1. 22nd hydrotech conference, 2015 Characteristics of siphon spillways R
22nd hydrotech conference,  Characteristics of siphon spillways R. Tadayon and A. S. Ramamurthy, Concordia univ.

2. Outline Summary Introduction Objective Experimental procedure
Brief outline of numerical modelling (RNG k-e)
Discussion of results
Conclusions

3. Summary: Siphon spillways control flood flows in reservoirs.
A Plexiglas lab siphon model was designed, fabricated and tested to determine its Cd.
Both existing test results and our data were used to validate numerical model developed.
The validated model permits evaluation of Cd for other flow configurations, without recourse to new expensive experimental procedures.

4. 2. Introduction Siphons are used for reservoir operation
2. Introduction Siphons are used for reservoir operation. They are conduit systems in the shape of an inverted U. During priming, at first, air enters conduit along with water. When reservoir level increases, conduit acts as a pipe (blackwater flow). Initial discharges (free flows): approximately proportional to 3/2 power. For siphon acting as a pipe, q/ft depends on ΔH (Fig. 1, Eq. 1): [1] q = Cdd(2gΔH)1/2, ΔH = H1 − H2 Following Rousselier & Blanchet (1951), many have studied siphon spillways [Head (1971), Charlton (1971), Houichi et al. (2006)]. Cd depends on u/s&d/s slopes,u/s depth&surface roughness.

5. Universal relations can not be provided for Cd
Universal relations can not be provided for Cd. It is influenced by ratio of radii of crest & crown Other factors altering Cd are entrance, outlet shape & nappe deflector. Form losses and not friction losses are dominant. 3. Objective: Cd for blackwater flow in the siphon spillways were modeled numerically and verified by using experimental data. RANS equations were applied to solve the siphon spillway flow. RNG k-ε turbulent model was adopted. 2-D finite-volume discretization: for simulation. VOF scheme modeled the water surface [details in Qu’s thesis (ref.).

6. Expérimental details
Plexiglas siphon model was set in a rectangular channel that was 25.1 cm wide (Fig. 1). Conduit depth: d = 11.1 cm. Radii of crest & crown: respectively 3.7 & 14.7 cm.
A deflector was set on lower leg to assure that siphon was air-regulated when conduit exit was not completely submerged.
The u/s & d/s channels
were 3.2 m and 8.7 m
long, respectively.
Flow rates: measured by V-notch (error < 3%).

7. Fig. 2. Head’s siphon spillway model (Head 1975, © ASCE).
Besides our test data, data of Head (1975) were used to validate the predictions of model.
Head used a model (Fig. 2) constructed of timber and plastic. The depth d at the conduit crest region was 12.0 cm. The radii of the crest and crown were 9.0 and 21.0 cm, respectively. All details of test procedure and results were available.
Validation of our model was
possible. Head’s (1975)’s
model ΔH/d range: 1 to 2.5.
Present test range: 2 to 6.
Fig. 2. Head’s siphon spillway model (Head 1975, © ASCE).

8. Numerical model
Numerical model: 2D Reynolds-averaged continuity and momentum eqns. for turbulent flow on the basis of 2-eqn. k-ε model (Wilcox 2007).
Using Bousssinesq eddy-viscosity approximation, for the components of the Reynolds stress tensor τij varying linearly with the mean rate of strain tensor, set eqn. 2:
[2] τij= 2νTSij − 2kδij
νT = kin. eddy viscosity; k = turb. kin. engry; ε=dissptn rate
In the RNG k-ε model (Yakhot and Smith 1992), the kinematic eddy viscosity is,
[3] νT= Cμk2 / ε . k and ε are found by the transport eqns.:
[4] ∂k/∂t +uj(∂k/∂xj) =τij(∂ui/∂xj) − ε+∂/∂xj [(ν+νT/σk)(∂k/∂xj)]
[5] ∂ε/∂t = uj(∂ε/∂xj) = Cε1(ε/k)τij(∂ui/∂xj) − Cε2(ε2/k) + ∂/∂xj[(ν+νT/σε)(∂ε/∂x)
Here, Cε1= 1.42; Cε2¼ = C*ε2+[Cμλ3(1 − λ/λ0)]/(1+ βλ3); C*ε2=1.68; λ = (k/ε)(2SijSji)1/2; β = 0.012; λ0
= 4.38; Cμ=0.085; σk= 0.72; and σε= 0.72

9. Computational Procedures
Modeling verified siphon discharge characteristics. Finite-volume method was used with pressure velocity coupling scheme (pressure implicit & splitting operators(PISO) algorithm (Issa1986)
First grid cell next to boundaries were within 30 <uτy/ν<100.
Results: checked for grid independence(coarse&fine grids).
Volume of fluid (VOF) scheme (Maronnier et al. 2003) gave free surface. Its shape was determined following procedure suggested by Ferziger and Peric (2002).
[6] ∂c/∂t=∂(cuj)/∂xj=0
c = filled fraction: changes from 1 for full cell to 0 for empty cell.
At wall, wall function approach (Launder & Spalding 1974) was used. At inlet boundary of u/s, water surface level was specified and uniform velocity distribution was used. k and ε at the inlet boundary were estimated by following eqns.:
[8] k = 3/2u2avgI2 and ε = Cμ3/2/(0.07Dh)
Here, I = turbulence intensity (1% to 5%), Dh = hydraulic diameter.
At outlet water surface level was prescribed & gradients (velocity and turbulence) were set to zero. After finding velocity, flow rate was determined to get Cd.

10. There is agreement between predictions & test data.
Results
Figs 3 & 4 show comparison of model predictions with test data (present & Head (1975)).
There is agreement between predictions & test data.
RMS values of deviation of Cd between predicted & data were of the order of 2% (Figs 3 & 4).
As stated previously, no universal graphs or empirical equations can be provided to obtain Cd. It is influenced mainly by form losses that depends on dimensionless radius of crest R1 (=d), entrance shape, outlet geometry, tail water depth, & more importantly size and position of nappe deflector.
Fig. 3. Cd vs. ∆H/d
Fig. 4. Cd vs. ∆H/d (Head 1975)

11. Conclusions
2-D k-ε model, along with VOF scheme, properly predicts Cd of flow through siphon spillways. Predictions agreed well with test data.
A properly validated model provides flow characteristics of siphons for flow configurations involving different field boundary conditions.
A numerical model demands less time & expense compared to physical models to predict siphon behavior with slightly altered geometric end conditions and flow configurations.
Development of models consumes less time and expense compared to test procedures.
Data relating Cd & ΔH/d indicate that predictions converge much closer to test data at higher Reynolds numbers (higher ΔH=d), as one would expect.