Evolution of pressure and cavitation in transition region walls for supercritical flow

Fr

Froude number

lateral expansion angle

Re

Reynolds number

density of water

the dynamic viscosity of water

u

the average flow velocity in the conduit

g

the acceleration of gravity

d

flow depth

P

flow pressure

cavitation index

minimum cavitation index

the allowable size of the lateral expansion angle

A, B, C, D, H

constant factors

S₁

section one

S₂

section two

flow velocity at the entrance of the transition region

flow depth at the entrance of the transition region

The occurrence of cavitation erosion in transition regions is strongly determined by hydraulic and geometric characteristics. Hence, it is required to thoroughly examine flow properties and conduit geometry. The hydraulic characteristics of different flow rates and expansion angles are complicated in transition regions (Jablonská et al. 2016). Asnaashari et al. (2015) investigated the effect of Froude numbers on the flow pattern in channel-expansive transitions. The results of this study indicated that bed-shear stress increased at the end of the transition region due to flow separation. Furthermore, turbulent strength was raised owing to a boost in the Froude number. Yamini et al. (2021) numerically studied Sardab Dam’s bottom outlet and Howell Bunger valve's hydraulic performance under different opening conditions. The results demonstrated various velocity values, ranging from 2 to 5 m/s at the branch entrance of the bottom outlet. Neither flow separation nor adverse conditions were observed in this area. In their study, Li et al. (2017) observed that the conduit wall encountered the cavitation phenomenon at all the lateral expansion angles for the Froude numbers of 5.77 and 6.59 (at any percentage of gate openings). Li et al. (2016a) also found that the cavitation index (σ) on the sidewalls of the conduit was less than 0.2 for Fr ≥ 3.89 at all the lateral expansion angles, even at α = 0°, when the gate was partially open. To prevent cavitation occurrence in transition regions, Thompson & Kilgore (2006) introduced the relevance of as a maximum expansion angle. Using this equation, the lateral expansion angle reached 4.9°, 3.2°, and 3° for the Froude numbers of 3.89, 5.77, and 6.59, respectively. Therefore, the results of Li et al. (2016a) are not in good agreement with those of Thompson & Kilgore (2006).

The k–ε (RNG) turbulence model and the Volume-Of-Fluid (VOF) method have been used in numerous studies to quantitatively measure high-speed flows in hydraulic structures. Yamini et al. (2022) studied the flow patterns, stream flow, vortex severities, pre-swirl principles, and performance acceptability criteria for pump intakes in different conditions. Jamali et al. (2022) explored the effect of flow conditions and chute geometry on shock waves on chute spillways. The impact of different slope ratios on the hydrodynamic characteristics of lateral withdrawal of water supply was examined by Zhaoa et al. (2022). Ebrahimnezhadian et al. (2020) investigated the impact of the flip bucket edge angle on flow hydraulic characteristics. The effects of pressure ratio conditions and plate stages on the characteristics of cavitation, energy transfer, and loss of perforated plates were numerically studied by Xiaogang et al. (2020). Li et al. (2016b) simulated aerated flows in curve-connective tunnels. Li et al. (2011) identified the hydraulic characteristics of flow downstream in radial gates whose conduit had a sudden lateral enlargement and a vertical bottom drop. Dargahi (2010) assessed the flow characteristics of bottom outlets with moving gates. Stamou et al. (2008) studied supercritical flows in gradual expansions. Krüger & Rutschmann (2006) modeled supercritical flows via an extended shallow water approach. High-speed flow in a spillway tunnel was also analyzed by Sha et al. (2006).

Based on what was mentioned, the results of previous studies on the allowable expansion angle need to be still improved. To contribute to this strand of research, the present study numerically explored the flow pressure distribution and cavitation index for various Froude numbers (2.56 ≤ Fr ≤ 20.1) and lateral wall expansion angles (0 ≤ α ≤ 10) at 10% gate opening using the RNG(k–ε) turbulence model and the VOF method in Flow three-dimensional (3D) software. Practical relevance for predicting the cavitation index and allowable lateral expansion angle in these areas is also introduced.

The pressure and velocity of flows are two fundamental parameters to describe the flows of transition regions. Since water is an incompressible Newtonian fluid, these parameters are determined by the classical Navier–Stokes equations. These equations are developed based on the physical laws of conservation of mass and momentum. The 3D Reynolds averaged Navier–Stokes (RANS) equations in the Cartesian coordinate system for incompressible and turbulent fluid flows are presented below (Chen et al. 2021):

where is the mean rate-of-rotation tensor in a rotating reference frame with the angular velocity ⁠. In general, is determined by the mean strain and rotation rates, the angular velocity of the system, and the turbulence fields.

The value of F is interpreted based on the type of problem. For a single fluid, F represents the fluid volume fraction. The F values of 1 and 0 show the presence or absence of the fluid, respectively. The locations in which F equals 0 are called voids; in other words, voids are regions without fluid mass. Therefore, a uniform pressure is assigned to these areas. In practice, these regions are filled with vapor or gas because the density of vapor or gas is insignificant compared to fluid density (Hirt & Nichols 1981; Zhang et al. 2011).

Seymareh Dam is a concrete double-arch dam located in Ilam Province, Iran, and is equipped with two bottom outlets. The entrances of the first and second bottom outlets are placed 620 and 640 m, respectively, above the sea level (20 and 40 m above the river bed). The plan and longitudinal parts of the first bottom outlet are depicted in Figure 2(a) and 2(b). This outlet is designed for a maximum discharge of 654/m³/s at 111.5 m upstream water head. With respect to flow conditions, it has two parts: the under-pressure part and the free-surface flow part with lengths of 36.5 and 8.9 m, respectively. As seen in Figure 2, a transition region with a length of 3.2 m is located at the end of the conduit.

The number of mesh increased as a result of applying the grid overlay boundary condition to the entrance of the bottom outlet, rather than to the upstream reservoir, in the investigated scenarios. Half of the flow was simulated due to the longitudinal symmetry of the conduit, which boosted the accuracy of flow simulation in the region. The grid size, ranging from 2 to 25 cm, decreased from the upstream toward the downstream of the conduit. This method generates a higher resolution for the transition regions. Table 2 indicates the numerical model sensitivity under various conditions.

The cavitation index and flow pressure were computed for 77 scenarios with 7 Froude numbers ranging from 2.56 to 20.1 and 11 expansion angles (0, 1, 2, …, 10) (see Table 3). Different Froude numbers were used to determine the possibility of cavitation and flow pressure at the entrance of the transition region. It is worth noting that the cross-section shape of this region was rectangular in all simulations.

In their study, Zhang et al. (2010) observed large negative pressures on the sidewalls of the transition regions for partial gate openings. In a similar vein, Davis & Sorensen (1984) found that air concentration reached its maximum level in small gate openings (the powder flow state). Hence, in the current study, the hydraulic characteristics of flows were investigated in 10% service gate opening as the critical percentage of gate openings. Moreover, the free-surface condition was used for the flow downstream of the gate in all the simulations.

Viscous forces are insignificant at large-scale flows for large Reynolds numbers (Ting 2016). Therefore, in this study, as the Reynolds number (16.52 × 10⁶ < Re < 30.94 × 10⁶) exceeds (Li et al. 2016a) was neglected. Additionally, the shape of the cross-sections was rectangular in all the simulations.

Pressure coefficients (⁠⁠) were computed for various entrance Froude numbers (⁠⁠) and lateral expansion angles (α) in S₁ (the entrance of the transition region where the geometry started to change) along the Z-direction (Figure 2). The results demonstrated that this area was prone to flow separation and local vortices.

Furthermore, there was a remarkable decline in the pressure for Fr₀ = 6.30. To explain it more clearly, negative pressures were obtained for all the lateral expansion angles, except for α = 0°. The minimum pressure coefficient value was −0.3 at α = 10°. With the increment of (6.3 < Fr₀ ≤ 20.10), the pressure reached stability.

Considering the cavitation index relevance, as another pressure coefficient, the entrance of transition regions is prone to cavitation. Jamali & Manafpour (2019) found that the cavitation phenomenon happened in the initial parts of the transition region. Consequently, in the present study, the cavitation index was measured for two sections, namely, and (2 m after ⁠), along the Z-direction on the side walls.

By increasing the to 14, the value of the cavitation index sharply decreased to less than 0.2 at ⁠, which suggests the high possibility of cavitation erosion. The minimum σ value was −0.15 for = 14 at α = 10°. By raising the from 4.49 to 14, the value reached 0.2 at α = 7°.

The values were substantially reduced to less than 0.2 for Fr₀ = 17.5 at all the lateral expansion angles, except for α = 0°. In general, when the was boosted to 20.10, the side walls faced the risk of cavitation even at α = 0°.

Further analysis indicated that there was a sharp decrease in the σ values at α > 7° for all the ⁠, which can be associated with several physical processes, such as flow separation and local vortex. Increasing α from 7° to 8° boosted the local vortex intensity in the flow field and strengthened the flow turbulence, which resulted in flow separation. However, further investigations into physical processes are required to address this issue more profoundly.

In this study, the flow pressure distribution and cavitation index on the side walls of the transition region for various Froude numbers and lateral expansion angles were numerically examined, and the possible relationship of with and was explicated. Ultimately, the allowable lateral expansion angle was measured. Based on the results, the following conclusions can be drawn:

• The comparison of the cavitation indices of S₁ (the entrance of the transition region) and (2 m after ⁠) revealed that S₁ was an area prone to local vortices, flow separation, and consequently, cavitation.

• By increasing the expansion angle (α), the pressure decreased for all the ⁠. Negative pressure was observed for Fr₀ = 2.56 at ⁠. For Fr₀ ≥ 4.49, negative pressure was reported at all the lateral expansion angles, except for α = 0°.

• Pressure coefficient values reached −0.2 and −0.25 for Fr₀ = 2.26 and 4.49, respectively, at α = 10°. By raising the from 6.3 to 20.1, the pressure values stabilized and the minimum pressure coefficient value was obtained (−0.3).

• The probability of cavitation formation on the side walls of the transition region was very low at all the lateral expansion angles for Fr₀ ≤ 4.49. The obtained results are well aligned with those of Thompson & Kilgore (2006). However, they differ from the results of Li et al. (2016a) who observed the occurrence of the cavitation phenomenon for Fr₀ ≥ 3.89 even at α = 0° in partial gate openings.

• With an increase in the to 14, the values of the cavitation index dropped to less than 0.2 at and reached 0.2 at α = 7°. To put it in other terms, the probability of cavitation erosion was very high under these conditions.

• Given that the values of the cavitation index were less than 0.2 at all the lateral expansion angles, the expansion of side walls in transition regions should be prevented for Fr₀ ≥ 17.5. Furthermore, the side walls held the risk of cavitation for Fr₀ = 20.1 even at α = 0.

• The increased local vortex intensity and flow separation led to a dramatic reduction in the σ values at α > 7° for all the Froude numbers.

These findings have great practical implications for designing and constructing transition regions. In this study, the capability of the CFD model to predict some flow characteristics, such as pressure distribution and flow separation, in transition regions was confirmed. However, gaining a deeper understanding of the issue requires carrying out experimental investigations under the aforementioned conditions at various gate openings (in the current research, the percentage of gate openings remained constant [10%]). Moreover, this study was limited to the case of rectangular cross-sections. Future research should scrutinize other shapes for cross-sections (e.g., the trapezoidal shape).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.