity (m/s)

u^*

Shear velocity (m/s)

$uu*$

Shear velocity at uniform flow condition (m/s)

U₉₀

Interfacial velocity at z = h₉₀ (m/s)

U_c

Average flow velocity at vena contracta (m/s)

U_crit

Critical flow velocity (m/s)

U_max

Maximum (interfacial) flow velocity in a profile (m/s)

u_rms

Root-mean-square of instantaneous velocity fluctuations (m/s)

U_u

Uniform flow velocity (m/s)

$W$

Weber number

$W$_crit

Critical Weber number

W_g

Gate width (m)

W_t

Tunnel width (m)

x, y, z

Reference system (m)

x_m,max

Location of C_m,max (m)

z₀

No-displacement plane (m)

α

Dip parameter (log-law)

Δx

Streamwise probe tip separation distance (m)

Δy

Transversal probe tip separation distance (m)

Δz

Vertical probe tip separation distance (m)

ζ

Air vent loss coefficient

κ

von Kármán constant

ν

Kinematic viscosity (m²/s)

ρ

Density (kg/m³)

σ

Surface tension coefficient between air–water (kg/s)

П

Wake parameter (log-law)

0

Computed/measured at inflow conditions (subscript)

AWCC

Adaptive window cross correlation

a

Air (subscript)

CP

Dual-tip phase-detection conductivity probe

c

Computed/measured at the vena contracta (subscript)

f, i

Lining setups (full- and invert-lined)

GVF

Gradually varied flow

LLO

Low-level outlet

LT

Leading tip

MLO

Mid-level outlet

R

Rough hydraulic regime (also known as “fully rough”)

R1, R2, R3

Roughness typologies (smooth, intermediate, and large roughness)

RVF

Rapidly varied flow

Smooth hydraulic regime

T

Transitional hydraulic regime (also known as “transitional rough”)

TT

Trailing tip

w

Water (subscript)

a

Gate opening (m)

A = a/a_m

Relative gate opening

A^*

Air vent parameter

a_m

Maximum gate opening (m)

C

Time-averaged void fraction

C_c

Contraction coefficient

C_m

Mean air concentration

C_m,max

Maximum mean air concentration

CS

Chord size (m)

CT

Chord time (s)

d₃₂

Mean Sauter diameter (m)

d_b,Hinze

Maximum bubble diameter before shear-induced splitting occurs (m)

$F$

Froude number

F

Interface frequency (1/s)

F_max

Maximum interface frequency in a profile (1/s)

g

Gravitational acceleration (m/s²)

H

Energy head at the gate (mWC)

h₉₀

Characteristic mixture flow depth, i.e., depth at which C(z) = 0.90 (m)

h_c

Contraction flow depth (m)

h_crit

Critical flow depth (m)

h_t

Tunnel height (m)

h_u

Uniform flow depth (m)

h_Umax

Flow depth at which U_max occurs (m)

h_w

Clear-water flow depth (m)

K_i

Empirical coefficients (with 1 ≤ i ≤ 17)

k_s

Equivalent sand roughness (m)

$ks+$

Roughness Reynolds number

L_t

Tunnel length (m)

$M$

Morton number

n

Power-law exponent

p_w,av

Average water pressure in the inflow tunnel (Pa)

q

Specific water discharge (m²/s)

Q_w

Water discharge (m³/s)

$R$

Reynolds number

R²

Coefficient of determination

R_h

Hydraulic radius for clear-water flow (m)

R_h,u

Hydraulic radius at uniform flow conditions (m)

S

Tunnel slope

U

Time-averaged streamwise interface velocity (m/s)

Ū

Mean flow velocity (m/s)

u^*

Shear velocity (m/s)

$uu*$

Shear velocity at uniform flow condition (m/s)

U₉₀

Interfacial velocity at z = h₉₀ (m/s)

U_c

Average flow velocity at vena contracta (m/s)

U_crit

Critical flow velocity (m/s)

U_max

Maximum (interfacial) flow velocity in a profile (m/s)

u_rms

Root-mean-square of instantaneous velocity fluctuations (m/s)

U_u

Uniform flow velocity (m/s)

$W$

Weber number

$W$_crit

Critical Weber number

W_g

Gate width (m)

W_t

Tunnel width (m)

x, y, z

Reference system (m)

x_m,max

Location of C_m,max (m)

z₀

No-displacement plane (m)

α

Dip parameter (log-law)

Δx

Streamwise probe tip separation distance (m)

Δy

Transversal probe tip separation distance (m)

Δz

Vertical probe tip separation distance (m)

ζ

Air vent loss coefficient

κ

von Kármán constant

ν

Kinematic viscosity (m²/s)

ρ

Density (kg/m³)

σ

Surface tension coefficient between air–water (kg/s)

П

Wake parameter (log-law)

0

Computed/measured at inflow conditions (subscript)

AWCC

Adaptive window cross correlation

a

Air (subscript)

CP

Dual-tip phase-detection conductivity probe

c

Computed/measured at the vena contracta (subscript)

f, i

Lining setups (full- and invert-lined)

GVF

Gradually varied flow

LLO

Low-level outlet

LT

Leading tip

MLO

Mid-level outlet

R

Rough hydraulic regime (also known as “fully rough”)

R1, R2, R3

Roughness typologies (smooth, intermediate, and large roughness)

RVF

Rapidly varied flow

Smooth hydraulic regime

T

Transitional hydraulic regime (also known as “transitional rough”)

TT

Trailing tip

w

Water (subscript)