Numerical investigation of alternating skimming ﬂ ow over a stepped spillway

This study aims to illustrate the in ﬂ uence of stepped spillway width on alternating skimming ﬂ ow development. A computational ﬂ uid dynamics (CFD) model in Ansys Fluent ® was established to simulate the ﬂ ow over stepped spillways, using a volume of ﬂ uid model (VOF) and Reynolds Averaged Navier – Stokes (RANS) turbulence model (SST k - ω ). The model was ﬁ rst validated by comparisons of velocity pro ﬁ les at step niches and water depth at step edges with existing measurements acquired by the bubble image velocimetry (BIV) technique and an ultrasonic sensor, in a 0.5-m wide stepped spillway physical model. The SST k - ω model gave good results for velocity and water depth, and the numerical predictions of the vorticity in the skimming and recirculating ﬂ ows were qualitatively adequate. The model was used to analyse the ﬂ ow regime for six different stepped spillway widths. The careful examination of ﬂ ow patterns at the different stepped spillway widths showed that the alternating skimming ﬂ ow appears for the stepped spillways wider than 0.35 m due to the asymmetrical distribution of vorticity patches that are generated in the step cavity. These vorticity patches are of uniform size and shape when the spillway width is less than 0.35 m, which does not produce an alternating skimming ﬂ ow. However, for wider stepped spillways, the vorticity increases, and an alternating skimming ﬂ ow appears closer to the crest. ﬂ ow, partially aerated ﬂ ow and fully aerated ﬂ ow.


INTRODUCTION
Stepped spillways (SP) are a famous technique used for their simplicity of shape to transfer high-energy water and to decrease flow velocities (Chanson ). However, some physics are not yet understood like the formation of an alternating skimming flow (Lopes et al. ). In this paper, the limit width, from which the formation of the alternating skimming flow occurs, is investigated, as well as the reasons and what is behind this formation.
The flow over SP occurs in three different regimes: nappe flow, transition, and skimming flow (Wan et al. ). The SP is typically designed to operate in a skimming flow regime (SK), which can be one of the two sub-regimes: SK1 and SK2 (Chanson ; Ohtsu et al. ). The SK regime is characterized by a main water flow over a pseudo-bottom, which connects all the steps where the development of a boundary layer along the length appears, reaching the free surface and allowing air-entrainment. This is usually called the inception point. Toro et al. () and Zabaleta et al. () presented details of the mechanisms leading to air entrainment in the skimming flow, as it is likely that the phenomenon occurs because of vorticity patches that were generated in between the steps, which then become released to the flow and reach positions close to the free surface. In the SK1 sub-regime, the mixing layer does not reach the step end, an undulant free surface appears, and in extreme cases, this effect is observed partially parallel to the step tread. The reason for this behaviour is the wake zone formed in each step downstream and the recirculation vortex underneath. In the SK2 sub-regime, there is an interaction between the wake and the recirculation vortex of one step and the wake formation of the subsequent one. In this paper, our focus is to continue the previous work of Lopes et al. (), and to understand the reason behind the formation of the alternating skimming flow and whether it is related to the vorticity patches generated between the steps discovered by Zabaleta et al. (); also it is to identify the critical width for this alternating skimming flow.
To achieve this purpose, we used the experimental data of Lopes et al. (), to validate the 3D numerical models established in Ansys Fluent ® .
Then, concentrating on the parameter of the width, we constructed similar models with different widths (w) and the same step height (s), expressing a non-dimensional ratio w/s to identify the widths for which the alternating skimming flow occurs. We also investigated the influence of the spillway width over the alternating skimming flow regime, and for doing this, the spillway was changed for different stages, starting from 0.1 m and increasing to 0.6 m. As a result, the alternating skimming flow regime in several steps can be defined by the alternated presence of the SK1 and SK2 sub-regimes.
The work is organized as follows: 'Experimental Facility' describes the physical experiment in the constructed facility to measure the flow depths and velocities. 'Numerical Model' presents the equations used in the numerical simulation research, the computational models (mesh generation and its analysis and the definition of constants and procedures), as well as its numerical validation using experimental data, and introduces the methodology fol-

NUMERICAL MODEL
In this research, the SP model studied experimentally is simulated using the commercial code Ansys Fluent ® , where To detect the free surface, a single set of mass and momentum conservation equations (Equations (1) and (2)) are solved for each phase, and then the volume of fluid at each cell of the domain is calculated. VOF defines an alpha (α) value, representing a fraction of volume of fluid in a cell, ranging from 1 to 0, representing a filled cell with water or fluid 1 (f 1 ), α ¼ 1, and empty cells or filled with fluid 2, air (f 2 ), a value α ¼ 0. In each case, the volume of the fluid fraction sum of air and water is unity, so the volume fractions representing air α a can be given as At each time step, α is updated using an advection equation for α (Equation (3)). The VOF model assumes that each phase is immiscible so it does not model air entrainment. It is common to use α ¼ 0.5 as first guess for where u is the Reynolds-averaged velocity components in x the mean velocity vector, ∇:(μ∇ u) þ (∇ u):∇μ the decomposition of the shear stress tensor, f [kg m À2 s À2 ] the volumetric surface tension force, and p* is a modified pressure. The density is ρ [kg m À3 ] (Equation (4)) and μ [kg m À1 s À2 ] (Equation (5)) is the dynamic viscosity and μ t (Equation (6)) is the turbulent viscosity at control volumes that are functions of water volume fraction α w and can be determined in each cell as follows: where k is the turbulent kinetic energy, ε is the turbulent energy dissipation, ω ¼ ε=k is the specific energy dissipation rate, and C μ is an empirically derived constant. The k, ε or ω are calculated by using transport equations for each of the variables with some particularities depending on the turbulence model used (ANSYS ).

Properties, boundary conditions and settings
The mechanical properties of the fluid used in this paper are: ρ w ¼ 998.78 kg m À3 and μ w ¼ 1.1094 × 10 À6 m 2 s À1 as density and kinematic viscosity of water, and ρ a ¼ 1.225 kg m À3 and μ a ¼ 1.4657 × 10 À5 m 2 s À1 for air. Regarding surface tension, it was defined as 0.072 kgs À2 . The discretization scheme for momentum, turbulent kinetic energy, and dissipation rate second-order methods were used as they are relevant when representing swirl flows. Finally, we used a SIMPLE coupled phase algorithm, to couple the pressure velocity. Moving to the boundary conditions, we considered the hydrostatic pressure for inlet on the left, with input water and air velocity, and outlet on the right. Also, the atmosphere was on the top and a no-slipping and stationary wall at the spillway. At the bottom, standard wall functions were applied to avoid the use of excessively fine meshes and to save computational resources while modelling the boundary layer. The initial conditions considered at t 0 are water just at the inlet with 0.1 m water depth (h) and velocity of 0.7 ms À1 (u x ).
Following these boundary conditions, we looked for a difference that is less than 1% between the inlet and outlet water flows. Also, we established as conditions the values k, ω, volume fraction, and mass staying constant (allowing a tolerance of 10 À6 for the residual). With these conditions and by using a time step variable to find the compatibility with the Courant number, we found that 22 seconds is the value for the steady-state solution (Kaouachi et al. ).

Grid convergence test
To optimize the process and reduce numerical problems, the best choice was to work using structured mesh algorithms.
They also offer more regular memory access and reduce significantly its latency (

Model validation
To validate the numerical model, we did two tests: (1)  and roughly equals 32 mm at step 5 and 6. We realized that at step 5 the water depth rises due to the air-entrainment.
The SST k-ω model is a hybrid model working as a k-ω at the wall and as a standard k-ε in the free stream. This is done to improve the performance in flows under adverse pressure gradients (such as those with separation as on a stepped spillway in skimming flow).      Free surface analysis along the width Figure 5 presents the cross-section free-surface profile in all stepped spillway models corresponding to different widths.

RESULTS
There is a gradual decrease in water depth along the steps, which is a safety factor and considered the main advantage of    Pressure distribution on the steps Figure 8 illustrates the pressure distribution on the step surfaces for the different stepped spillway widths along the length. In SSW0.10 (Figure 8(a)), the pressure on different horizontal step surfaces is uniform, but it seems very clear that the pressure increases longitudinally along the x-axis.
In SSW0.20 (Figure 8 In the particular cases SSW0.50 and SSW0.60, as shown in Figure 8(g) and 8(h), after step 12, the pressure distribution presents alternating oscillations with two or three peaks along the step width until step 19, leading to an increase of pressure intensity, reaching the maximum at step 19.
Transversal velocity field at the step edge For the stepped spillway SSW0.10, the free surface shape is steady at all the steps (Figure 5(a)). The pressure field shows a uniform distribution along each section (Figure 8(a)). The velocity field (Figure 6(a)) shows perfect similarity in both profiles' W/4 and W/2, which leads to identical values of kinetic energy turbulence at the free surface and in step cavities (Figure 7(a)).
The SSW0.20 model shows that the free surface along a cross-sectional profile is also quasi-stable at all the steps ( Figure 5(b)). The pressure on different horizontal step surfaces shows uniform distribution (Figure 8(b)), as well as velocity longitudinal distribution (Figure 6(b)). However, looking to the k values, we can see a larger rise in this model than in SSW0.10 simulations, especially for step 7, where it reached its higher value (Figure 7(b)).
The SSW0.30 model also shows regular velocity until step 6/8, then a change in the velocity values near the wall can be observed (Figure 9), which corresponds to a change in the pressure distribution and in the kinetic energy near the surface from step 6 (Figures 8(c) and 7(c)). A bellshaped distribution of the free surface starts from step 8 across the model width ( Figure 5(c)). However, a perfect similarity of the main flow velocity shows up in the longitudinal profiles W/4 and W/2 (Figure 6(c)). An identical k value from the crest to step 5 is shown in the profiles W/4 and W/2 (Figure 7(c)). There is an increase of 20% of k in W/2 compared with W/4 for steps 6 to 28. Focusing on the step cavity, a slight drop of k is found in odd steps for W/2.
The SSW0.35 model shows regular velocity distribution until step 12 ( Figure 10). After that, a velocity irregularity starts to appear on both sides in all the steps, which may be noted in step 13. In the pressure contour (Figure 8  This is consistent with the cross-section free-surface profile, where we observe intensive oscillation ( Figure 5(g)).
The longitudinal section of velocity W/4 ( Figure 6(g)) demonstrates that the flow does not skim until the end of each step (SK1). This effect is not seen for W/2 where, from step 10 and on, the flow reaches the end of the odd steps (SK2). In even steps, the mixing layer does not reach the end of the step (SK1). On the horizontal step faced, there is a large pressure differential ( Figure 8(g)). This is a phenomenon to be taken into consideration during the design of masonry stepped spillways as described by Winter et al. (), as a pressure variation Regarding the TKE profiles in SSW0.40 for W/4 and W/2 (Figure 7(e)) starting from step 6, the k value is higher in W/2 than the k value at W/4 by 20%, and the k value of odd steps at W/4 is equal to the k of even steps at W/2. In SSW0.50, k profiles at W/4 and W/2 (Figure 7(g)), the value of k increases in direction to the free surface. For the W/4 section, the k in even steps is higher than in odd steps. At SSW0.60 the k value increases compared with the previous cited models, but a high similarity shows up in the profile's W/4 and W/2 (Figure 7(h)).
In summary, when the relative stepped spillway width is low, no differences of pressure and velocity across the channel width are noted. With increasing width, some differences across the width appear, first in kinetic energy near the free surface, pressure and velocity field.

CONCLUSIONS
Towards the understanding of the formation of alternating skimming flow, noted by Lopes et al. (), as well as to achieve a width limit for its occurrence, we did numerical research on the skimming flow regime in different stepped spillways with various widths. using CFD with fully threedimensional turbulence model closure SST k-ω and employing the VOF method for free-surface detection. Then we used laboratory data from a specific stepped spillway width to validate the model. The following conclusions can be retrieved from this work: • CFD based on VOF and RANS is a valid methodology to simulate the flow over a stepped spillway in the laboratory with an appropriate computational mesh, and the SST k-ω gives good results.
• While when the stepped spillway width is less than 0.35/ 0.06 (w/s ¼ 6), no differences of pressure and velocity across the stepped spillway width are relevant, from that limit, a clear alternating pattern can be identified in the distribution of velocity, kinetic energy and pressure.
• The relative stepped spillway width of w/s ¼ 0.35/0.06 ¼ 6 (w between 0.35 and 0.4), can be identified as a transition (critical limit) since it leads to the formation of vortices along the width which are still of uniform size and still keep the similarity of the main flow velocity.
• When increasing stepped spillway width, an alternating skimming flow and inception point appear early.
• Asymmetrical distribution of vortices in terms of shape and size are present.
• A dependence from the ratio of width and oscillation length is suggested, however more simulations are needed.
Future work on alternating skimming flow will focus on research of more simulations using different widths and conjugate with more flow rates. Research on the influence of air concentration profiles and bubble behaviour relation with the alternating patterns is also an interesting topic.