Numerical Modeling of a Stepped Spillway 523 DESIGN

DESIGN APPROACHES AND NUMERICAL MODELING OF A STEPPED SPILLWAY UNDER HIGH TAILWATER CONDITIONS Laura Shearin-Feimster P.E.1, Brian M. Crookston, Ph.D., P.E.2, Stefan Felder, Ph.D3 ABSTRACT Rehabilitation of existing embankment dams often includes addressing deficiencies in the spillways and outlet works. One common cost-effective solution is to armor the embankment using RCC, forming a stepped chute that increases spillway capacity while also offering energy dissipation. With the wealth of knowledge on the subject, it can be challenging to identify suitable methodologies for a specific rehabilitation project. Additional challenges arise when the design methodologies and research do not specifically account for site conditions and constraints that impact a specific spillway’s hydraulics. Schnabel recently completed an embankment dam rehabilitation project with a stepped chute subject to backwater effects. The design was based upon previous experimental studies identifying design parameters for embankment dam stepped spillways including flow depth, inception point of air entrainment position, and residual energy at the downstream end of the stepped chute. Moreover, CFD modeling of this specific stepped spillway geometry and hydraulic conditions was performed and the results were compared to the predicted results from published design methodologies. This paper presents an overview of a unique rehabilitation project with high tailwater, a review of appropriate design methodologies, and a discussion regarding the applicability and limitations of the current design parameters to the described project, including tailwater effects. This paper also discusses the current challenges of modeling numerically stepped spillways. These insights should aid practitioners in the design of stepped chutes for embankment dams. INTRODUCTION The most common type of dam used today is the embankment dam. Insufficient spillway capacity, upstream and downstream development, and aging continue to justify rehabilitation and upgrading projects. One common cost-effective solution is to armor the embankment using RCC or conventional concrete, forming a secondary stepped spillway that increases spillway capacity while also offering energy dissipation (see 1

Senior Engineer, Schnabel Engineering, 11A Oak Branch Drive, Greensboro, NC, USA 27407. Phone: (336) 274-9456, Email: [email protected] 2 Project Engineer, Schnabel Engineering, 1380 Wilmington Pike, Suite 100, West Chester, PA, USA 19382. Phone: (610) 696-6066, Email: [email protected] 3 Lecturer, University of New South Wales, Water Research Laboratory, School of Civil and Environmental Engineering, Sydney NSW 2052 Australia. Phone: (+612) 9385-5562, Email: [email protected]

Numerical Modeling of a Stepped Spillway

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Figure 1), which is one advantage of stepped chutes. It is not uncommon to cover a stepped chute with topsoil and turf, where these surface treatments can be inexpensively replaced as required (below right).

Figure 1. Examples of embankment dams with stepped chutes (courtesy Schnabel) The use of stepped chutes to rehabilitate embankment dams has encouraged detailed studies regarding the hydraulics of moderate-sloped stepped chutes during the past 12 years. Detailed research has been conducted throughout the world, including hydraulic laboratories located in Australia, Switzerland, Portugal, Germany, Japan, Italy, and the USA. A selection of published literature specific to moderate-sloped stepped spillways and the effects of steps on energy dissipation includes Peyras et al. (1992), Chanson (2002), Chanson and Toombes (2002), Ohtsu et al. (2004), Gonzalez and Chanson (2007), Felder and Chanson (2009, 2011, 2013), Meireles and Matos (2009), Hunt and Kadavy (2010), Bung (2011), Frizell and Svoboda (2012) and Hunt et al. (2014), For example, the multi-year research conducted at the USDA-ARS hydraulic laboratory was of stepped chutes at or near prototype scale (Hunt and Kadavy 2014). The general flow characteristics of a moderate-sloped stepped chute are presented in Figure 2. As shown there is a non-aerated flow region and an aerated flow region, which begins downstream of the inception point, defined as the location where the boundary layer reaches the water surface; the length from the downstream edge of the crest to this location is Li. Q is discharge, θ is the spillway slope, Ho is the total energy head at the crest to the point of interest, V 2/2g is the velocity head component where V is the mean velocity and g is the acceleration constant of gravity. The approach depth is yo. The step size is defined by the step height, h, and the step length, l. The equivalent clear water depth is ycw. Regarding energy dissipation, H is residual energy and ΔH is total energy lost at a distance L from the downstream edge of the crest.

Figure 2. Moderate-sloped stepped chute hydraulic overview 524

USSD 2015 Annual Conference

Despite the wealth of research, there is still lack of consensus in the research community regarding several key aspects of stepped chute hydraulics, including the occurrence of uniform flow conditions, the occurrence of the state of equilibrium, and roughness height. Also, it can be challenging to identify and compare suitable design methodologies for a specific project that may include unique geometric or hydraulic components, including the inclusion of various weir types upstream of the stepped chute. Furthermore, the application of computational fluid dynamics (CFD) appears to produce favorable results in the clear-water (non-aerated) region, but significant challenges have been encountered when modeling the two-phase flow region. Schnabel has encountered numerous sites where a stepped spillway has been the preferred option for rehabilitation. Many of these sites are estimated to have a high tailwater condition for large storm events. This paper presents an overview of a unique rehabilitation project with high tailwater, a review of appropriate design methodologies, and a discussion regarding the applicability and limitations of the current design parameters to the current project, including backwater effects. This paper also discusses the current challenges of modeling numerically stepped spillways. These insights should aid practitioners in the design of stepped chutes for embankment dams. BASIS OF CASE STUDY Fox Creek Multi-Purpose Structure Number 4 (Fox Creek 4) was used as a basis for examining the various methods noted herein. This particular site was selected due to its availability of site data and presence of high tailwater during flood flow conditions. Background Fox Creek 4 is located near Fleming, Kentucky, USA. It was originally designed by NRCS for the purpose of flood control and recreation. The dam was originally constructed in 1968 as a significant hazard structure. Later, the dam was considered to be high hazard based on downstream structures. In 2010, Schnabel was contracted to design the selected upgrade of a stepped auxiliary spillway over the embankment to meet the NRCS requirements for high hazard dams. Construction was completed in 2012; two photographs of the constructed spillway are presented in Figure 3.

Figure 3. Fox Creek 4 Spillway (photos courtesy Schnabel Engineering)


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Site Data For the purposes of this paper, Fox Creek 4’s dimensions and some site characteristics were used to compare the cited published methods with the CFD model. This paper does not intend to analyze Fox Creek 4, but rather to form a basis of comparison of the various cited methods. The scenarios analyzed do not necessarily match site conditions used for design. The RCC stepped spillway at Fox Creek 4 features an ogee crested weir upstream of the steps. The stepped chute is comprised of four 4-ft high steps, with the upper-most step at 3 ft and the final step at 5 ft. In addition, it is a 15° converging chute, with the top width of the spillway of 295 ft and a bottom-width of 263 ft. The chute length along the slope (L) is 64.4 ft. Additional pertinent project information is summarized in Table 1. Table 1. Project Overview Parameter Value Top of Dam Elevation 761.8 ft Spillway Crest Elevation 757.5 ft Spillway Width, Top 295 ft Spillway Width, Bottom 263 ft Spillway Slope 2.5H:1V (22°) Basin Apron Elevation 728.0 ft, 722 ft Two flow conditions were analyzed for energy dissipation for this study; the spillway design flood (SDF) that is the probable maximum flood (PMF) and about 2/3 of the PMF. When the spillway is passing the PMF and operating at full capacity, the peak flow rate was estimated to be 37,040 cfs (headwater elevation 767.3ft, tailwater elevation 748.4 ft). During the 2/3 PMF, the flow was estimated 24,705 cfs (headwater elevation 765 ft, tailwater elevation 744.3 ft). STEPPED CHUTE RESIDUAL ENERGY PREDICTIVE METHODS Four stepped chute energy dissipation predictive methods, summarized in Table 2, were selected for this study. Meireles and Matos (2009); Hunt and Kadavy (2010), Hunt et al. (2014); Gonzalez and Chanson (2007); and Felder and Chanson (2011, 2013, 2014) were used to calculate the energy loss along the stepped spillway. These methods base the energy dissipation capacity of the steps on air-water flow data. Note that the stepped chute for Fox Creek 4 is estimated to be too short for inception to occur for the 2/3PMF and PMF flow conditions. The clear-water flow regime, with a hydraulic jump on the steps followed by high tailwater in the stilling basin, is anticipated.

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Table 2. Stepped Chute Energy Dissipation Predictive Methods Flow Reference Slope dc/h Regime Meireles and Matos (2009) 26.6° Skimming 1.27 to 2.85 Hunt and Kadavy (2010), Hunt et al. 10° to 30° Skimming 0.035 to 2.27 (2014) Gonzalez and Chanson (2007) 10° to 25° Skimming 1.0 to 3.2 Felder and Chanson (2011, 2013, Transition & 3.4° to 26.6° 0.61 to 4 † 2014) Skimming † energy prediction method based upon air-water flow data

Meireles and Matos Meireles and Matos (2009) focused on using the inception point to evaluate energy dissipation. This experimental setup used a 2H:1V slope, which is similar to our case. This setup focused on the non-aerated flow regime, or the area upstream of the inception point. The researchers found the energy dissipation to be less than what had been found by others for uniform or fully developed flow conditions. To calculate the location of the inception point, Meireles and Matos used a relationship developed by Chanson based on the roughness Froude number, F* defined as F* =

q

(g sin (θ )k )

3 12

(1)

s

where ks is the roughness height perpendicular to the pseudo-bottom, defined as k s = h cos(θ )

(2)

The location of the inception point, Li, and the depth of flow at this location, di, are calculated using Li = 5.25k s F *0.95

(3)

d i = 0.28k s F *0.68

(4)

where di is the flow depth at the inception point. The relationship between the clearwater depth d and the flow depth at the inception point was proposed as d = 0.971 + 0.891e −3.41( L Li ) di

(5)

The resulting relation developed by Meireles and Matos to account for energy head loss along the stepped chute is presented as Eq. 6. As with the clear-water depth, head loss was found to be dependent on the normalized length, L/Li.


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2.5   L   ΔH  = 1 − exp − 0.3  (non-aerated flow region) (6)  Ho Li      where ΔH = Ho – H, the upstream total head at the crest of the stepped chute Ho= Hdam +1.5dc with dc the critical flow depth and the residual energy at any point along the nonaerated spillway: V 2   H o = y cw cos(θ ) + α  (7) g 2   In Eq. 7, ycw is the equivalent clear water flow depth, α is the energy coefficient, V is mean velocity computed as V = q y cw .

Hunt and Kadavy Hunt and Kadavy (2010) and Hunt et al. (2014) provide a very similar design approach design procedure that was developed for flat embankment slopes in the non-aerated flow region (14° ≤ θ ≤ 26.6°). A portion of their method utilizes the computed inception point location to estimate energy dissipation. Hunt and Kadavy present multiple ways to calculate the location of the inception point, Li, depending on specific site characteristics. For our case study, the following relation was utilized.

Li 0.89 = 5.19(F *) ks

(8)

In our case study, the chute is not long enough for the flow to reach the inception point for the 2/3 PMF or PMF design points. However, Hunt and Kadavy (2010) found a linear trend for energy dissipation upstream of the inception point (clear water flow region): L ΔH = 0.30  (non-aerated flow region) (9) Ho  Li  Hunt et al. (2014) also utilize Eq. 7 to compute energy dissipation (see for additional information). Gonzalez and Chanson Gonzalez and Chanson (2007) discusses a design procedure based on an experimental setup tailored toward stepped chutes over embankment dams. The paper presents separate design procedures depending on whether aerated flow conditions are reached above the toe of the dam. The procedure is valid for slopes ranging from 10° to 25° and a critical depth to step height ratio (dc/h) from 1.0 to 3.2. Due to the case study site conditions, the procedure for small embankment spillways with non-aerated flow conditions was used. The ideal velocity for this condition is estimated using the Bernoulli equation Vmax = 2 g (H o − d cos(θ ))

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(non-aerated flow region) (10)


where d is the depth of flow at the base of the spillway. This equation is combined with the following relation d=

q Vmax

+

δ


N +1

where the velocity distribution exponent, N, is estimated to be approximately 5 for stepped chutes. The boundary layer in this relation, δ, is estimated using the following:  L    = 0.0301 0.11  L sin (θ )  h cos(θ ) 

δ

1

−0.17


From the unit flow rate and calculated depth at the spillway exit, the residual energy head can be calculated with Eq (7). Felder and Chanson Often stepped spillways are aerated when the chute is long enough to reach the inception point of air entrainment. In such a case, the design must be based upon air-water flow estimates of the residual energy of the spillway toe to accurately predict the energy to be dissipated in the stilling structure. Felder and Chanson (2014) developed a simple design guideline for slopes ranging from 8.9° and 26.6° based upon all available aerated flow data comprising both aerated transition and skimming flows. They identified characteristic slope groups for which the energy dissipation performance was similar and linked the residual energy to the flow rate via the critical flow depth at the upstream end. A embankment dam chute slope of 21.8° (2.5H:1V) is suggested to be the most efficient system for energy dissipation. The design guideline is valid for the aerated flow region downstream of the inception point and not for the non-aerated flow region. Based upon air-water flow data, the residual energy at the downstream end can be calculated using: Y90

H=

 (1 − C )cos(θ )dy +

qw

2

(aerated flow region) (13) 2 Y90   0 2 g   (1 − C )dy   0  where C is the void fraction and Y90 is the flow depth where the void fraction is 90%. [Note that the air-water flow Eq. (13) is equivalent to Eq. (7)]. For our case study, the recommended design guidance is a ratio of residual energy head to critical depth (H/dc) of 3.37. Because the spillway in our case study is not of sufficient length for the occurrence of an air-water flow region downstream of the inception point for the flow rates of interest (PMF, 2/3 PMF), this method may not be directly applicable as far as estimating head losses for the design discharges. However, Felder and Chanson (2011) investigated stepped spillways with varying step heights, as is the case with this case study. Overall, their conclusions were that the dissipation was very similar when comparing uniform and non-uniform steps heights. It should be emphasized that, discharges less than 2/3 PMF (not included herein) on the present stepped spillway would


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be aerated resulting in nappe flow regime and also in transition flows. Transition flows can cause instable flows in particular in step geometries with non-uniform or unconventional step designs. Method Comparison

Table 3 presents the computed results from the stepped chute methodologies discussed previously, as applied to Fox Creek 4. The calculations were performed separately for the main section of the spillway (the high section) and the section lowered to handle the principal spillway outflow (the low section). For the 2/3 PMF, the estimated energy head at the top of the slope is 36 ft for the high section and 42 feet for the low section. For the full spillway capacity, the estimated energy head at the top of the slope is 38.7 ft for the high section and 44.7 ft for the low section. Table 3. Summary of calculated residual energy head at the base of the spillway Residual Energy Head† (ft) 2/3 PMF PMF Method† High Section Low Section High Section Low Section Meireles and Matos 24.6 24.6 33.3 35.2 Hunt et al. 24.8 24.8 30.0 32.4 Gonzalez and 29.3 29.3 33.6 37.3 Chanson Felder and Chanson 21.0 (aerated flows) 27.6 (aerated flows) †does not include tailwater effects NUMERICAL MODELING

Although the aforementioned stepped chute methods were identified to be most applicable to Fox Creek 4, no method was fully applicable to Fox Creek 4 (clear water flow regime, varying step height, ogee weir upstream of chute, high tailwater, converging chute). Therefore, numerical modeling of Fox Creek 4 was performed to • •

Evaluate spillway hydraulics for this specific geometry and site conditions Compare the numerical results to the previously cited stepped chute methods

Model Characteristics

The commercially available CFD solver, FLOW-3D (by Flow Science) was used to develop numerical models of Fox Creek 4. FLOW-3D solves the Reynolds-average Navier-Stokes equations (RANS) with modified algorithms to model flow past solid objects (e.g., weirs and steps), model the volume fraction of fluid in each discretized cell (VOF model), track the free surface, and model air void fraction in the flow. The RANS equations, with modified algorithms for continuity and momentum, are presented as Eqs. 14 and 15, respectively.

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∂ (uAx ) + ∂ (vAy ) + ∂ (wAz ) = 0 ∂x ∂y ∂z

(14)

∂U i 1  1 ∂P ∂u  + + gi + fi U j Aj i  = −   ∂t VF  ∂x j  P ∂xi

(15)

Numerical modeling conducted for this study utilized the Renormalized Group (RNG) turbulence model. The Fractional Area/Volume Obstacle Representation (FAVOR) algorithm was used to embed the spillway and adjacent regions (drafted in Civil3D) in the computational mesh. Because air entrainment due to surface turbulence and a hydraulic jump is anticipated, simulations were performed with and without the Air Entrainment, Density Evaluation, and Drift-flux models. These three models are a parameterized resolution that simulates the volumetric fraction of entrained air in the numerical flow. An entrainment rate coefficient of 0.5 and surface tension coefficient of 0.005 were selected. Also, a drag coefficient of 0.5, average bubble radius of 1 mm, and the Richardson-Zaki coefficient of 1 were selected. Although a spatially-dependent bubble size distribution in the stepped chute would be expected, several bubble diameters were tested and the selected bubble diameter appears to be reasonable in this instance. Flow domain boundary conditions were specified (pressures, symmetry, walls, etc.) based upon hydrologic routings and a tailwater rating curve estimated using the USACE 1dimensional HEC-RAS program. No embedded mesh blocks were incorporated to minimized numerical interpolation and truncation at mesh block boundaries. The cross-sectional models were 1 cell wide perpendicular to the flow. Sectional models were more computationally expensive at 64 ft in width. Both models also included flux surfaces and history probes to record numerical results at locations of interest within the models (see Fig. 4). Simulation times were based upon the flow field and achieving solution convergence under steady-state conditions. Each simulation included multiple iterations to analyze runtime diagnostics and time-history plots. Final simulation characteristics are summarized in Table 4. By way of example, an isometric view of the section model (simulations 5 and 8) is presented in Figure 4.


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Table 4. Numerical Simulations Test Matrix Total Cell Simulation Computation Count Time Time (ft) () (s) (hr:min:s) 1 X 1 11,562 120 00:06:32 2 X 0.5 45,120 120 00:37:42 2/3 PMF 3 X 0.25 180,480 120 07:38:50 4 X 0.5 45,120 120 00:50:57 5 S 0.5 5,775,360 40 16:23:31 6 X 0.5 45,120 120 00:42:22 PMF 7 X 0.5 45,120 120 01:06:28 8 S 0.5 5,919,744 40 20:51:16 †X = cross-section, S = section, g = gravity, ν = viscosity, T = turbulence closure Entrainment, DF = Drift Flux, DE = Density Evaluation Simulation

Event

Mesh Size

Physics† g,ν,T g,ν, T g,ν,T g,ν,T,AE,DE,DF g,ν,T,AE,DE,DF g,ν,T g,ν,T,AE,DE,DF g,ν,T,AE,DE,DF method, AE = Air

As a reference for recorded computation times reported in Table 4, simulations were performed using a workstation with 32 cores @ 3.4 GHz (Xeon® processors) and 64 GB of RAM. As shown in Table 4 and Figure 5, three cell sizes for the meshed domain were simulated to refine the grid and investigate mesh convergence and solution independence.

Figure 4. Sectional model overview

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Figure 5. Cross-section models with mesh size at 1.0 ft (A), 0.5 ft (B), and 0.25 ft (C) Due to the non-conforming hexahedral mesh in FLOW-3D, the solution is dependent upon discretization of the domain, specifically the selected mesh size. For the purposes of this study, and in view of the level of uncertainty in the hydrologic analysis used to develop the spillway design flood, a detailed mesh refinement study and quantification of numerical uncertainties were not performed. However, computed discharge at four locations during the simulation time (120 sec) for simulations 1, 2, and 3 are presented in Figure 6. Note that flux surface E1a was located at the endsill of the stilling basin and flux surface E1 downstream, locations in close proximity to the hydraulic jump.

Figure 6. Discharge results for grids with 1 ft, 0.5 ft, and 0.25 ft cell size. The percent difference in computed discharges, relative to the finest cell size of 0.25 ft is presented in Figure 7. As shown, discharges measured upstream of the stepped chute were approximately 7% greater and 3% greater for the 1 ft and 0.5 ft cell sizes. In


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addition, each step was represented with 8 cells in the vertical direction for the 0.5 ft cell size, which was viewed to be an adequate mesh resolution for this study.

Figure 7. Percent difference relative to 0.25 ft cell size Numerical Results

No field data or physical model experimental results exist for calibration of the CFD models. Therefore, the results have not been verified. Computed unit discharges are presented in Table 5. Also included is the ratio of the numerically computed qCFD versus the original q estimated during the hydrologic analysis. Good agreement exists between the empirical estimate and the CFD results (within 3% to 5% for the finer meshes). Also, the sectional model results agreed well with the cross-section model results (3.5% and