INCEPTION POINT CHARACTERISTICS OF STEPPED SPILLWAYS

Robert M. Boes1 and Hans-Erwin Minor2

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

Laboratory of Hydraulics, Hydrology and Glaciology (VAW)

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

Tel. +41 1 632 68 23, Fax: +41 1 632 11 92, E-mail: boes@vaw.baug.ethz.ch

Abstract: Experimental investigations on the location of the inception point of air entrainment on stepped spillways and on the flow depth at this characteristic point were conducted at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW). The results are compared to literature model data and empirical fits proposed by several authors, and new approximations are presented for chute inclination angles between 26° and 75°. These fits reliably predict both the vertical distance from the spillway crest to the inception point as well as the corresponding flow depth.

Keywords: air-entrainment, boundary layer, cavitation, inception point, stepped spillways, two-phase flow

1    INTRODUCTION

Like many other high-speed flow configurations in hydraulic engineering, stepped spillway flows are characterized by the large amount of self-entrained air. The macro roughness of the steps leads to a sharp increase in the thickness of the turbulent boundary layer. Once the boundary layer reaches the free surface, air is entrained at the so-called inception point of air entrainment (Fig. 1 ), which is located further upstream than on con­ventional smooth spillways.

Hydraulic model investigations on skimming flow were conducted at VAW, ETH Zurich, Switzerland to study the energy dissipation characteristics and typical features of the two-phase flow on stepped chutes (Boes 2000a, Schläpfer 2000).

2    EXPERIMENTAL SET-UP

All experiments were conducted in a prismatic rectangular channel of width b=.50 m and length l=5.7 m with bottom inclination angles from the horizontal of f=30°, 40° and 50° or slopes (V:H) of 1:1.73, 1:1.19 and 1:0.84. The pseudo-bottom was used as the reference level for flow depths (Fig. 2 a). Three step heights s=23.1, 46.2 and 92.4mm were investigated for the 30° cascade, steps of 31.1 and 93.3 mm for the 50° chute and of 26.1 mm for f= 40°. Referred to a typical prototype value of s = 0.61 m, the model scales were 1:26.4, 1:13.2 and 1:6.6 for f = 30°, 1:19.6 and 1:6.5 for f = 50° and 1:23.4 for f = 40°. Only the spill­way face with constant bottom slope and step size was modeled (Fig. 1 a). The discharge Q was provided with a jetbox (Schwalt & Hager 1992), which trans­formed the pressurized approach flow in a free surface open channel flow of pre-calibrated approach flow depth h_o and approach velocity u_o (Fig. 1 a). Its advantage over a conventional free overfall was an independent variation of both h_o and of the approach Froude number Fr_o = u_o/(g h_o)^1/2, where g denotes the acceleration of gravity. The inductive discharge measurement had an accuracy of ±2%.

A fiber-optical probe with two tips was used to measure both local air concentrations C and mixture flow velocities u_m in selected cross sections at step corners. This novel fiber-optical instrumentation is described by Boes & Hager (1998) and Boes (2000a). The errors of both local air con­centration and velocity measurements were examined in preliminary experiments and are esti­mated to be less than 5%. Both C and u_m data could be reproduced with only small devia­tions. A single sampling sequence usually stopped after 30 s, or after de­tection of 4000 air bubbles by any of the probe tips. As the bubble frequency in the fully aerated zone was of the order of 1 kHz, the signal acquisition was accomplished within only a few seconds. The vertical translation of the probe was controlled by a fine ad­justment travelling mechanism connected to a metric scale unit. The accuracy on the vertical probe posi­tion was estimated at ±0.25 mm. The probe was mounted on a trolley travelling parallel to the channel bottom. The error in the longitudinal probe posi­tion was ±0.5 mm.

3    INCEPTION POINT OF AIR ENTRAINMENT

At the inception point the degree of turbulence is large enough to entrain air into the black water flow. In the model experiments, the inception point was defined as the location where the pseudo-bottom air concentration C_b = 0.01 which agreed well with visual observation. The location of the inception point on stepped spillways is significantly closer to the spillway crest than on smooth chutes because the growth of the boundary layer depends on the bottom roughness (Chanson 1994).

A main advantage of the significant aeration along stepped spillways is the reduction of the cavitation risk potential. In the case of high velocities, the hydrodynamic pressures on the step surfaces or at the step corners may fall below the vapour pressure, resulting in cavitation which might cause severe damage to the spillway concrete. Based on the fundamental work of Peterka (1953), a bottom air concentration of about 5% is considered sufficient to avoid cavitation damage because the compressibility of the air-water mixture can absorb the impact of collapsing vaporized bubbles. Knowing the location of the inception point is thus important to have an idea of the unaerated spillway zone which is potentially prone to cavitation damage (Boes & Minor 2000). Although steps form large offsets away from the flow direction and thus inhibit cavitation from residing on the boundary (Frizell & Mefford 1991), the placement of an aerator to artificially entrain air might be of interest in the black water region of a stepped spillway. Air entrainment can also be achieved by bridge-supporting piles downstream of the spillway crest (Mateos Iguácel & Elviro García 1992) or by a flap-gate on top of the crest (Minor & Boes 2001).

Inception location

The results of several authors on the location of the inception point were analyzed and compared with the model data obtained at VAW (Boes 2000a, Schläpfer 2000). The inception point location is usually expressed either in terms of the length L_i of the black water reach or by the vertical distance z_i from the spillway crest ( Fig. 2 a). The latter is chosen here in analogy to Mateos Iguácel & Elviro García (1997), because besides the advantage of determining the flow velocity more directly from the elevation difference it allows a general comparison of data from different spillway slopes.

In Fig. 3 the experimental data in terms of the non-dimensional parameter z_i/s of Boes (2000a), Schläpfer (2000) and Wahrheit-Lensing (1996) are plotted as a function of a so-called roughness Froude number Fr_* =q/(g sinf s³)^1/2, where q is the unit discharge. Also plotted are approximations proposed by Chamani (2000), Chanson (1994), Mateos Iguácel & Elviro García (1997) and Matos (1999, see also Matos et al. 2000). Where necessary, the alternative roughness Froude number Fr_K=q/(g sinf K³)^1/2 used by many authors was transformed adequately. The only difference between Fr* and Fr_K comes from the definition of the roughness height K=s∙cosf perpendicular to the pseudo-bottom (Fig. 2 a), i.e. Fr*= Fr_K (cosf)3/2 and Fr*=0.5Fr_K for f= 51°.

Most data and approximations stem from model experiments with the gravity dam type slope of about 1:0.8 or f≈ 51°. Only the data of Boes (2000a) and Schläpfer (2000) were obtained for different inclination angles of 30° and 40°, respectively, and the function proposed by Chanson (1994) refers to two data points with f= 26.6°, whereas the remaining 47 experimental results of various authors he took into account were all obtained with models of f≈ 52°. In addition, out of the ten scale models underlying the approximation given by Mateos Iguácel & Elviro García (1997), one featured a steep downstream chute angle of f= 71.6°. Whereas Chamani (2000), Mateos Iguácel & Elviro García (1997) and Wahrheit-Lensing (1996) had maximum Froude numbers of Fr_* ≈ 10 and Matos (1999) of only Fr_* ≈ 7, the data of Boes (2000a) and those analyzed by Chanson (1994) extend to roughness Froude numbers Fr_* > 40 (Fig. 3 ). For a given Fr_*, the approximation proposed by Chamani (2000) yields the largest vertical distance from the spillway crest, whereas according to Matos (1999) the flow down stepped chutes becomes aerated further upstream.

Because of the special experimental arrangement without a spillway crest of Boes (2000a) and Schläpfer (2000), the blackwater drawdown curve obtained downstream of the jetbox was extrapolated in the upstream direction assuming a similar surface roughness and friction behaviour as observed on the constantly sloped channel part. The fictitious location of the crest was then defined as the point where the non-aerated flow depth hw was equal to the critical depth hc (Fig. 2 b). With the so-obtained length Li the vertical distance was taken as zi ≈ Li cosf (Fig. 2 a). The present data agree very well with the data from crest profile spillways. Moreover, even the results from the 30° and 40° models follow the gravity dam type data very closely, so that all data obtained at VAW can be approximated by the formula given by Boes (2000b) with a correlation coefficient of R= 0.986 (Fig. 3 ). The data obtained at VAW agree well with those of Mateos Iguácel & Elviro García (1997) and suggest slightly smaller zi/s-values than given by Wahrheit-Lensing (1996) or even Chamani (2000) for Fr* <10. For large rougness Froude numbers, the approximation is in close agreement with that proposed by Chanson (1994). For the overall data, an approximation is thus (Fig. 3 )

                            (1)  

Eq.(1) should not be applied outside chute inclination angles of 26°<f <75°.

Inception flow depth

In analogy to the inception point location, the normalized inception flow depth is plotted in Fig. 4 for the data of Boes (2000a), Schläpfer (2000) and Wahrheit-Lensing (1996) as well as the approximations of Chanson (1994) and Matos (1999). For the data obtained at VAW, distinction should be made between the mixture flow depth h₉₀ =h(C=0.9) and the equivalent clear water flow depth h_w = h₉₀ (1–C_mean), where C_mean denotes the depth-averaged air concentration. The average difference of 35% between the two characteristic flow depths can be explained by the definition of the inception point location. Because this is taken where the pseudo-bottom air concentration C_b =0.01, the flow is already affected by aeration and the average air concentration at the inception point amounts to approximately C_mean,I =0.26. Fig. 4 shows that the h_90,i/s-values of Boes (2000a) and Schläpfer (2000) are closer to the data of the other authors than the relative clear water depths h_w,i/s. Obviously, the inception flow depths measured in the mentioned studies were mixture flow depths h_m,i, a conclusion similar to that drawn by Matos et al. (2000). An approximation taking into account all data relating to mixture flow depths and 26°<f <55° is (Fig. 4 )

                          (2)

Mixture flow depths should generally be considered for the design of chute training walls (Boes & Minor 2000).

4    CONCLUSIONS

New formulae to compute the location of the inception point of air entrainment and the corresponding flow depth are proposed based on a large number of model data. The good agreement suggests that the inception point characteristics can be reliably predicted from model results nowadays. The approximations show that the distance from the spillway crest as well as the flow depth increase with increasing unit discharge. Aeration of the black water flow may be provided with an aerator downstream from the spillway crest or a flap gate on top of the crest to inhibit cavitation damage. The steeper the spillway, the closer to the crest the aeration inception occurs and the smaller the flow depth for a given discharge and step height. Finally, an increase in step height leads to slightly higher flow depths at the inception point, but moves this point towards the spillway crest for a given discharge and bottom slope.

References

[1]    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).

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

[3]    Boes, R.M. & Hager, W.H. (1998). Fiber-optical experimentation in two-phase cascade flow. Proc. Intl. RCC Dams Seminar, Denver, USA (K. Hansen, ed.).

[4]    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.

[5]    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.

[6]    Chanson, H. (1994). Hydraulic design of stepped cascades, channels, weirs and spillways. Pergamon: Oxford, UK.

[7]    Frizell, K.H. & Mefford, B.W. (1991). Designing spillways to prevent cavitation damage. Concrete International 13(5): 58-64.

[8]    Mateos Iguácel, C. & Elviro García, V. (1992). The use of stepped spillways in energy dissipation. Intl. Symposium on Dams and Extreme Floods, ICOLD, Granada, Spain: 241-250.

[9]    Mateos Iguácel, C. & Elviro García, V. (1997). Initiation of aeration in stepped spillways. Proc. 27th IAHR Congress, ASCE, San Francisco, USA (F.M. Holly & A. Alsaffar, eds.) D: 589-594.

[10]   Matos, J. (1999). Emulsionamento de ar e dissipação de energia do escoamento em descarregadores em degraus (Air entrainment and energy dissipation on stepped spillways). Research Report, IST, Lisbon (in Portuguese).

[11]    Matos, J., Sánchez, M., Quintela, A. & Dolz, J. (2000). Air entrainment and safety against cavitation damage in stepped spillways over RCC dams. Proc. Intl. Workshop on Hydr. of Stepped Spillways, VAW, ETH Zu­rich (H.-E. Minor & W.H. Hager, eds.). Balkema, Rotterdam: 69-76.

[12]    Minor, H.-E. & Boes, R.M. (2001). Hydraulic design of stepped spillways. 29^th IAHR Congress, Beijing, China: submitted.

[13]    Peterka, A.J. (1953). The effect of entrained air on cavitation pitting. Proc. 5th IAHR Congress, Minneapolis, USA: 507-518.

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

[15]    Schwalt, M. & Hager, W.H. (1992). Die Strahlbox (The jetbox). Schweizer Ingenieur und Architekt 110(27-28): 547-549 (in German).

[16]    Wahrheit-Lensing, A. (1996). Selbstbelüftung und Energieumwandlung beim Abfluss über treppenförmige Entlastungsanlagen (Self-aeration and energy dissipation in flow over stepped spillways). Doctoral Dissertation. University of Karlsruhe, Germany (in German).

(a)

(b)

Fig. 1    Skimming flow over steps of s= 31.1 mm,  f =50°, Fr_o =4.9, h_o = 60 mm: (a) View of jetbox, ap­proach flow and inception point; (b) Flow detail of inception point at x≈1.5 m downstream of jetbox.

(a)

(b)

Fig. 2    (a) Side view of a stepped spillway: ( --- ) pseudo-bottom, flow region with(  ▓) equivalent clear water depth h_w and ( ▒) mixture flow depth h₉₀, (  ∙—∙—∙) energy head, and (○) inception point;
(b) Determination of L_i from drawdown curves.

Fig. 3    Non-di­mensional vertical distance z_i/s from spillway crest to inception point as a function of roughness Froude number Fr_* for different spillway slopes.

Fig. 4    Non-di­mensional flow depth hi/s at inception point as a function of roughness Froude number Fr* for different spillway slopes.