Three-Dimensional Computations of Water-Air Flow in a Bottom Spillway During Gate Opening

Engineering Applications of Computational Fluid Mechanics Vol. 8 No. 1, pp. 104–115 (2014)

104

THREE-DIMENSIONAL COMPUTATIONS OF WATER–AIR FLOW IN A

BOTTOM SPILLWAY DURING GATE OPENING

Ting Liu *

† and James Yang *

#

*Hydraulic Engineering, Royal Institute of Technology (KTH), SE-100 44, Stockholm, Sweden #Vattenfall Research and Development (R&D), SE-814 26, Ä lvkarleby, Sweden

†E-Mail: [email protected] (Corresponding Author)

ABSTRACT: Undesired entrainment of air in a bottom spillway often leads to problems in both safety and

operational functions. A numerical analysis of a transient process of air entrainment into bottom spillway flows

when a spillway gate is opened was conducted in this study. The Volume of Fluid (VOF) model was used. The 3D

computational domain consisted of a spillway conduit, a moving bulkhead gate, a gate shaft, an upstream reservoir

and a downstream outlet. The large number of cells, together with the dynamic mesh modelling of the moving gate,

required substantial computational resources, which necessitated parallel computing on a mainframe computer. The

simulations captured the changes in the flow patterns and predicted the amount of air entrainment in the gate shaft

and the detrainment downstream, which help in the understanding of the system behaviour during opening of the

spillway gate. The initial conduit water level and the gate opening procedure affect the degree of air entrainment in

the gate shaft. To release the undesired air, a de-aeration chamber with a tube leading to the atmosphere was added

to the conduit. Despite the incomplete air release, the de-aeration chamber was found to be effective in reducing

water surface fluctuations in the downstream outlet.

Keywords: bottom spillway, moving gate, air entrainment, two-phase flow, CFD

1. INTRODUCTION

A bottom spillway can be designed for such

purposes as flood discharge during dam

construction, emptying of a reservoir, or removal

of sediment. Though less frequently used than a

surface-type spillway, a bottom spillway must be

properly maintained to guarantee that it will

function as expected when needed. In a survey of

38 bottom spillways in Sweden, operational

problems were observed in nine. Eight of these

problems were caused by entrained air, which

mainly occurred during gate opening. For reasons

of both structural safety and personnel security,

many of these spillways are no longer in use

(Dath and Mathiesen, 2007).

A typical layout of a bottom spillway consists of a

gate (bulk head or radial), a gate shaft and a

conduit leading downstream. When the water

from the gate plunges into the gate shaft, air

becomes entrained in the water. The air

entrainment starts when a turbulent flow regime

develops and the energy of the surface eddies is

greater than the energy content of the surface

tension (Ferrando and Rico, 2002). Air pockets

and bubbles are entrained at the interface and

trapped at discontinuities between the impinging

jet flow and the receiving water body (Chanson,

2008). At the plunge location, the air entrainment

is caused by surface disturbance due to the jet, the

air boundary layer or the free surface shear layer

(Ervine, 1998).

Some general-purpose empirical formulas are

available to determine the amount of air

entrainment associated with different

mechanisms. Ervine (1998) showed that the

maximum aeration rate per unit jet width was a

function of the jet velocity and could be expressed

as a simple polynomial with factors for

consideration of all of the mechanisms. However,

it is difficult to determine which of the

mechanisms is relevant and to determine some of

the constants in the formulas (Khatsuria, 2005).

Due to the complexity of the problem, which also

depends on site-specific details, numerical

modelling and experiments are often essential in

understanding the air entrainment process and

considering it in design.

For two-phase flows, the geometry of the flow

field or the distribution of the phases commonly

refers to flow patterns or flow regimes, mainly

depending on the flow rates of the fluids. The

Baker chart is one of the flow pattern maps often

used to determine the flow regime and predict the

transitions between the flow patterns. Fig. 1

shows various flow patterns in a horizontal pipe

(Baker, 1954). Among these flow patterns, the

degree of destructivity to the structure increases

from annular and bubbly flow to stratified flow

and further to slug and plug flow (Rahimi, 2010).

Received: 10 Jun. 2013; Revised: 24 Sep. 2013; Accepted: 11 Nov. 2013

mailto:[email protected]

Engineering Applications of Computational Fluid Mechanics Vol. 8, No. 1 (2014)

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When the entrained air in a bottom spillway forms

air pockets, a plug or slug flow pattern often

develops. When moving liquid slugs suddenly

encounter obstructions such as valves and pipe

bends, shock loads are generated (Thorley, 1991).

Enclosed air pockets in a pipe flow can cause pipe

vibrations, blowout, blowback and even a risk of

cavitation.

Fig. 1 Two-phase flow patterns in horizontal pipes

(Baker, 1954).

A de-aeration device is often installed to prevent

air-related problems in bottom spillways. Such a

device is usually located immediately

downstream of the gate shaft to remove undesired

air from the system at an early stage. A typical

de-aeration device, consisting of a de-aeration

chamber and a vertical vent tube leading to the

atmosphere, is typically suggested. According to

Sigg et al. (2004), this type of device offers high

de-aeration efficiency for horizontal or nearly

horizontal conduits. Its function is normally tested

through scaled experiments prior to its installation

in the prototype.

As an option to costly prototype tests and

laboratory experiments that suffer from scale

effects with respect to air entrainment,

computational fluid dynamics (CFD) simulations

lead to both qualitatively and quantitatively

reasonable results. Many mathematical models

have been developed to simulate multiphase flows

in diversified engineering applications. The VOF

model, an Eulerian–Eulerian approach, is suitable

for simulating two or more immiscible fluids and

interface tracking, e.g., water–air flows with free

surface or under pressurised condition (Haun et

al., 2011; De Schepper et al., 2008). In this

approach, the water–air flow is best represented

as a continuous medium, which is the situation in

a bottom spillway.

The finite volume code FLUENT has been

validated for simulations of multiphase and

turbulence flow problems. De Schepper et al.

(2008) successfully used FLUENT to simulate

flow regimes predicted by the Baker chart. Baylar

et al. (2009) employed FLUENT to model the air

injection rate of Venturi flows and achieved good

agreement with experimental results. With

FLUENT, Liu and Yang (2011) modelled air

pocket transport in pipe flows. Politano and

Carrica (2007) and Politano and Arenas (2011)

adopted the software to compute dissolved gas

dynamics downstream of a dam.

Some previous numerical studies of flow in

bottom spillways have focused on only part of the

waterway or steady-state solutions or have been

performed in 2D (Dargahi, 2010; Hong et al.,

2011; Haun et al., 2011; Liu and Yang, 2011). To

examine the overall behaviour of a bottom

spillway when its gate opens, it is essential to

carry out transient simulations in three

dimensions.

In this study, numerical simulations were

conducted to examine a process of air entrainment

and detrainment following the opening of the

spillway gate. The purpose of this study was to

gain insight into the air flow behaviour in the

bottom spillway that would be of use in managing

its operation. The issues discussed include the

characteristics of the two-phase water–air flow

and the influences of contributing factors, such as

the initial water level in the conduit and the gate

opening procedure. This study also included an

examination of the function and efficiency of a

de-aeration chamber.

2. BOTTOM SPILLWAY OF LETAFORS

DAM

The Letafors Dam, located in the municipality of

Torsby in Sweden, is an earth-fill dam with a

maximum structural height of 25m. The full

reservoir water-stage elevation (FRWS) is

+349m. The reservoir has a regulated water depth

of H = 19.1m. There is one surface spillway and

one bottom spillway. The bottom spillway, which

lies below the dam, includes a bulkhead gate, a

gate shaft, a horizontal conduit and a downstream

outlet (Fig. 2).

The discharge of the bottom spillway is controlled

by the bulkhead gate, which has a width of 3.05m

and a maximum opening hmax of 5m. The height

of the gate opening is the gap between the lower

edge of the gate and the reservoir bottom, denoted

by hg. The ordinary gate opening velocity is vg =

1cm/s. The gate shaft, which is open to the


106

+349.00

+329.90

+320.95

+328.90

Letten Lake

Gate shaft

Letafors

Dam

Fence

Retaining wallOutlet

Half-filled

Inital water level:

Fully-filled

Empty

Fig. 2 Bottom spillway of Letafors Dam.

Fig. 3 Blowout at downstream outlet in prototype test.

Air inlet:

Pressure inlet

( P = 0)

Downstream outlet:

Pressure outlet

( P = 0)

Flow

Moving

gate

Wall

Water inlet:

Pressure inlet

( H =19.1 m)

M NCentreline of gate shaft

B C DA

ff

2.55m1.275m

Half-filled9.225m

25m 50m 25m 4m

Initial water level:

Fully filled

Fig. 4 Computational domain and boundary conditions of numerical model.

(a)

(c) (d)

(b)

Fig. 5 3D views of grid showing regions: (a) around gate, (b) at downstream outlet without de-aeration device, (c) at

downstream outlet with de-aeration device and (d) of de-aeration chamber with vertical tube.

Engineering Applications of Computational Fluid Mechanics Vol. 8 No. 1, pp. 104–115 (2014)

107

Fig. 6 Mass imbalance in conduit for mesh with

varying sizes for t = 5–15s.

Fig. 7 Dynamic layering method for modelling of gate

opening: (a) initial height of first-layer cell, (b)

cell height increasing with time and (c) cell

splitting into two layers.

Fig. 8 Water flow through gate and flow into river as

function of gate opening.

Table 1 Description of the scenarios modelled.

Air entrainment

scenario Gate opening

VOF

discretisation

scheme

Initial water

level

in conduit

Without de-

aeration device

Continuous

Explicit Half full

Implicit

Empty

Half full

Full

Instantaneous Explicit Full

With de-aeration

device Continuous Implicit Half full

x=40m x=50m

stratified flow

plug flow

bubbly flow

(e)

(f)

(g)

Fig. 9 Contours of air volume fraction in bottom

spillway at gate opening velocity of 1 cm/s.

Water released from the reservoir causes mixing

and air entrainment in the gate shaft, with free

orifice flow at (a) t = 15s and (b) t = 35s and for

submerged conditions at (c) t = 85s and (d) t =

105s, corresponding to gate shaft water levels of

6.29 and 8.27m, respectively, above the

reservoir bottom. The flow pattern in the

conduit undergoes a change from stratified flow

at (e) t = 65s to plug flow at (f) t = 70s to bubbly

flow at (g) t = 75s. In the lower part, air

discharges from the downstream outlet at (h) t =

25s, (i) t = 55s and (j) t = 75s.

(c)

(d)

(h)

(i)

(j)

Engineering Applications of Computational Fluid Mechanics Vol. 8 No. 1 (2014)


108

atmosphere, is located immediately downstream

from the bulkhead gate. The conduit, which has a

circular cross section, has a diameter of 2.55m

and a length of 104m. Its bottom elevation is

30.6m below the FRWS. The gate shaft connects

to the horizontal conduit through a 90-degree

bend. The bottom of the river is 20.1m below the

FRWS.

During the opening of the gate, a blowout was

observed to occur due to entrained air that was

released downstream (Fig. 3). At the time, no de-

aeration structure existed in the prototype

spillway.

3. NUMERICAL MODELING

A numerical model was set up to simulate the

water–air flow in the bottom spillway of the

Letafors Dam during the gate opening. The VOF

model, combined with the k–ε turbulence model,

was used. It was assumed that mass transfer does

not occur between the air and water, which is

usually a valid assumption as air entrainment

from the gate shaft is mainly caused by turbulent

mixing and pressure reduction. The amount of

entrained air in the flow is dominant, while the

amount of dissolved air is negligible. The

entrained air in the conduit tends to accumulate at

the top of the conduit, forming air pockets.

3.1 VOF model

In the VOF model, all of the phases share a single

momentum equation and velocity field. In each

control volume, the fluid properties are

determined by the presence of the component

phases, i.e., the properties of the mixture in each

control volume. A control volume is filled with

air, water or a mixture of the two; no void volume

is allowed. The momentum equation is written as

Fguupuuut

T

)()(

where αa is air volume fraction in a control

volume, αw is water volume fraction, ρ and µ are

mixture density and viscosity, respectively, ρ =

αaρa + (1 – αa)ρw, μ = (αaρaμa + αwρwμw)/ρ, ρw is

water density, ρa is air density, u

is velocity

vector, t is time, p is static pressure, g is

acceleration of gravity and F is volumetric force

at the interface resulting from the surface tension.

The VOF model is a direct solution for the motion

of the two phases that does not track the moving

boundary of the interface (ANSYS Inc., 2011).

The volume fraction of air is obtained by solving

its continuity equation, written as

0)()(

u

ta

(2)

The volume fraction of water is calculated as

αw = 1 – αa (3) The VOF equations are time-discretised using

either an implicit or an explicit scheme. The explicit scheme is only used to compute a time-

dependent solution, whereas the implicit scheme is used for both time-dependent and steady-state

solutions. To improve convergence and allow large time

steps, the implicit scheme is used, rather than the

explicit scheme, to discretise the VOF equations (Chau and Jiang, 2001 and 2004). Simulations are

performed to examine the difference between the two schemes and also to confirm that the implicit

VOF yields sufficiently high accuracy. With the volume fraction of each phase calculated

in each computational cell, the geometry of the interface is configured through the interface

reconstruction method (ANSYS, 2011). In the explicit VOF, the water–air interface is calculated

using a piecewise linear interface calculation (PLIC) procedure, the so-called Geo-reconstruct

routine in FLUENT (De Schepper et al., 2008). This procedure has the highest accuracy of the

interface reconstruction techniques available in the code. A second-order reconstruction scheme

(the Compressive routine) is used for the implicit

VOF. The interface is almost as sharp as the solution from the PLIC method.

3.2 Numerical solver

The equations are solved using an unsteady solver. The code uses a second-order upwind

scheme and the PISO pressure–velocity coupling routine. For the interface computations, Geo-

Reconstruct is used for the explicit VOF and Compressive is used for the implicit VOF.

3.3 Simulations

3.3.1 Geometry and grid

The reservoir in the model is limited to an area of

50m×50m. The gate is centrally located at the downstream boundary of the reservoir. Fig. 4

displays the cross-sectional profile of the 3D computational domain (here without the de-

aeration device). Due to its irregular geometry, the downstream outlet in the river is somewhat

simplified and treated as a rectangular pool. When

pressurised, air pockets attach to the top of the conduit and are discharged close to the upper

(1)


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vertical boundary of the pool. The simplification

of the lower boundary might have some slight effect on the air flow. Our intention is that, with

the data from the Letafors Dam and the generalisation of the outlet geometry, the blowout

behaviour can be used as a reference to illustrate other similar outlets. A de-aeration chamber was added to the conduit in the model to demonstrate its effectiveness in mitigating the blowout in the prototype. The de-aeration chamber is situated 92m downstream of the gate shaft, at an existing hole in the prototype. Based on Wickenhäuser and Minor’s (2008) recommendations for the minimum dimensions of a de-aeration chamber, the height, length and width were set to 28%, 27% and 50% of the conduit diameter, respectively. The chamber length must be optimised because an excessively long chamber could exert an unfavourable influence on the flow. The vertical tube connecting the chamber to the atmosphere has a diameter of 0.3m. Fig. 5 shows a 3D view of a typical grid for

simulations with and without a de-aeration

device. The domain is discretised using a multi-block grid with hexahedral cells. The typical

mesh sizes are between 6×106 and 8×10

6 cells for

the cases of air entrainment and de-aeration. The

grids are refined near the solid boundaries, near the gate opening and near the de-aeration

chamber, where large velocity gradients are expected. In the conduit, the height of the first

layer mesh is 1.3cm, i.e., 0.5% of the conduit diameter. The maximum skewness of the mesh is

0.65. The ordinary mesh, with n = 6×10

6 cells, is

denoted by G. To show grid independence, a fine mesh G1 and a coarse mesh G2 were generated,

with n = 9×106 and 3×10

6 cells, respectively. The

mass imbalance ηm in the conduit refers to the net

flux through the conduit, i.e., the difference

between the flow rates at cross sections M and N,

ηm = (QM – QN)/QM·100% (4)

Fig. 6 shows the change in ŋm with time. The

value of ŋm falls below 0.2% when the solutions

for meshes G0 and G1 converge to 15s. The mass

conservation is best achieved with mesh G1. The

change in the net mass flux with time is almost

the same for G0 as for G1, implying that the

overall mass balance in the conduit is achieved

and that the solution for the ordinary mesh G0 is

grid-independent.

3.3.2 Boundary conditions

The symmetry boundary condition applies to the

central plane of the computational domain so that

only half of the geometry is simulated. The

hydrostatic pressure boundary applies to the

upstream end of the reservoir (Fig. 4). It is

implemented by means of a user-defined function

(UDF) in which the hydrostatic pressure p(z) at a

water depth of z is defined as

p(z) = (ρw – ρa)gz (5)

The gate shaft is open to the atmosphere and

serves as an air inlet, allowing air to flow freely in

and out. The same boundary condition applies to

the downstream outlet and to the de-aeration tube

in the de-aeration simulation. The remaining

boundaries are treated as walls.

3.3.3 Modelling moving gate

A dynamic mesh is created to model the upward

gate movement, and a dynamic layering method is

used to avoid mesh distortion.

The domain of the gate opening, i.e., the volume

connecting the upstream reservoir and the gate

shaft, is defined as the dynamic zone. The lower

edge of the bulkhead gate, moving vertically

upwards, is treated as a rigid body. The layering

method is activated for the dynamic zone, where

new layers of mesh are added following the lower

gate edge. A ‘profile’ text file of the gate opening

velocity specifies its moving velocity, which is

constant in this case. The cell height in the

dynamic zone is specified as hmin = 0.05m. A

close-up view of the mesh in the rectangular box

in Fig. 5a is shown in Fig. 7. Figs. 7a−b show the

growth of the dynamic zone with time. A split

factor of αs = 0.4 is used, indicating that the cell

adjacent to the rigid body splits into two layers of

cells when the cell height is larger than (1+ αs)hmin

(Fig. 7c).

3.3.4 Computations

The transient process of air entrainment and de-

aeration in the bottom spillway is calculated for

two gate opening scenarios, i.e., continuous

opening at a constant velocity and instantaneous

opening to a designated position. For the former

scenario, the gate, with a prescribed initial

opening of hini = 5cm, opens at a constant velocity

vg. The gate opening height at time t is hg = hini +

vgt. For the latter scenario, the gate opens

instantaneously from the closed position to a

designated opening height h.

In practical situations, the conduit may be dry or

partially full after maintenance. For this reason,

different initial water levels in the conduit are


110

modelled to demonstrate their influences on air

entrainment (Fig. 4).

The implicit and explicit VOF models were

applied to compare the two. Simulations were

performed for the existing bottom spillway layout

without the de-aeration structure. The time step

was 0.001s for the continuous gate opening and

0.005s for the instantaneous opening. The

modelled scenarios are summarised in Table 1.

Parallel computing was performed on a

mainframe computer with 4–8 CPUs. The

residual values of the mass, velocity and volume

fraction were the convergence criteria. A

simulation was considered to have converged to a

solution when the scaled residuals were lowered

by at least three orders of magnitude.

4. RESULTS AND DISCUSSIONS

4.1 Gate opening at constant velocity

In this scenario, the bulkhead gate is opened at a

velocity of 1cm/s up to a height of 1.8m.

Considering that air entrainment ceases when the

gate becomes submerged in the gate shaft, full

gate opening is not necessary.

4.1.1 Water–air flow

The initial water depth in the conduit is 1.275m,

half of the conduit diameter. As the gate opening

increases, the water flow rate from the gate

increases (Fig. 8). The water outflow rate into the

river is zero at the beginning of the simulation.

When the water level at the downstream outlet is

higher than the riverbed elevation, the water starts

to run into the river. The outflow rate undergoes a

sudden increase at hg ≈ 0.85m. For gate openings

hg = 1–1.8m, the water level in the gate shaft

increases, and the water flow through the gate is

greater than the outflow into the river.

At each time step, the dynamic mesh due to the

gate movement is updated automatically. The

step-by-step gate opening gives rise to small

pressure perturbations in the reservoir, which in

turn affects the gate discharge. This explains the

somewhat ‘wavy’ shape of the calculated water

flow rate from the gate.

The simulated flow rate through the gate is

compared with empirical results. The points in

Fig. 8 correspond to the discharge through a

rectangular orifice under free and subsequently

submerged conditions, as determined by the

formula

hgCbhQ g 2 (6)

where Q is flow rate, C is discharge

coefficient, b is gate width, ∆h is difference in

elevation between the reservoir water surface

and the water surface at the centreline of the

gate opening, for free-orifice flow, or

between the reservoir water surface and the

gate shaft water stage for submerged

conditions. The comparison indicates a good

agreement up to a gate opening of 1.1m;

thereafter, the simulated flow rate is slightly

higher than that indicated by the empirical

results. The discrepancy is most likely due to

the assumption of a constant discharge

coefficient in the empirical formula, which

should increase with the gate opening. This

coefficient is also affected when the discharge

from the opening undergoes a change from

the free to the submerged outflows. Figs. 9a-d show the air volume fraction in the

upper part of the bottom spillway, from 5m

upstream to 35m downstream of the gate shaft

centreline. The water jet hits the wall of the gate

shaft and plunges into the pool of water (Fig. 9a).

Air is entrained into the bottom spillway mainly

by the turbulent mixing in the gate shaft. Once the

downstream water level at the outlet becomes

higher than the conduit top, the initially filled air

in the conduit becomes trapped as well. The

entrained air from the gate shaft, together with the

trapped air, accumulates under the conduit roof,

forming air pockets in the bottom spillway (Fig.

9b). Once the gate opening becomes submerged,

little air is entrained into the bottom spillway

(Figs. 9c-d).

During the filling of the conduit, the flow pattern

in the middle of the conduit 40–50m downstream

of the gate shaft undergoes a change from

stratified flow (Fig. 9e) to plug flow (Fig. 9f) and

then to bubbly flow (Fig. 9g). The flow pattern

transition that occurs as the rate of water flow

increases during the gate opening agrees

qualitatively with that predicted by the Baker

chart (Baker, 1954).

Figs. 9h-j show the results for the lower part of

the conduit. In the outlet region, air pockets

escape from the water body. In the meantime, the

released air pockets blow up the water surface

immediately downstream of the conduit (Fig. 9h).

When the resulting air pocket penetrates into the

downstream water body, the next blowout is

prepared (Fig. 9i). The blowout height depends on

the air pocket size. As the air flow rate decreases,

the entrained air is discharged in the form of


111

small air pockets, resulting in small water surface

fluctuations (Fig. 9j).

Two cross sections in the conduit, at distances of

2.55 and 103.9m from the centreline of the gate

shaft, are denoted by M and N, respectively (Fig.

4). The air flow through cross section M, i.e., air

flow into the conduit, is the entrained air from the

gate shaft. When the gate opening reaches

0.605m, the air entrainment slows down. When

the gate opening reaches 1.088m, the air inflow

drops to almost zero (Fig. 10). The air flow

through cross section N, i.e., the air outflow,

consists of the entrained air and the initially filled

air. When the opening is between 0.605 and

1.088m, spikes occur in the air outflow, which

suggests that air is present in the form of air

pockets.

The simulated results for air entrainment shown

in Fig. 10 are similar to the air flow conditions

observed at a Morning Glory spillway intake

(Khatsuria, 2005). As shown in Fig. 11, the air

inflow rate through the intake increases in Region

I, decreases to zero in Region II when the air

passage at the lower end is sealed by water and

remains zero in Region III for a submerged

discharge. The air flow rates in Regions I, II and

III correspond to the air entrainment for gate

openings of less than 0.605m, between 0. 605m

and 1.088m and more than 1.088m, respectively.

4.1.2 Air pocket release downstream

The flow in the conduit exhibits a plug flow

pattern when a series of air pockets is built up

(Fig. 9f). The air pockets are carried with the flow

and then escape downstream of the conduit,

blowing up the water surface locally (Fig. 9h).

The downstream water level, i.e., the water level

in the middle of the rectangular pool, keeps rising

until it is higher than the river bed elevation. The

blowout height corresponds to an air volume

fraction of 0.5. The water surface fluctuation

corresponds to the blowout height at the wall

above the downstream water level. The release of

an air pocket corresponds to one spike of the

fluctuations in Fig. 12, the size of which can be as

large as 4.54m. The frequency of occurrence is

approximately 6s during the interval t = 40–70s.

After t = 70s, the air outflow continues and

generates an average rise in the water surface of

0.69m. Blowouts were observed and documented

during the opening of the prototype bottom

spillway.

As shown in the 2nd

and 4th pictures in Fig. 3, two

of the highest consecutive blowouts following the

start of the gate movement had approximate

heights of 2.5 and 4.5m, respectively. The

corresponding heights predicted by the simulation

were 2.96 and 4.81m, respectively, at positions in

the blowout where the air content is 90%. At the

locations with air contents of 99%, the heights are

only 0.1–0.2m greater. The opening procedure in

the prototype test was not precisely defined;

presumably, the gate was opened at a constant

velocity as in the numerical model. As the air

pocket size is the dominant factor in this context,

the comparable blowout heights indicate that the

prototype tests and the CFD model generate air

pockets of similar dimensions up to a gate

opening of 1.8m.

The pressure fluctuations in the conduit are

mainly the combined result of the water level rise

in the gate shaft, entrained air and air pocket

release downstream. The pressure fluctuations at

four points along the top of the conduit, A, B, C

and D, are shown in Fig. 13, at distances of 2.55,

25, 75 and 100m from the gate shaft centreline

(Fig. 4), respectively. The transient pressures are

recorded every 0.2s during the gate opening. The

pressure at point A is lower than that at point B,

and the pressure drop is caused by the 90-degree

bend. The amplitudes of the pressure oscillations

above the mean pressure are ≈104Pa for points B,

C and D and ≈2×104Pa for point A during the

filling of the conduit. The oscillations at point A

are larger than those at the other points.

4.1.3 Implicit VOF

For the same mesh grid used in this study, the

implicit VOF requires approximately 25% of the

computational time that the explicit VOF

requires. The use of the implicit VOF improves

the convergence, and the calculations are

performed more efficiently. The air flow rate, a

parameter of interest, is chosen for comparison of

the two methods.

For the implicit and explicit VOF, Fig. 14 shows

the total air flow passing through the air inlet of

the gate shaft and the flow through the

downstream outlet. A larger difference exists for

the air inlet than for the downstream outlet. The

peak values of the air flow rates also differ. The

implicit VOF results in maximum differences of

79% and 28% in the air flow rate through the gate

shaft and the downstream outlet, respectively.

There is also a phase shift in time: the time to the

peak is shifted by approximately 2s for the gate

shaft air inlet and 8s for the downstream outlet.

The accuracy of the previously mentioned

polynomial given by Ervine (1998) for calculating

the maximum aeration rate is likely to be ±30%


112

(Khatsuria, 2005). Hence, the implicit VOF gives

reasonable results with acceptable accuracy,

especially with respect to the air flow through the

downstream outlet and the peak values of air

flow.

Fig. 10 Air flow rate through upstream and

downstream ends of conduit versus gate

opening.

Reservoir level

Air

dis

char

ge

Region I Region II Region III

Air discharge Air passage at

lower end sealed

Submerged discharge

Fig. 11 Air discharge versus reservoir level of Morning

Glory intake (Khatsuria, 2005).

Fig. 12 Water surface fluctuations in downstream

outlet caused by release of air pockets for t =

40–100s.

Fig. 13 Transient pressure along the top of conduit

during gate opening.

Fig. 14 Air flow rate for (a) inflow through gate shaft

and (b) outflow through downstream outlet,

using implicit and explicit VOF.

Fig. 15 Entrained air flow rate versus gate opening for

initially full, half-filled and empty conduit.

Fig. 16 Entrained air flow rate for t = 0–40s for

instantaneous gate opening ratios β of 10%,

20%, 30%, 40% and 50%.

Fig. 17 Air flow rate in the bottom spillway with a de-

aeration device during gate opening.

4.1.4 Initial water level in conduit

The initial water level in the conduit affects the

amount of air entrainment. Therefore, three initial

conduit water levels were simulated to

quantitatively evaluate this effect. Initial water


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heads of 0, 1.275 and 10.5m, measured from the

conduit bottom and corresponding to an empty, a

half-filled and a fully filled conduit, respectively,

were simulated.

As Fig. 15 shows, the amount of air entrained

during the gate opening is the greatest for the

initially empty conduit and slightly lower for the

initially half-filled conduit. For the fully filled

conduit, a limited amount of air is entrained into

the conduit. When the gate opening is larger than

0.4m, little air entrainment is detected, which

indicates that a submerged conduit can effectively

reduce air entrainment during the gate opening.

Fig. 18 Water surface fluctuations downstream with

and without a de-aeration device for t = 40–

120 s.

Fig. 19 Transient pressure along the top of the conduit

during gate opening with a de-aeration device

installed between Points C and D.

4.2 Instantaneous gate opening

Five instantaneous gate opening cases, in which

the gate opens instantaneously from a closed

position to a designated position, were simulated

for the initially fully filled conduit. The flow in

the conduit was calculated until a steady-state

flow condition was reached, i.e., up to t = 40s.

The opening ratio β was defined as β = hg/hmax.

The instantaneous gate opening cases were

hypothetical scenarios. The air entrainment into

the conduit during gate opening varies with the

opening ratio (Fig. 16). For a larger opening ratio

(β), more air is entrained into the conduit; the

process is more intense at the beginning and

terminates earlier. In order words, the larger the

gate opening is, the more air is entrained into the

bottom spillway, because of the higher flow rate

and more intense mixing in the gate shaft. If a

blowout occurs, allowing a small gate opening

velocity may help improve the situation.

4.3 De-aeration

As in the situation without any de-aeration

structure, the entrained air accumulates at the top

of the conduit and forms air pockets. After the

conduit becomes submerged at hg = 0.51m, the air

in the conduit becomes trapped. Most of the

entrained air is released through the de-aeration

tube, and a small amount is carried downstream

by the flow. The largest de-aeration rate of the de-

aeration structure is 4.4m3/s (Fig. 17).

The simulation results showed that during the

filling of the bottom spillway, complete de-

aeration is not achieved because the incoming air

flow rate is higher than the value measured in the

experiment for the determination of the de-

aeration chamber layout (Wickenhäuser and

Minor, 2008). The results indicate that the

chamber dimensions needs to be larger to achieve

higher de-aeration efficiency during the gate

opening.

As Fig. 18 shows, the addition of the de-aeration

chamber mitigates the water surface fluctuations

downstream. Without the de-aeration structure,

the average water surface fluctuation is 3.23m for

10% of the highest peaks. With the de-aeration

structure, the corresponding average fluctuation is

only 0.91m.

The transient pressure change with time at points

A, B, C and D along the conduit is shown in Fig.

19. Compared with the case without de-aeration,

the magnitudes of the pressure oscillations at the

beginning of the gate operation are larger. The

reason for this is that air release from the de-

aeration structure introduces wave propagations

in the conduit and gives rise to somewhat larger

pressure fluctuations.

5. CONCLUSIONS

The flow jet plunging into the pool of the gate

shaft causes strong mixing of air and water and

leads to entrapped air in the bottom spillway

conduit, which characterises the opening process

of the spillway gate. A 3D numerical model was

set up in FLUENT for the purpose of modelling

the water–air flow in the bottom spillway with an

upward-moving bulk-head gate. A dynamic mesh

was devised to simulate the gate movement. The

VOF model was employed in combination with

the k–ε turbulence model. The simulated flow rate

through the gate agreed well with the empirical


114

results. The documented and simulated blowout

heights at the downstream outlet were compared

and also agreed well.

During the filling of the conduit, the conduit flow

pattern evolves from stratified flow to plug flow

to bubbly flow. The transition of the flow patterns

indicates that less air is entrained through the gate

shaft as the gate opens. That is, when the water

level in the gate shaft increases and the gate

finally becomes submerged, the air entrainment

slows down and finally ceases. The initial water level in the conduit is an

important factor in the amount of air entrained.

The total amount of entrained air during the gate opening is similar for an empty and a half-filled

conduit, while a fully filled conduit results in much less air entrainment. Furthermore, a small

gate opening reduces air entrainment.

With the addition of a de-aeration chamber, most

of the entrained air can be released through the

chamber. During the early phase of the gate

opening, pressure fluctuations in the conduit are

larger because the chamber introduces

propagating pressure waves as air is released

through it. Compared with the case without any

de-aeration chamber, the height of the water

surface fluctuations downstream of the conduit

are reduced by 60–70%.

For hydropower purposes, most simulations of air

entrainment are performed with the ordinary VOF

model. The affordability of CPU resources are

most likely to be the limiting factor, as it requires

considerable computational effort to model a

long, transient process in 3D. Furthermore, the air

entrainment caused by the plunging jet in the gate

shaft and the air blowout downstream are not

exactly separated two-phase flow problems with

well-defined interfaces. Free-surface penetration

by plunging water gives rise to air bubbles that

are initially transported in discrete clusters. This

behaviour is attributed to bubble entrapment by

shear vortices (i.e. large eddies) travelling in the

edge of the jet (Thomas et al., 1984). A refined

CFD model, such as the multi-fluid VOF model

or the dispersed multi-phase model, would most

likely yield more accurate results. These models

are characterised by better physics in dispersed

regions and improved accuracy of interfacial

exchanges.

ACKNOWLEDGEMENTS

This study was conducted as part of a PhD

programme in the area of hydraulic design

financed by the Swedish Hydropower Centre

(Svenskt Vattenkraftcentrum, SVC), Stockholm.

SVC was established by the Swedish Energy

Agency, Elforsk and Svenska Kraftnät, together

with Luleå University of Technology, the Royal

Institute of Technology, Chalmers University of

Technology and Uppsala University.

www.svc.nu. The authors are indebted to Mr Rolf

Steiner, Fortum Generation, for providing the

necessary prototype data for the study. The

computations were performed on the mainframe

computer at the PDC Centre for High

Performance Computing (PDC-HPC) at the Royal

Institute of Technology, Stockholm.

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Three-Dimensional Computations of Water-Air Flow in a Bottom Spillway During Gate Opening

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