HYDRAULIC DESIGN OF STEPPED SPILLWAYS

Hans-Erwin Minor 1 and Robert M. Boes 2

1 Director, Professor Dr.-Ing., 2 Research Engineer, Dr. sc. techn.

Laboratory of Hydraulics, Hydrology and Glaciology (VAW)

Swiss Federal Institute of Technology (ETH), CH-8092 Zurich, Switzerland

Tel. +41 1 632 40 90, Fax: +41 1 632 11 92, E-mail: minor@vaw.baug.ethz.ch

Abstract: The hydraulic design of stepped spillways follows the planning sequence of con­ven­tional chutes: Flood analysis, selection of design flood and safety check flood, selection of weir type and width, design of weir crest, definition of the stage-discharge curve, calculation of reservoir retention.

Additionally, for stepped spillways the selection of step height and the analysis of flow regime and air entrainment are necessary. The design of the sidewalls has to take into account the bulging of the air-water mixture, and the energy dissipation profits from higher energy losses along the steps compared to smooth chutes. The latter is however smaller than often assumed. Higher steps have a positive effect on the hydraulic performance of the spillway.

Keywords: Stepped spillway, energy dissipation, RCC dam, air-water mixture, air concentration

1    INTRODUCTION

Stepped spillway chutes can be economically integrated into the downstream face of gravity dams, especially if roller compacted concrete (RCC) is used for the con­struction. Another common application is the use of stepped overlays on embankment dams as emergency spillways. In both cases, a careful hydraulic and structural design of the complete spillway, including the energy dissipator, is necessary to ensure safe operation over the whole lifetime of the dam structure.

Based on hydraulic model investigations at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW) of ETH Zurich, Switzerland (Boes 2000a, Schläpfer 2000) and the results of the International Workshop on the Hydraulics of Stepped Spillways carried out at VAW in March 2000 (Minor & Hager 2000), as well as practical experience with design and construction of spillways, a design procedure for stepped spillways is presented and guidelines for the hydraulic design are formulated.

2    PLANNING SEQUENCE

The planning sequence of stepped spillways follows in general the one for conventional spillways. The subsequent text follows this sequence.

Flood analysis to find the peak flows and volumes for floods of various recurrence intervals and the selection of the design flood and the safety check flood, as well as the decision whether the weir structure should be gated, are common for all spillway types (Minor 1998).

3    SELECTION OF WEIR WIDTH

The choice of the weir width is made considering the crest length of the dam, the width of the riverbed downstream, and a possible reduction of effective weir width because of piers or effects at the abutments and the entrance at the sidewalls. A very important value for all spillways is the allowable rise of the water level in the reservoir above normal operation level caused by the considered flood event. A larger rise results in a more pronounced dampening of the peak discharge. A higher head on the crest on the other hand increases the specific discharge q. Since for stepped spillways q is usually limited to 25-30 m³/s×m (Minor 2000), this condition leads to fairly wide crests. The reason for this limitation is the fear of cavitation damage. It can be shown (Boes & Minor 2000) that cavitation risk can be ruled out for much higher specific discharges than the limit given above once the flow is aerated.

For uniform flow the bottom air concentration is high enough to avoid cavitation for specific discharges up to almost q_max = 140 m³/s×m for steep chutes and high steps ( Table 1 ). But also for smaller step heights and flatter slopes the allowable values are fairly high. These values are on the conservative side for design purposes, because the aeration tends to be more pronounced in the prototype than suggested by model results due to a higher de­gree of turbulence (Boes 2000a and b).

4    DESIGN OF WEIR CREST

Most spillways are designed using the crest shape given in chart 111-1 of the Hydraulic Design Criteria (COE/WES 2000). Mateos Iguácel & Elviro García (2000) propose a transition between the crest and the stepped chute in which the steps grow continuously from a relatively small height to the constant step height in the chute.

At fill dams the crest is much wider and therefore on top of this wide crest often an additional wall is installed, creating an overfall. The flow is then already aerated there if the air supply at the sides is sufficient and the crest is not too long.

As stated above, the specific discharge may be chosen much higher if it is ensured that the flow is aerated already at the top of the spillway. This can be achieved if a conventional weir shape is chosen (smooth concrete surface) and an aerator is introduced near the point of tangency. Another possibility is a flap gate on top of the crest. The advantage of such a solution would be that the flow is aerated and the rise of the water level in the reservoir can be limited.

5    VERIFICATION OF MAXIMUM OUTFLOW

Once the crest design is defined, the corresponding key curve can be computed. In case of gated structures, the operation rules have to be taken into account. The key curve, inflow hydrogramme and reservoir volume curve are needed to calculate the outflow hydrogramme giving the maximum discharge. The resulting specific discharge q has to be compared with the assumption made above.

6    SELECTION OF STEP HEIGHT

The height of the steps has to be chosen taking into account the construction procedure. Very often RCC dams are constructed in layers of 30 cm and formwork-heights of 60 cm to 1.20 m. From that standpoint a step height between 30 cm and 1.20 m would be convenient. Results of model tests show (Boes & Minor 2000) that higher steps seem to have an advantage over smaller steps considering hydraulics.

Regarding the cavitation risk high steps allow a markedly larger specific discharge than small steps (Table 1 ). As shown in Fig. 1 , the uniform bottom air concentration, which governs the safety against cavitation damage, is considerably higher for larger slopes. Furthermore, higher steps are also preferable since more energy is dissipated than with smaller steps (Boes & Minor 2000, Boes 2000c). Steps smaller than 60 cm are not recommended.

7    FLOW REGIME AND AIR ENTRAINMENT

Two distinct flow regimes are found on stepped spillways. Whereas in nappe flow the steps act as a series of overfalls with the water plunging from one step to another, the water flows as a coherent stream over the pseudo-bottom formed by the step corners in skimming flow. Generally speaking, nappe flow is found for low discharges and large steps. For small steps or larger discharges such as the design discharge, the water skims over the step corners. The transition from nappe to skimming flow can be expressed by the ratio of critical flow depth h_c and step height s. Ac­cording to Boes (2000a), skimming flow sets in for ratios larger than

                          (1)

This is in approximate agreement with results of other authors (Boes 2000c) and is applicable for chute inclination angles of approximately 26° <f < 55°.

Nappe flow is generally well aerated, whereas in skimming flow the turbulent boundary layer has to reach the free surface before air may be entrained from the atmosphere. This occurs at the inception point. According to Boes (2000c) the unaerated length L_i from the spillway crest to the inception point can be described by

L_i = 4.93    for 26° < f < 55°         (2)

Whereas the step height s has limited influence on the length of the unaerated zone, the specific discharge dominates this value.

Chamani (2000) reports that at the inception point, the suddenly increased turbulence at the water surface leads to a "rooster tail", which results in a markedly higher flow depth over some length. Boes (2000a) did not find such a pronounced flow structure in his tests. The mixture flow depth showed a local maximum only for higher steps.

The entrained air, together with the energy dissipation on the chute, lead to a bulging of the flow, which has to be taken into account in designing the side walls of the chute. The characteristic mixture flow depth h₉₀ with a surface air concentration of 90% serves as a guide for the design in the aerated or white water region. Starting from the inception point of air entrainment, the air-water mixture is de­scribed by a supercritical backwater curve. In non-dimensional form, the mixture flow depth can be expressed by (Hager & Boes 2000)

   for 26° < f < 55°     (3)

where L_i is calculated from eq.         (2) , x is the streamwise coordinate with its origin at the spillway crest and h_90,u denotes the uniform mixture flow depth that is function of the roughness Froude number Fr* = q/(g sinf s³)^1/2. The uniform mixture flow depth is found to be

             for 26° < f < 55°                (4)

The exponent p = 0.54 for f = 30° and 40°, and p = 0.59 for f = 50° (Boes 2000c). The volumetric portion of the water phase is almost negligi­ble above h₉₀, but the developing spray can lead to fog or ice formation in winter.

The mixture depth h₉₅ is about 12% larger than h₉₀, whereas h₉₉/h₉₀ ≈ 1.4 (Boes 2000a). It is recommended to consider the erosion potential in the non-overflow section of a dam when de­signing training walls. The proposed design height h_d for sidewalls reads

h_d = h·h₉₀,                              (5)

with a safety factor h = 1.2 for concrete dams with no con­cern of erosion on the downstream face and h = 1.5 in case of emergency spillways on embankment dams prone to erosion. The safety factors take into account the increase of the spray height in the prototype due to a higher turbulence degree, as compared to the model results (Boes 2000 a).

8    ENERGY DISSIPATION AT THE DAM TOE

One of the advantages of stepped spillways put forward by many authors is the higher energy dissipation along the chute compared with conventional, smooth chutes. The energy dissipation is different if the flow is aerated or not.

If the chute length is superior to the length L_u measured from the crest to the location where uniform flow is attained (Boes 2000c)

L_u ≈ (57-13/sinf) h_c              for 26° < f < 55°                                     (6)

then the rela­tive residual energy head H_res/H_max at the toe of an ungated stepped spillway can be cal­culated from (Chanson 1994)

                  (7)

where H_max denotes the reservoir head, f_w,u is the Darcy-Weisbach friction factor for uniform equivalent clear water flow and a  ≈ 1.21 (Boes & Minor 2000) is the kinetic energy correction coefficient. Conservative friction factors of f_w,u = 0.09 for f = 30° and f_w,u = 0.06 for 40° and 50° are proposed (Boes 2000c). These values are smaller than previously suggested by most authors, so that the energy dissipation of aerated skimming flow on stepped spillways is smaller than given in many publications. Fig. 2 shows the relative residual energy head as a function of the relative dam height for different spillway slopes.

With the knowledge of the residual energy head Hmax at the toe of the dam, the energy dissipator may be designed. In most cases this is a conventional stilling basin. For the dimensioning of this structure, the knowledge of the downstream water level is needed. It is very often not easy to find the stage-discharge curve for a natural stream, but it is important to have a reliable function. The calculation of backwater curves starting all from a point where the stage-discharge curve can easily be defined may become necessary. This of course has to be done for various discharges covering the total range of the considered discharges.

The stilling basin, too, has to be checked for the whole range of discharges. Depending on the tailwater, the maximum discharge might not be the most critical case.

9    STRUCTURAL DESIGN INCLUDING FINAL DEFINITION OF SHAPES

Once the basic dimensions and shapes of the spillway are given, the detailed structural design has to follow. The construction and contraction joints have to be at the right position. Decisions have to be taken about reinforcement. RCC dams are unreinforced and an application of reinforcement at the stepped chute would complicate construction. Sometimes reinforced precast elements are used as lost formwork. Experience with existing stepped spillways show that the steps withstand the stresses due to overflow corresponding to the currently applied specific discharge. Further investigations are needed to confirm that this is also true for higher discharges and higher steps.

A critical point is always the edge of the steps. Very often this edge breaks, giving an unpleasant appearance and may initiate further damage. It is therefore proposed to phase off the edges over 2 to 3 cm.

Shapes of the upstream part of the sidewalls and piers have to be chosen so that the reduction of the inflow capacity is minimized.

10    CONCLUSIONS

Stepped spillways are a very convenient means to pass floods safely over RCC as well as fill dams. The maximum specific discharge applied today, i.e. 25-30 m3/s×m, may be increased if aeration of the chute is assured from the top. Larger steps have a positive effect on the hydraulic performance.

The energy dissipation along the chute is exaggerated in many publications. It is recommended to use the values given above (see also Boes 2000a). For the design of the stilling basin, in addition to the calculation of the residual energy head at the toe of the dam, the tailwater rating curve is of utmost importance. The maximum discharge might not be the most critical case for the energy dissipation.

Much care must be taken when defining the right shapes of all contours that are in contact with water, since these may affect the capacity and the performance.

References

Boes, R.M. (2000a) Zweiphasenströmung und Energieumsetzung auf Grosskaskaden (Two-phase flow and energy dissipation on cascades). Doctoral Dissertation N^o 13510. ETH Zurich (in German).

Boes, R.M. (2000b). Scale effects in modelling two-phase stepped spillway flow. Proc. Intl. Workshop on Hydr. of Stepped Spillways, VAW, ETH Zu­rich (H.-E. Minor & W.H. Hager, eds.). Balkema, Rotterdam: 53-60.

Boes (2000c). Two-phase flow and energy dissipation on stepped spillways. Proc. ASDSO Annual Conference on Dam Safety, Providence, USA: CD-ROM.

Boes, R.M & Minor, H.-E. (2000). Guidelines for the hydraulic design of stepped spillways. Proc. Intl. Workshop on Hydr. of Stepped Spillways, VAW, ETH Zurich (H.-E. Minor & W.H. Hager, eds.). Balkema, Rotterdam: 163-170.

Chamani, M.R. (2000). Air inception in skimming flow regime over stepped spillways. Proc. Intl. Workshop on Hydr. of Stepped Spillways, VAW, ETH Zu­rich (H.-E. Minor & W.H. Hager, eds.). Balkema, Rotterdam: 61-67.

Chanson, H. (1994). Comparison of energy dissipation between nappe and skimming flow regimes on stepped chutes. Jl. of Hydr. Res. 32(2): 213-218.

COE/WES (2000). Hydraulic Design Criteria. Corps of Engineers / Waterways Exper. Station. http://chl.wes.army.mil/library/publications/hydraulic_design_criteria/

Hager, W.H. & Boes, R.M. (2000). Backwater and drawdown curves in stepped spillway flows. Proc. Workshop Hydr. of Stepped Spillways, VAW, ETH Zu­rich (H.-E. Minor & W.H. Hager, eds.). Balkema, Rotterdam: 129-136.

Mateos Iguácel, C. & Elviro García, V. (2000). Stepped spillway studies at CEDEX. Proc. Intl. Workshop on Hydr. of Stepped Spillways, VAW, ETH Zurich (H.-E. Minor & W.H. Hager, eds.). Balkema, Rotterdam: 87-94.

Minor, H.-E. (1998). Report of the European R&D Working Group “Floods”. Proc. Intl. Symp. on New Trends and Guidelines on Dam Safety, Barcelona (L. Berga, ed.). Balkema, Rotterdam: 1541-1550.

Minor, H.-E. (2000). Spillways for high velocities. Proc. Intl. Workshop on Hydr. of Stepped Spillways, VAW, ETH Zurich (H.-E. Minor & W.H. Hager, eds.). Balkema, Rotterdam: 3-10.

Minor, H.-E. & Hager, W.H., eds. (2000). Proc. Intl. Workshop on Hydr. of Stepped Spillways, VAW, ETH Zurich. Balkema, Rotterdam.

Schläpfer, D. (2000). Treppenschussrinnen (Stepped Spillways). Diploma Thesis. VAW, ETH Zurich (in German).

      Table 1    Maximum specific discharge q_max in m³/s×m that can be spilled without risk of  cavitation damage for uniform flow as a function of spillway slope and step height.

Fig. 1    Bottom air concentration in uniform flow C_b,u as a function of roughness Froude number Fr* = q/(g sin f s³)^1/2 for different spillway slopes.

Fig. 2    Relative residual energy H_res/H_max as a function of relative dam height H_dam/h_c for different spillway slopes.