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The increase of water demand needs for human multi-purposes and the environmental improvement aspects have necessitated the construction of new dams and their related structures, especially spillways. Spillways often are run over moderate to steep slope channel ranges to convey excess water from higher to lower elevations. The flow over such channels is characterized with high velocity and kinetic energy. Therefore, surplus water should be released as safely as possible; otherwise it would cause a series damage of cavitation and erosion to the structure itself and the surrounding area. This could be achieved by protecting the surface of the structure in such a way that provides a hydro-dynamically rough surface with extremely high turbulence which has the ability to slow down the rapid flow and dissipate a significant portion of the overflowing energy. In order to enhance their effectiveness, many techniques have been investigated by hydraulic researchers such as stepped spillways riprap, stone pitching, gabions, stepped spillways ...etc.
Stepped channels and spillways have been used for more than 3500 years. The ease of construction and the design simplicity have led this structure to be more popular since 1980 (Chanson, 2002). A stepped spillway can be defined as that hydraulic structure in which a series of steps of different shapes, dimensions, and arrangements is built into the spillway surface at some distance from the spillway crest and extended to the toe and the dissipation of the overflowing energy would take place on the structure. Chanson (2002) points out that energy dissipation is caused by steps on the spillway. These steps could produce high turbulence and aeration due to the strong interaction between the overflowing water and the surrounding atmospheric air. Although, the energy dissipation efficiency would be significantly enhanced by stepped spillways, the need for a large size of the energy dissipator structure at the downstream of the spillway could be greatly reduced as well (Chanson, 2002). The development of new construction technology has regained the design of such structure. Accordingly, an intensive investigation on stepped spillways has been conducted to determine the flow behaviour and to provide standardized design criteria, especially associated with roller compacted concrete (RCC) for both embankment and gravity dams.
2- Literature Review:
Stepped spillways have become popular in recent years due to the development of new construction technology. A series of steps is introduced into the spillway profile by which a significant portion of the overflowing energy may be dissipated. High turbulence could be created by the steps which can be considered as the key point of the energy dissipation process. The flow over stepped spillways are characterized with high velocity, high turbulence and self-aerated. Various step geometries have been tried to enhance the turbulence and consequently the performance of the structure to dissipate the flow energy as much as possible. The steps might be horizontal, pooled, inclined upward and inclined downward.
A brief comprehensive review of citations in the literature regarding the various elements associated with the definition of the hydraulics of stepped spillways is explained in this section. In addition, due to the limited number of investigations conducted on the inclined steps, the hydraulics of flow over stepped spillways is concentrated on the flow characteristics over horizontal steps.
2-1 flow regimes and conditions over stepped spillways.
Depending on the discharge and step geometry, three different flow regimes could be distinguished over a stepped spillway of any slope. Nappe flow regime with small flow rates, skimming flow regime corresponds with high overflowing discharges and the transition flow regime between the first two regimes with intermediate discharges. Although, each regime has its own flow characteristics, the flow behaviour is characterized with highly turbulent and self-aerated. A detail characteristic of the flow regimes is discussed in the following section.
2-1.1 Nappe flow regime:
Generally, with small flow rates and relatively large step heights, a series of over-falls comprises the spillway profile. The free falling nappe at the upstream end of each step, an air cavity, a pool of recalculating water, and with or without a hydraulic jump on the steps are the flow characteristics of this regime. The formation of a hydraulic jump downstream the nappe impact depends on the step geometry and the flow conditions. Fig (1) shows a nappe flow regime on a single-step model.
The design of a spillway to operate specifically for this regime is limited for low dams with larger steps and flatter slopes. Nappe flow will not be investigated in this study. More details for this flow regime can be found in Horner (1969), Chamani and Rajaratnam (1994),Chanson (1996), Toombes and Chanson (2002) and Toombes (2002), Andre`(2004), El-Kamash et al (2005) and Toombes and Chanson (2008).
Fig (1) Photograph of nappe flow regime on a single-step model (from Toombes, 2002).
2-1.2 Skimming flow regime:
For relatively high discharges, the overflowing water skims over the pseudo bottom as a coherent stream and cushioned by intense re-circulating cavities underneath which are trapped inside the steps and rotating in a rounded triangular vortex (Diez-Cascon, Blanko, Revilla and Garcia 1991; Chanson 1994; Chamani and Rajaratnam 1999; Pegram, Officer and Mottram 1999). Pseudo bottom refers to a hypothetical line parallel to the chute slope and formed by passing through the steps outer edges. Filled pool with water inside the steps re-circulating vortices are the main characteristics of this flow regime. Fig (2) presents a skimming flow regime over a stepped spillway. The transmission of shear stress from the fluid flowing sustains the vortices. This mechanism is considered to be the predominate contribution of the energy dissipation in this regime. In the present study the flow behaviour under skimming flow regime will be investigated extensively over upward inclined stepped chutes with moderate slopes.
In skimming flow regime and based on the vortices characteristics and the chute slope, Chanson (2002) proposed three sub-regimes as follows:
SK1- Wake-step interference sub-regime. This might be occurred in flat chutes (θ < 12) and can be discriminated by the formation of a wake region downstream the step edge which does not extend over the full step length and skin friction drag could be occurred on the step downstream of the wake.
SK2- Wake-wake step interference sub-regime. For slopes (12 < θ < 25) the drag skin fraction is small which could be due to the extension of the wake region nearly over the full length of the step and the interference with the wake of the successive step.
SK3- Re-circulating cavity flow sub-regime. For steep slopes (θ > 25) a quasi-stable re-circulation may be occurred in the step corners and the re-circulating eddies in the cavity are large-scale vortices.
Fig (2) Photograph of skimming flow regime over a stepped spillway (from Gonzalez 2005).
2.1-3 Transition flow regime:
This regime would be appeared with moderate discharges. The flow in this regime has not been investigated extensively; hence the current knowledge about this regime is limited. Basically, the flow is chaotic and significant number of water droplets would be ejected for heights about 3 to 8 times the step height (Chanson and Toombes 2004). For this regime, nappe flow associated with the presence of air cavities can be observed on some steps and skimming flow with re-circulating vortices on the other steps. Fig (3) displays the transition flow regime over a stepped chute. Transition flow regime will not be investigated in this study. More detail on the transition flow regime can be found on (Chanson, 2002), Toombes and Chanson (2002) and Gonzales (2005).
Fig (3) Photograph of transition flow regime over a stepped chute (from Gonzalez 2005).
2.1-4 Onset of skimming flow regime:
The complexity of the flow structure, due to high turbulent and air entrainment, has led the analytical solution regarding the clear definition of boundaries between the flow regimes to be difficult. The visual interpretation based on the appearance of smooth and glassy water surface and step cavities filled with water corresponding for skimming flow regime and a series of free-falling nappe and air pocket within step cavities for nappe flow regime would be greatly subjective. Hence many empirical equations have been proposed based on the visual interpretation. Dimensional analysis showed that the flow regime prediction depends on the discharge defined by the dimensionless parameter (dc/h) and the spillway slope defined by the dimensionless step geometry (h/l), where, dc is the critical depth and h and l are the height and length of the step (Essery and Horner 1978, Peyras et al 1992).
Rajaratnam (1990) proposed the first criterion to define the onset of skimming flow in terms of the ratio between the (dc/h) to be greater than o.8 for values of (h/l) between 0.4 and 0.9. While, Chanson (1994) suggested the following equation to predict the onset of skimming flow for 0.2 < h/l < 1.25 based on fitting the experimental data:
On the other hand, Yasuda and Ohtsu (1999) proposed the following equation for the prediction of the lower limit of skimming flow regime for stepped spillways of slopes lower than 55:
Furthermore, Chamani and Rajaratnam (1999) presented the following equation to determine the onset of skimming flow regime based on the observation that in skimming flow the jet and the spillway slope are parallel:
-1 ............................... (4)
While, Chanson (2002) stated that the onset of skimming flow regime can be predicted from the following equation:
The derivation of the above equation is based on the re-analysis of a large of experimental observation of change in flow regimes.
Finally, the experimental results of Boes and Hager (2003a) on stepped spillways of slopes 30, 40, and 50 showed that the following equation can be used to predict the onset of skimming flows which was in agreement with data of others.
It should be noted that the above empirical equations were derived for horizontal steps. In the present study upward inclined steps will be investigated and may result in the derivation of different empirical equations.
2.2 flow regions over steeped spillways in skimming flow regime:
In the following section, the description of natural air entrainment mechanism is only related to skimming flow regime. For a skimming flow regime down a slopped stepped spillway, four distinct regions can be observed as follows:
As water flows down a stepped spillway, the flow is accelerated and the boundary layer develops and grows rapidly next to the crest which has no a significant effect on the water surface. This region is known as non-aerated flow and the flow depth is known as the clear water depth because air is not entrained yet and the flow is seemed to be smooth and glassy.
Natural air entrainment or self-aeration mechanism initiates as the outer edge of the turbulent boundary layer reaches the water surface and this point is called inception point and white water appears clearly. In addition, the turbulent energy of the vortices acting next to the free surface becomes large enough to overcome both the surface tension and buoyancy effects. Downstream this point a layer containing air and water extends the flowing fluid which increases the air concentration and the free surface becomes wavy. Further downstream and near the downstream end the quasi-uniform flow may be achieved, particularly in relatively long slope chutes, where the flow parameters are nearly constant on each step. Fig. (4) illustrates typical flow regions and air entrainment down a stepped spillway under skimming flow regime. Indeed, Felder, S. And Chanson (2009) observed seesaw pattern for the air-water flow properties at the fully developed aerated region downstream a 21.6 steeped spillway. They believed that the interaction of successive step cavities and their interference with the free surface cause this pattern to take place. Therefore, they stated that the concept of uniform flow on spillways may be restricted to smooth spillways and further investigations needed for the validation of uniform flow for skimming flow on stepped spillways.
2.3- Inception point of air entrainment:
As mentioned above, the intersection point of turbulent boundary layer with the water surface is known as the point of inception. Based on visual interpretation, it corresponds to the point of the apparition of white water (Chanson, 1994; Chamani, 2000; Matos, 2000). The exact location of this point is necessary for the designer, because the upstream reach of this point is subjected to cavitation risk. Different empirical equations have been proposed by investigators regarding the length to and the flow depth at this point. Their main conclusions are: the steps lead the boundary layer to grow faster and therefore shifting up the inception point further upstream; the discharge is the dominate parameter affecting the inception point with a slight effect of the chute slope.
Fig. (1) Typical flow regions and natural air entrainment down a stepped spillway under skimming flow region.
The re-analysis for a wide range model data of (Beitz and Lawless1992; Bindo et al 1993; Frizell and Mefford 1991; Sorenso 1985; and Tozzi 1992), Chanson (1996) proposed the following two equations to predict the length to, and the depth at, the point of inception for spillway slopes from 6.8<< 59:
Where, Li, di are the length to, and the depth at, the inception point respectively;, is the chute slope; ks is roughness height measured perpendicular to the flow direction: ks = h cos,; h is the step height; and F* is the Froude number defined in terms of the roughness height deduced from:
Where, q is unit discharge, and g as the gravitational acceleration.
Combining equation (11) and equation (12) yields:
While, Chamani (2000) obtained the following equation to estimate the length from an ogee crest shape to the point of inception and the chute slope range (50<< 60:
Where Fi is the Froude number defined in terms of inception point calculated from:
In which, l is the step length.
Matos (2000) conducted experiments on 53 slope of crested stepped flume and found the following equations:
In which, dw and Ci are the clear water depth and the mean air concentration at the point of inception respectively.
On the other hand, the experimental investigation by Boes and Hager (2003) on gated stepped flumes with the extension results for 26<< 55 crested stepped spillways yields:
Where, Fs is the step Froude number defined as:
Additionally, Meireles and Matos (2009) studied the skimming flow regime in the non-aerated region down a 26.6 stepped spillway and obtained the following two equations which nearly best fit the data of some other investigators:
Although, different empirical formulas have been acquired by different investigators and the subjective criterion is still lacking, it can be concluded that the flow prosperities at this point depends on the step geometry and the discharge. The same conclusion was made by LANE (1939) on smooth slopes.
In the present study the point of inception is thought to be shifted further upstream when compared with horizontal steps which may be due to reduction of the turbulent boundary layer thickness produced by upward inclined steps. This could be considered as another advantage of these steps which shorten the length subjected to the cavitation risk.
2-4 flow over stepped spillways with upward inclined steps:
Few experimental investigations have been performed on stepped spillways with upward inclined steps. Researchers studied the efficiency of inclined steps in terms of energy dissipation or the residual energy at the downstream end of the spillway without details of air-water flow prosperities. The key feature of upward inclined steps is to enhance the turbulence and consequently the energy dissipation.
Horner (1969) classified the flow over upward inclined steps into three categories: subcritical with relatively low discharges, supercritical with higher discharges, and a transition category in which a mix of subcritical and supercritical flow may occur. With subcritical category, the flow leaves the steps would pass from subcritical to supercritical through a critical depth and the may return to subcritical by means of a hydraulic jump. Furthermore, he considered the entire cascade to be acted as a uniform zone. While, the flow characteristics in the supercritical category were thought to behave similar to that on horizontal steps. However, with transition category he observed subcritical flow in the transitory zone and supercritical flow in the uniform zone.
Essery and Horner (1978) stated that with the same slope and step size and at any discharge, lesser number of steps on cascades with inclined steps is required than on cascades with horizontal steps.
Chinnarasri and Wongwises (2004) investigated the flow over stepped spillways with upward inclined steps. The chute slopes were 30, 45, and 60 whilst the step inclinations were 10, 20, and 30 Twenty identical steps with step heights 7.5cm, 10.6cm, and 13.0cm respectively were tested with a discharge ranged from 0.01 to 0.17 m2/s. A pitot tube was used for the outlet velocity measurement at a section which was located downstream to the lowest step face about 3 to 4 times the step length where it was thought that the air entrainment was lessened significantly.
Three flow regimes were observed; nappe, transition, and skimming flow as the same on the horizontal steps. In nappe flow, at small amount of discharges, they observed free falling nappe at the step brink of the inclined step and the development of hydraulic jump on the step face. While the free jet was vanished in the transition flow regime, with intermediate discharges, and the free surface became wavy with spray. On the other hand, smooth water surface with small air entrainment were the characteristics of skimming flow with large discharges. Moreover, they studied the effect of step inclination on the upper limit of nappe flow and the lower limit of skimming flow in order to define the transition regime. They observed that the upper limit of nappe flow is not influenced by the step inclination. While they observed a slight increase of the lower limit of skimming flow as the step inclination is increased. This was attributed to the fact that the increase of the step inclination increases the relative height of the outer edge and consequently both the pool height and air pocket under the falling jet of the nappe flow are increased as well. However, the following equation was presented to predict the minimum critical depth required for the onset of skimming flow on chutes with inclined steps for ( :
With as the step upward inclined angle.
Their experimental results demonstrated that, for the same flow conditions, a higher energy dissipation of about 6% of the total drop height HT was presented by upward inclined steps, particularly in skimming flow regime, compared with horizontal steps. They also concluded that the energy loss is increased by the increase of the step inclined angle which may be due to the production of more spray caused by the steps obstruction to the flow direction. On the other hand, in nappe flow regime, they observed a rapid decrease of energy loss as the drop number increases. While, in skimming flow regime, this decrease of energy loss was lessened and approached a constant value. Furthermore, in terms of velocity ratio defined as , this ratio was increased as the drop number increases for the chute slopes and step angles tested and the following empirical equation was proposed to estimate the kinetic ratio:
Finally, they stated that the slight difference in the construction cost would lead the upward inclined steps to be considered as a more effective alternative to horizontal steps. However, Chanson (2002) argued that such technique may exert more hydraulic loads and increase the cost of the spillway and more researches are needed for this purpose.
Chinnarasri and Wongwises (2006) studied the performance of stepped chutes having steep slopes 30, 45, and 60, in terms of energy dissipation, with different step geometries: horizontal steps, endsills, and upward inclined 10, 20, and 30 steps. They introduced the characteristic height (m) defined as the height of the endsill and the incremental height due to the upward inclined step above the horizontal step. In nappe flow, significant energy dissipation was caused by the formation of the hydraulic jump and impact of jet on the step face implying a slight effect of the characteristic height (m) on the relative energy loss. While, the increase of the characteristic height (m) increases the relative energy loss in both transition flow and skimming flow regimes. However, the results demonstrated that steps with end sills produced higher relative energy loss than steps with upward inclined steps. This is because the flow is disturbed and obstructed by the end sills more than that by the inclined steps and resulting in the development of smaller re-circulating vortices in the latter.
In contrast, the experimental results of Barani et al (2005) showed that the more energy dissipation was produced by steps with upward inclined than steps with end sills in 41.41 stepped spillways with 21 identical steps of 4cm height and four inclined slopes 15, 26, 36, and 45. Moreover, the following empirical equations were proposed to estimate the relative energy loss as a function of the relative critical depth ():
Where, EL is the energy loss of flow, Eo is the total head of flow, , and are coefficients defined as follows:
For inclined steps:
While, for steps with end sills:
Barani et al (2005) stated that for small dams with low flow rates corresponding to nappe flow regime, most of the energy can be dissipated with inclined steps and the need for stilling basin at the downstream end of the spillway can be neglected. For large dams and high discharges, stepped spillways with inclined steps can produce higher energy loss than steps equipped with end sills. They also concluded that the loss of energy is increased as the angle of step inclination increases. The same conclusions were made by Karim El-Jumaily and Mariam Al-Lami (2009) when used scale 1:20 to study Bastora stepped spillway laboratory. The stepped spillway slope was 42 and three upward slopes 14, 28, and 42 for the steps were tested with 92 identical steps of 4.5cm height and 5cm length.
It can be observed that few experiments have been performed on the hydraulics of flow over stepped spillways with inclined steps. These studies are restricted to steep slopes and experiments on moderate slopes are still lacking. On the other hand, the hydraulics of flow in terms of the pressure on the steps, the location of the inception point where the self-aeration process initiates and the region where the boundary layer develops, upstream the inception point, that is subject to the cavitation risk have not been investigated yet.
In the present study the hydraulic characteristics of the flow over moderate sloped stepped spillways with upward inclined topography bottom are investigated by taking into consideration various elements governing the phenomenon.
3- Numerical simulation and its role in solving engineering problems:
Recent advances in computers, hardware technology and the development of robust computational fluid dynamics (CFD) software have led the numerical simulation, for solving practical scientific problems, to be used widely and be considered as a more powerful alternative tool. Despite the expensive cost and time consuming of experiments, the direct measurements needed in some of such experiments are often, even with high accurate and developed instruments, could not be acquired easily and becomes problematic. Two fundamental approaches have been used to achieve this goal are; Grid-based methods and Particle-based methods.
Various computational fluid dynamic models have been utilized by hydraulic researchers and engineers to investigate the hydraulic characteristics of flow over stepped spillways based on Grid-based methods. In the present study this problem is investigated thoroughly using Particle-based method for the first time. Therefore, a comprehensive review related to Particle-based methods is presented in more details.
3-1 Grid-based methods for simulating flow over stepped spillways:
This method requires pre-defined grid to descretize the problem domain. The complexity of the hydraulic behaviour of the flow over stepped spillways, as described in the literature which is characterized by the wavy free surface, high turbulence, self-aerated due to the air entrainment process, and droplets ejected into the atmospheric air surrounding the flow, might cause the realistic numerical modelling of such flows be somewhat difficult. A few numerical simulation of flow over stepped spillways has been investigated using various techniques such as Finite difference method, Finite Volume method, Finite Element method, Volume of Flow method (VOF), FLUENT, ......etc. this section illustrates some important research conducted in this field.
Mehdi, A. (1997) developed the SPIL-FLOW programme for a 2D finite element method algorithm to find the location of the free water surface profile over an ogee type stepped spillway. In this study the trial and error procedure based on Bernoulli's equation was used. The well-known and generally accepted U.S Army Corps of Eng. formula was applied to introduce the effects of the air entrainment on the flow characteristics. It was observed that the model can be applicable for the date set where the uniform flow condition over the model was achieved. Moreover, it was also found that the numerical results of the water surface profiles were always underestimated the experimental results by a maximum of 10%. It was thought that this could be due to the unsatisfactory application of the air entrainment equation in this study.
Chen et al (2002) investigated flow, including free surface profile, velocity profile, pressure on the steps and energy dissipation rates, over an ogee type stepped spillway experimentally and numerically. In this study VOF with unstructured grid is used to track the free surface composed of water and air. While, the k-e turbulence model was used to simulate eddies forming at the step corners by which a significant portion of the overflowing energy can be dissipated. The hydrostatic pressure distribution was modelled by using a modified resulting pressure term. The experimental and numerical results were about the same, except in tracking the water surface profile where a little discrepancy was observed especially at the end of the spillway. This was attributed to the fact that in this region the air concentration is relatively high and the free surface is fluctuated strongly leading the accurate measurements be difficult to conduct. On the other hand, the fluid is considered as a single-phase mixture and the momentum and continuity equations are solved based on this assumption. Therefore, the validity of the results is still needed to take into consideration the effects of the air entrainment on the flow.
Andre`, S. (2004) developed a quasi-2D numerical model of two phase flow over stepped spillways of moderate and steep slopes based on the finite volume approach by applying the classical depth-averaged simplified Navier-Stokes equations with the consideration of the stepped bottom . A transport equation of depth-averaged air concentration was used to model the process of air entrainment. While, Bossinesq correction coefficient was applied to achieve turbulent flow structures such as internal jets and recirculating cells for transition and skimming flow regimes over stepped spillways. Although the comparison of the numerical and experimental results were promising and encouraging and demonstrated that the model is adequately simulated the flow, the model is still needed to be improved in terms of the turbulent structures in the vertical plane and waves on the free surface.
Chatila and Tabbara (2004) and Tabbara et al (2005) studied the flow behaviour, over smooth an ogee spillway and a stepped spillway respectively, experimentally and numerically. ADINA-F software was used in both studies by applying the finite element approach. The results of the physical and numerical models, for tracking the water surface profile, of both models were in good agreement especially for the stepped spillway case. The omission of the air entrainment effects at the mid-section of the smooth ogee type spillway led some discrepancies to be noticed.
The flow characteristics, in terms of the interaction between entrained air bubbles and cavity recirculation in the skimming flow regime, velocity distribution and pressure profiles on the step surface, over stepped spillways were simulated by CHENG Xiangju et al (2006) by applying the combined mixture model of air-water two-phase flow and RNG k-e turbulence model using the finite volume CFD module of the FLUENT software with unstructured grids. The experimental results of Chen (2002) were used in this study. Promising results were achieved in this study and considered to be the fundamental tool for assessing the cavitation risk which might take place at the upstream to the inception point.
Zong Dong, Q. et al (2006) applied four different turbulence models, namely: realizable k-e model, SST k-w model, V2-f model and LES model, to simulate the flow over a steep slope stepped spillway model. The simulation included the comparison between the experimental and numerical results in terms of the growth of turbulent boundary layer, the mean flow velocity and the spanwise vorticity behaviour. It was noticed that more reasonable results can be obtained with realizable k-e turbulent model for simulation flow over stepped spillways. This was attributed to the fact that the rotation tensor, which is involved in the flow over stepped spillways, is represented by this model, but is ignored in the subgrid-scale stress in the Smagorinsky-Lilly of LES turbulence model. Indeed, other models provided inaccurate estimations. This is because higher turbulence levels were produced with SST k-w model especially in stagnation regions and regions with strong acceleration. Moreover, the flow could not be well predicted in the recirculation region near the wall with the v2-w turbulence model.
The experimental and computational results of the velocity and pressure field flow were also compared based on the realizable k-e model. The comparison shows this model can predict the flow field adequately. It should be noted that in this study the author simulated the flow characteristics in the non-aerated region where the air entrainment is not initiated. However, more investigations would be needed to include the effect of self-aeration on the flow over stepped spillways in the aerated region.
Amarin, T. et al (2009) assessed various turbulence models, Large Eddy Simulation with Smagorinsky-Lilly subgrid scale model, the non-linear turbulence model of Craft et al.(1996) and the modified non-linear turbulence model, to predict multiphase recirculating turbulent free-surface flows over the chute spillway with air entrainment. It was observed that higher accurate results can be predicted with non-linear turbulence models than with linear ones. Furthermore, it was concluded that although the results of both LES and modified non-linear turbulence models were close, the latter can be run with faster computational time up to four times than the former. Finally, for spillways wither higher slopes and step heights, it was found that higher accurate results can be predicted with modified non-linear turbulence models than LES.
Fabian, B. et al (2010) studied the flow behaviour, experimentally and numerically, in the non-aerated region of the skimming flow regime in a relatively large scale stepped spillway model. The air-water flow properties, the boundary layer development and the free surface and velocity profiles were the main aspects involved in their study. The finite volume technique of the FLOW-3D software with multi-block grids is applied. Furthermore, k-e and TruVOF methods were utilized to introduce the effects of the turbulence and air entrainment in the flow over stepped spillways respectively. The numerical results demonstrated a very good satisfactory to the corresponding experimental results. However, some differences were observed with respect to the boundary layer growth with the experimental data and expressions presented by Chanson (2006) and Amador A. (2005), especially near the spillway crest.
3-3 Particle-based methods for simulating flow over stepped spillways:
It is common to solve fluid dynamic problems, especially free surface flows, by using grid-based methods as described in the previous section. Hydraulic engineers often encounter problems from these methods concerning to the lack of creating interactive fluid simulations together with detailed flow surfaces. Nowadays, the equations of motion of interactive fluids are solved more effectively by using various techniques of particle-based methods known as Smoothed Particle Hydrodynamics (SPH). This method is characterized with a meshfree and 3D Lagrangian approach.
Basically, in (SPH) the physical properties of a fluid element can be represented by smoothed particles. The physical quantities of the fluid elements are averaged over an extended volume using Kernel function to create the smoothed particles (Cha, S.-H. et al 2003).
Originally, this method was presented by Lucy (1977) and Gingold and Monaghan (1977) to solve the tree dimensional astrophysical problems. It has been developed extensively to suit with other physical applications, for example free surface flows. Free surface flow and flow over hydraulic structures, such as weirs; have been simulated by this method, but up to now this method has not been applied to simulate the flow properties over stepped spillways.
Initially, Monaghan (1994) studied the free surface flow by discretizing the continuum into a finite set of particles distributed arbitrarily. Each particle carries the physical properties, including density, pressure and velocity, of the fluid and moves along the streamline in the space and at a certain time. Problems involving large deformations can be successfully simulated by this without any mesh refinement like other classical grid-based methods.
Ferrari, A. (2010) developed a new (SPH) scheme to simulate the overtopping flow over a sharp-crested weir. The mathematical model of this study was based on the weakly compressible Navier-Stokes equations with the Tait equation of state. The model was solved by using the 3D parallel SPH scheme, to accurately achieve both tracking the free surface profile and computing the pressure field, proposed by the same author (2009) for free surface flows. The numerical results were compared with the experimental data of Scimemi (1930) in which a very good quantitative agreement was obtained.
4- Problem Statement:
In view of above, it can be noticed that the hydraulics of stepped spillways with upward inclined steps has not been understood well, particularly with moderate and flat slopes, regarding to upstream reach of the inception point where the cavitation might be taken place, the growth of boundary layer and the flow resistance of such steps to estimate the energy dissipation/ residual energy accurately.
In the present study the characteristics of skimming flow on stepped spillways of moderate slopes, typical to embankment dams, with upward inclined steps will be studied experimentally and numerically.
5- The aim of the present study:
The objective of this research is to:
Conduct an experimental investigation on stepped spillways of moderate slopes typical to embankment dams with upward inclined steps for a wide range of flow rate regarding to skimming flow regime.
Examine the flow properties upstream of the inception point which is prone to the potential of cavitation risk and the development of boundary layer.
Investigate the flow resistance of upward inclined steps.
Examine the performance of such steps on the energy dissipation.
To compare the results of this investigation with others utilizing other step geometries.
Develop a numerical model using smoothed particle hydrodynamics (SPH) as a new technique for simulating the flow over stepped spillway.