Numerical Modeling of Self-Aeration in High-Speed Flows over Smooth Chute Spillways

Abstract

In chute spillways, self-aeration occurs downstream of the inception point, where the turbulent boundary layer edge approaches the free surface, if they are long enough. Downstream of the inception point, a layer containing an air-water mixture extends gradually through the flow with the bulking effect. Flow bulking is essential in terms of sidewall freeboard design. In addition, the introduction of enough air quantity near the solid boundaries prevents cavitation damage. In the present work, a 2D numerical model was developed for the prediction of self-aeration and air concentration profiles across the depth and the free-surface location, together with flow bulking along the smooth chutes. The developed model deals with the solution of the one-way direction parabolic equations of mixture continuity, air mass, and air-water mixture momentum conservation. These equations are solved accompanied by the dynamic equation for the free surface, utilizing the marching technique and Prandtl's mixing length turbulent model. The experimental data obtained by prototype measurements and laboratory tests were used to assess the accuracy of the numerical model. The relevant results were compared in terms of the induced inception point of the boundary layer development, air concentration profiles within self-entrained flows, and the consequent bulking of the flow. The capability of the numerical model for practical purposes is signified in accordance with the fairly accurate obtained results, shedding light on new horizons for further research.

Full text

Numerical Modeling of Self-Aeration in High-Speed
Flows over Smooth Chute Spillways
Mohmmadreza Jalili Ghazizadeh1; Amir R. Zarrati2;
and Mohammad J. Ostad Mirza Tehrani3
Abstract: In chute spillways, self-aeration occurs downstream of the inception point, where the turbulent boundary layer edge approaches
the free surface, if they are long enough. Downstream of the inception point, a layer containing an air–water mixture extends gradually
through the flow with the bulking effect. Flow bulking is essential in terms of sidewall freeboard design. In addition, the introduction
of enough air quantity near the solid boundaries prevents cavitation damage. In the present work, a 2D numerical model was developed
for the prediction of self-aeration and air concentration profiles across the depth and the free-surface location, together with flow bulking
along the smooth chutes. The developed model deals with the solution of the one-way direction parabolic equations of mixture continuity, air
mass, and air–water mixture momentum conservation. These equations are solved accompanied by the dynamic equation for the free surface,
utilizing the marching technique and Prandtl’s mixing length turbulent model. The experimental data obtained by prototype measurements
and laboratory tests were used to assess the accuracy of the numerical model. The relevant results were compared in terms of the induced
inception point of the boundary layer development, air concentration profiles within self-entrained flows, and the consequent bulking of the
flow. The capability of the numerical model for practical purposes is signified in accordance with the fairly accurate obtained results, shedding
light on new horizons for further research. DOI: 10.1061/JHEND8.HYENG-12914.© 2022 American Society of Civil Engineers.
Author keywords: Smooth chute spillways; Inception point; Self-air entrainment; Free-surface flow; Flow bulking; Numerical modeling.
Introduction
Acceleration of flow down a chute spillway results in the develop-
ment of a boundary layer. At a location defined as the inception
point, bubble entrainment takes place where turbulence shear
stresses (destabilizing) are more significant than surface tension
and gravity (stabilizing) forces per unit area (Chanson 2009,
2013b;Valero and Bung 2016,2018;Zhang and Chanson 2017).
The self-aeration region starts downstream of the inception
point, where a layer containing a mixture of air and water is gradu-
ally extended through the fluid. The slow growth rate of the
air–water mixture layer leads to gradual variation of the air concen-
tration distribution along the spillway, approaching a uniform flow
condition at a sufficiently long distance. The adequate presence of
air near the solid boundary prevents cavitation damage. On the
other hand, air entrainment causes bulking of the flow and needs
to be considered when designing spillway and chute sidewalls
(Falvey 1980;Wood 1991).
Numerous studies have been conducted to specify the location
and flow properties of the inception point (Bauer 1954;Campbell
et al. 1965;Meireles et al. 2012;Wood et al. 1983). Some research-
ers defined the inception point based on visual observation where
air bubbles appear in the flow or intersection of the free surface and
the outer edge of the developing boundary layer (e.g., Amador et al.
2009;Meireles and Matos 2009;Meireles et al. 2012). Valero and
Bung (2018) proposed a new definition for prediction of the incep-
tion point, verified by observations of Cain (1978) on the Avie-
more dam.
Prediction of self-aeration and, consequently, the behavior of the
downstream air–water mixtures is essential to estimate the aerated
flow depth and prevent cavitation damage in hydraulic structures,
namely tunnels, smooth and/or stepped chutes (Falvey 1990).
Straub and Anderson (1958) were among the first researchers to
conduct experimental studies on self-aeration in supercritical flows.
Later on, extensive experimental investigations regarding self-
aeration and downstream flow bulking were conducted on smooth
invert chute spillways (Bung 2010;Cain 1978;Cain and Wood
1981;Chanson 1993,1996,2013a;Hager 1991;Matos et al. 2001;
Rao and Gangadharaiah 1971;Rao and Kobus 1971;Wilhelms and
Gulliver 2005;Wood 1991) and stepped chute spillways (Bung
2013;Chanson and Toombes 2001;Ostad Mirza et al. 2016,
2018;Pfister and Hager 2011). These studies investigated charac-
teristics of self-aerated flows such as air–water surface definition,
air concentration distribution, bubble characteristics, and flow
bulking. Nevertheless, reproduction of the naturally observed self-
aerated flows in physical models is challenging due to scale effects
(Chanson 2013a;Heller 2011). Moreover, the expensive instrumen-
tation required for measurements in air-water flows and its relevant
drawbacks is another issue (Borges et al. 2010;Felder and Pfister
2017;Jalili Ghazizadeh and Zarrati 2004;Kramer et al. 2020).
Therefore, numerical simulation could be an alternative for study-
ing self-aerated flows (Valero and García-Bartual 2016).
1Associate Professor, Faculty of Civil, Water and Environmental Engi-
neering, Shahid Beheshti Univ., Tehran 177651719, Iran (corresponding
author). ORCID: https://orcid.org/0000-0002-8242-7619. Email: m_jalili@
sbu.ac.ir
2Professor, Dept. of Civil and Environmental Engineering, Amirkabir
Univ. of Technology (Tehran Polytechnic), Tehran 1591634311, Iran.
ORCID: https://orcid.org/0000-0002-8483-3186. Email: zarrati@aut.ac.ir
3Assistant Professor, Faculty of Civil Engineering, K. N. Toosi Univ. of
Technology, Tehran 1996715433, Iran; formerly, Postdoctoral Research
Fellow, Dept. of Civil and Environmental Engineering, Amirkabir Univ. of
Technology (Tehran Polytechnic), Tehran 1591634311, Iran. ORCID: https://
orcid.org/0000-0002-5162-6332. Email: mohammad.tehrani@kntu.ac.ir
Note. This manuscript was submitted on May 15, 2021; approved on
September 30, 2022; published online on December 21, 2022. Discussion
period open until May 21, 2023; separate discussions must be submitted
for individual papers. This paper is part of the Journal of Hydraulic En-
gineering, © ASCE, ISSN 0733-9429.
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Computational fluid dynamics (CFD) has been applied to ana-
lyze self-aerated flows over spillways. As one of the first attempts,
the 2D Cartesian model of Hooping and Hoopes (1988) for sim-
ulation of self-aeration phenomena over smooth invert chutes
can be mentioned. André et al. (2003) and Sabbagh-Yazdi et al.
(2008) also developed a depth-averaged model for simulation of
self-aerated flows on stepped and smooth chutes, respectively.
They implemented an experimental equation for air concentration
distribution across flow depth. Numerous studies have also used
commercial software such as FLOW-3D (Flow Science, Inc., Santa
Fe, New Mexico) for the simulation of aerated flows with volume
of fluid (VOF) formulation to track the air–water free surface, in-
cluding self-aerated flow over both smooth inverts (Hirt 2016;
Hohermuth et al. 2021;Marofi et al. 2012) and stepped chutes
(Dong et al. 2019;Lopes et al. 2017;Valero and Bung 2015). Lopes
et al. (2017) developed a numerical model for studying self-aerated
flow over stepped spillways using Open FOAM, as an open-source
platform. They added to the VOF formulation an extra advection-
diffusion equation in conjunction with a source of air at the free
surface to facilitate simulation of the dispersed bubble phase in
self-aeration phenomena.
Despite some numerical studies on self-aerated flow, lack of val-
idation and verification is still an issue for air–water flow modeling
pointed out by several researchers (Chanson 2013a;Felder and
Chanson 2017;Ma et al. 2011;Valero and García-Bartual 2016).
In addition, 3D models are extremely time-consuming and require
experienced modelers and compelling and expensive computers,
especially in cases considering the prototype scale. Therefore, further
research on numerical air entrainment models is still required to pro-
vide robust and fast tools to analyze the characteristics of self-aerated
flows and bulking effects to be used by researchers and engineers.
In the present study, a 2D in-depth numerical model was devel-
oped to predict self-aeration and air concentration profiles within
the flow and the free-surface bulking along smooth chutes. The
model solves the one-way coordinate parabolic equations of mix-
ture continuity, air mass, and mixture momentum conservation,
together with the dynamic equation for defining the free surface.
Various available field and experimental data were then used to
verify the accuracy of the model.
Numerical Modeling of Air–Water Flow
Air and water in a mixture are discontinuous phases in space. The
continuum description may be adapted to distinguish air being
transported by the flow as bubbles (“entrained”air) and air trans-
ported with the flow in the roughness or waves of the water surface
(“entrapped”air) (Elghobashi et al. 1984;Wilhelms and Gulliver
2005). Since the components are discontinuous, in addition to con-
servation equations for each component, relationships are also re-
quired to describe the conservation of mass and momentum across
the interface separating the components (Elghobashi et al. 1984).
However, the spatial and temporal distribution of air and water is so
complex that it is almost impossible to solve these equations with-
out a computational grid containing sufficient nodal points to
resolve individual bubbles. Additional complexities are introduced
by bubble coalescence and breakup.
A macroscopic description of the mixture was proposed to ob-
tain a feasible solution. In this approach, the air and water compo-
nents of the flow are treated as continua that coexist everywhere in
space. At a particular given point, instantaneous properties for each
phase are obtained by averaging over a small space. However, this
process does not reduce the model’s validity because the average
flow properties are sufficient for all engineering purposes. For ex-
ample, in the case of air bubbles, the volume of bubbles in a small
mixture volume (i.e., air concentration) is mainly of interest.
The volume required for spatial averaging is assumed to be large
relative to individual bubbles but small compared to the overall do-
main. Hence, the continuum is macroscopic as far as the bubbles
are concerned, while regarded as microscopic for the flow problem
as a whole. To this end, it was assumed that air in a specified
volume is distributed in small bubbles (Hooping and Hoopes
1988).
Governing Equations
The governing equations describing the instantaneous mass and
momentum distribution in water flow may be derived by assuming
a small elementary control volume and expressing the conservation
of mass and momentum within that space. It is also assumed that air
bubbles are small and evenly distributed, and components of air and
water are treated as continua that coexist everywhere in space.
Therefore, the conservation of air mass and mixture flow is written
as follows, respectively (Jha and Bombardelli 2010;Soo 1967;
Zarrati 1994):
∂
∂tðρφÞþ ∂
∂xiðρφuaiÞ¼0ð1Þ
∂ρ
∂tþ∂
∂xiðρuiÞ¼0ð2Þ
where ρ= mixture density; and uai, and ui(i¼1, 2, 3) = compo-
nents of air and mixture velocities, respectively (Fig. 1).
Air concentration cand air mass fraction ϕfor the small elemen-
tary control volume are defined as
c¼ΔVa
ΔVð3Þ
φ¼ΔMa
ΔMð4Þ
ρφ ¼ρacð5Þ
ρ¼ρw
1þφðρw=ρa−1Þð6Þ
where ΔVaand ΔMa= volume and mass of air; ΔVand ΔM=
volume and mass of the control volume, respectively; and ρaand
ρw= air and water density, respectively.
The air and water velocity components may be defined based
upon the mixture velocity as
uai ¼uiþð1−φÞusi ð7Þ
Averaging Volume
(1 )
iai wi
uu u
ai
u
si ai wi
uuu
wi i
uu
wi
u
Fig. 1. Definition of the corresponding velocity components.
(Reprinted from Hooping and Hoopes 1988, © ASCE.)
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uwi ¼ui−φusi ð8Þ
where usi = slip velocity component of air relative to water. It is
worth mentioning that the value of φis negligible for air concen-
trations less than 90%; thus, water velocity may be assumed as
being equal to the mixture flow velocity.
The momentum conservation for air–water mixture is written
(Soo 1967)as
∂ðρuiÞ
∂tþ∂ðρuiujÞ
∂xj¼−∂p
∂xiþμð∂2uiÞ
∂x2
jþρgið9Þ
where gi= component of gravitation acceleration; μ= dynamic
viscosity of water; and p= pressure.
Considering a 2D flow in the streamwise and normalwise direc-
tions, ui,usi,uai ,ρ,φ, and pare unknowns in this set of equations.
These unknown parameters can be calculated using Eqs. (1), (2),
(6), (7), and (9), and the slip velocity of air relative to water (usi )is
estimated from the available experimental data (bubble rise veloc-
ity), as will be explained in the Bubble Slip Velocity in the Numeri-
cal Model section. The bubble slip velocity component can be
neglected compared to the flow velocity in the longitudinal flow
direction.
Free-Surface Modeling
For finding the location of the free surface, the method of body-
fitted coordinate with variable mesh height was utilized, which
has also been used in some previous studies (Casulli and Walters
2000;Stansby and Zhou 1998). This method deals with the appli-
cation of the kinematic condition of the free surface and the Leibniz
rule and integration of the continuity equation over the flow depth
(Casulli and Walters 2000;Vreugdenhil 1994). By considering the
density of air–water flow as a variable parameter and steady-state
condition, one can write (Morovvat et al. 2021)
∂
∂xZη
yo
ρudy −ρu∂y
∂xη
yoþZη
yo
∂
∂yðρvÞdy ¼0ð10Þ
where yoand η= bottom and free-surface elevations with respect to
a datum that is assumed parallel to the channel bed, respectively;
and uand v= components of the mixture velocity. Eq. (10) can be
simplified to
∂
∂xZη
yo
ρudy −ðρuÞη
∂η
∂xþðρvÞη¼0ð11Þ
Noting that ∂η=∂xis the water surface slope and that velocity
vector at the free surface is parallel to the free surface, it is obvious
that the last two terms cancel each other. In the air–water mixture,
the bulking effect is included in calculating the water surface
elevation.
Time-Averaged Flow Equations
The Reynolds-averaged Navier–Stokes equations (RANS) are
the most widely used approach to model turbulence fluctuations.
Adopted after Reynolds (1895), the instantaneous dependent
variables are assumed to equal the sum of time-averaged and
fluctuating components. Time-averaging is performed by two ap-
proaches: conventional and mass-weighted (Jones 1979). In con-
ventional averaging, an instantaneous value such as uis defined
(Jones 1979)as
u¼¯
uþu0;where ¯
u¼1
tZtoþt
to
udt ð12Þ
The averaging time, t, is long compared with the time scale of
the turbulent motion. The notation used for the mass-weighted
averaging (Jones and McGuirk 1979)is
u¼~
uþu00;where ~
u¼ρu
¯
ρð13Þ
The reader may note that the overbar signs imply conventional
time averaging. Mass-weighted averaging reduces the number of
unknown second-order correlations in terms of those having a prod-
uct including density (Jones 1979). Conventional averaging is used
for the rest of the terms, (i.e., pressure and density). According to
Jones and McGuirk (1979), the implementation of the time-average
method on Eqs. (1) and (2), together with considering a steady flow
yield the following.
For the conservation of the air mass (Jones and McGuirk 1979)
∂
∂xð¯
ρ~
ua~
φÞþ ∂
∂yð¯
ρ~
va~
φÞ¼−∂
∂xð¯
ρu00φ00Þ−∂
∂yð¯
ρv00φ00Þð14Þ
Substituting the air velocity components from Eq. (7) yields
∂
∂xð¯
ρ~
u~
φÞþ ∂
∂yð¯
ρ~
v~
φÞ¼−∂
∂xð¯
ρu00φ00Þ−∂
∂yð¯
ρv00φ00Þ
−∂
∂xð¯
ρusð1−¯
φÞ¯
φÞ
−∂
∂yð¯
ρvsð1−¯
φÞ¯
φÞð15Þ
where usand vs= components of the slip velocity in the streamwise
and depthwise directions, respectively. The fluctuations of slip
velocity (u0
sand ν0
s) are neglected to simplify the model.
Conservation of mixture flow (continuity) is represented by
∂
∂xð¯
ρ~
uÞþ ∂
∂yð¯
ρ~
vÞ¼0ð16Þ
and the momentum equations [Eq. (9)] take the form (Jones and
McGuirk 1979)of
∂
∂xð¯
ρ~
u~
uÞþ ∂
∂yð¯
ρ~
v~
uÞ¼−∂p
∂x−∂
∂x¯
ρu00u00 −μ∂u
∂x
−∂
∂y¯
ρu00v00 −μ∂u
∂yþ¯
ρgxð17Þ
∂
∂xð¯
ρ~
u~
vÞþ ∂
∂yð¯
ρ~
v~
vÞ¼−∂p
∂y−∂
∂x¯
ρu00v00 −μ∂v
∂x
−∂
∂y¯
ρv00v00 −μ∂v
∂yþ¯
ρgyð18Þ
Also, averaging the free-surface equation [Eq. (11)] is written as
follows:
∂
∂xZη
yo
~
ρ~
udy ¼0ð19Þ
Turbulence Model
A turbulence model, by which the unknown fluctuating correlations
are approximated in terms of the mean flow quantities, has to be
introduced to close the system. Among many different turbulence
models suggested in the literature, Prandtl’s mixing length model,
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which is based on the Boussinesq hypothesis, is used herein. This
model has been successfully used for various applications in open-
channel flows and boundary layer development (Czernuszenko and
Rylov 2000;Namin et al. 2001;Nezu and Nakagawa 1993;Zhang
and Chanson 2017). It should be noted that, for more simplicity, the
air–water flow is assumed as continua coexisting everywhere in
space, similar to a fluid with variable density; therefore, using
the usual turbulence models are justified.
Based on this model, the kinematic eddy viscosity υtis com-
puted as
υt¼μt
ρ¼L2
mJð20Þ
where μt= dynamic eddy viscosity; (Lm) = Prandtl’s mixing
length; and Jis defined (Nezu and Nakagawa 1993) as follows:
J¼2∂u
∂x2
þ2∂v
∂y2
þ∂u
∂yþ∂v
∂x20.5
ð21Þ
In the present study, the Nikuradse equation is used for Lm
(Beattie 1972;Czernuszenko and Rylov 2000)as
Lm
h¼0.14 −0.081−y
h2
−0.061−y
h4
ð22Þ
where h= flow depth; and y= coordinate perpendicular to the flow
direction.
Solving the Parabolic Equations by Using the
Marching Technique
In the present study, the high-speed supercritical air–water flow
over smooth invert chutes is assumed as a one-way coordinate
parabolic flow. Under such flow conditions, diffusion can be ne-
glected in the streamwise direction (Patankar 1980;Towne and
Colonius 2015). However, the diffusion term cannot be neglected
perpendicular to the flow direction, representing two-way coordi-
nate ellipsoidal flow equations. Accordingly, the solution of the
one-way coordinate parabolic flow domain could be performed
by marching integration due to the independence of the down-
stream flow condition (Patankar and Spalding 1972). The mesh size
in both streamwise and depthwise directions was refined to achieve
mesh-independent results.
Sigma Coordinates System
In the present study, due to the variation of the free surface, and
considering that free-surface location is not known from the begin-
ning, the σcoordinates system (Gross et al. 1998;Stansby and
Zhou 1998) was employed based on the following transformation
rules:
∂
∂x¼∂
∂xσ
−1
hσ∂h
∂xσþ∂η
∂xσ∂
∂σð23Þ
∂
∂y¼1
h
∂
∂σð24Þ
where σ¼ðy−ηÞ=h. Accordingly, the transformed air mass con-
servation [Eq. (15)], mixture continuity [Eq. (16)], and mixture
momentum [Eq. (17)] in σcoordinates system considering the tur-
bulence model, neglecting all over bar signs for simplicity (Jalili
Ghazizadeh 2003), are as follows.
Air mass conservation is
∂ðhρuφÞ
∂x−∂ðσρuφ∂h
∂xÞ
∂σ−∂ðρuφ∂η
∂xÞ
∂σþ∂ðρφvÞ
∂σ
−∂
∂σμt
h
∂φ
∂σþ∂ðρvsφÞ
∂σ¼0ð25Þ
Mixture continuity is
∂ðhρuÞ
∂x−∂ðσρu∂h
∂xÞ
∂σ−∂ðρu∂η
∂xÞ
∂σþ∂ðρvÞ
∂σ¼0ð26Þ
Mixture momentum in x-direction is
∂ðhρuuÞ
∂x−∂ðσρuu ∂h
∂xÞ
∂σ−∂ðρuu ∂η
∂xÞ
∂σþ∂ðρuvÞ
∂σþ∂ðhpÞ
∂x
−∂ðσp∂h
∂xÞ
∂σ−∂ðp∂η
∂xÞ
∂σ−∂
∂σμt
h
∂u
∂σþρhgx¼0ð27Þ
It should be noted that the distribution of pressure in depth is
assumed to be hydrostatic; therefore, the momentum equation in
the depthwise direction [Eq. (18)] is not solved. However, the flow
velocity in the depthwise direction is calculated according to the
mixture continuity equation [Eq. (26)]. In the presence of air in
the flow, the pressure in depth is assumed hydrostatic, for example,
pressure at the bed may be obtained (Chanson 1996)as
p¼gh cos θZ0
−1
ρdσð28Þ
where θ= channel slope.
The dynamic equation for defining the free surface along the
steady-state flow is
∂
∂xZ0
−1
hρudσ−∂
∂σσ∂h
∂xZ0
−1
hρudσ
−∂
∂σ∂η
∂xZ0
−1
hρudσ¼0ð29Þ
Finally, Jin Prandtl’s mixing length turbulent model [Eq. (21)]
is defined (Jalili Ghazizadeh 2003)as
J¼2∂u
∂x−1
hσ∂h
∂xþ∂η
∂x∂u
∂σ2
þ21
h
∂v
∂σ2
þ1
h
∂u
∂σþ∂v
∂x−1
hσ∂h
∂xþ∂η
∂x∂v
∂σ20.5
ð30Þ
Boundary Conditions
Free-Surface Boundary Condition with Considering
Self-Aeration
Air entrainment from the free surface needs to be simulated after
the inception point. Thereupon, in the vicinity of the inception
point, the following boundary conditions are applied at the free
surface:
∂u
∂y¼0
v¼0
vs¼0
φ¼φair-reservoir ð31Þ
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where φair-reservoir = air mass fraction at the uppermost cell across
depth, which is set to 0.00892, corresponding to C¼90% follow-
ing Boes and Hager (2003), Chanson (1997), and Wood (1991).
The air concentration introduced at the uppermost cell center is
then diffused and transported downstream due to the convection
and diffusion mechanism. On the other hand, the bubble buoyancy
forces push the bubbles toward the free surface. The air entrainment
continues until the resultant turbulent forces and the bubble buoy-
ancy forces are equal.
Bed Boundary Condition
The so-called wall law is implemented to deal with high-velocity
gradients near the solid wall (Launder and Spalding 1974;Vo n
Karman 1930). Based on the law of the wall function, the bottom
parallel velocity is defined (Schlichting 1979)as
uB
u¼1
κln9.05uy
υð32Þ
if uBks
υ<5ða smooth hydraulic bedÞ
and
uB
u¼1
κln30y
ksð33Þ
if uks
υ>70 ða rough hydraulic bedÞ
where uB= velocity at a near-wall node; u= bed shear velocity;
κ= Von Karman constant coefficient (=0.4); υ= kinematic viscos-
ity of water; and ks= roughness height. Because uBand uare
codependent, an iterative procedure was carried out in the present
model for calculation of u(Jalili Ghazizadeh 2003).
Upstream and Downstream Boundary Conditions
The velocity profile for the water phase was used at the upstream
section. Due to the parabolic domain of the flow, no downstream
condition was necessary, implying independence from the down-
stream flow condition.
Bubble Slip Velocity in the Numerical Model
The slip velocity of bubbles in the flow is needed to develop the
numerical model. Air rising velocity is considered the velocity of
air bubbles in the gravitational direction (vr), while slip velocities
are assumed to be components of rising velocity in computational
coordinates. Bubble rise velocity in an air–water mixture flow de-
pends on the size of the bubbles and air concentration (Chanson
1996). According to the measurements conducted on the Aviemore
dam chute, for air concentration ranging between 4% and 20%, the
corresponding bubble size varied between 0.5 and 3 mm, as re-
ported by Cain and Wood (1981) and Chanson (1994).
On the other hand, the air bubbles’size mostly depends on air
concentration and flow conditions. Different studies have been
carried out for measurement of the rising velocity of air bubbles
in various situations such as still (Haberman and Morton 1956) and
turbulent (Kramer and Hager 2005;Tekeli and Maxwell 1978;
Zarrati and Hardwick 1991) water flows. In the present study,
the assumed bubble rise velocity is selected based on these refer-
ences, summarized in Table 1. The value of rising velocity was
obtained by linear interpolation for the other ranges of air concen-
tration in between. Moreover, the rising velocity of a bubble in a
cloud of bubbles is less than a single bubble. Following (Chanson
1996), the rising velocity given in Table 1was therefore corrected
by multiplying it to ffiffiffiffiffiffiffiffiffiffiffi
1−c
p.
Algorithm of Computation
An iterative algorithm should be followed to solve the governing
equations. The following steps are used:
1. Knowing all variables in an upstream section, the governing
equations are solved for the next section downstream, assuming
all variables are the same as the upstream section for the first
step.
2. The mixture continuity equation [Eq. (26)] is solved for the
velocity in the depthwise direction (v).
3. The momentum equation [Eq. (27)] is solved to calculate
streamwise velocities (u). In this step, an internal loop is imple-
mented to satisfy the bed boundary condition defined in the Bed
Boundary Condition section.
4. Turbulent viscosity is calculated by solving Eq. (20).
5. If boundary layer depth approaches the free surface, the incep-
tion point is recognized; from this section, an air concentration
of 90% is set at the uppermost cell center of the depth.
6. After the inception point, the air mass conservation [Eq. (25)] is
solved to calculate φvalues. The new ρvalues are calculated
by Eq. (6).
7. The flow depth (h) is updated by integration of the dynamic
equation [Eq. (29)].
8. Calculation is repeated from step 2 to reach convergence in the
new section.
9. Calculation proceeds to the next section.
Numerical Model Verification
The numerical model was tested with available data for both labo-
ratory experiments and field measurements as follows:
1. (a) Velocity profiles (Keller 1972); and (b) boundary layer de-
velopment in turbulent flow over the Aviemore dam (Cain and
Wood 1981).
2. Air concentration and water surface profiles, including bulking
effect based on the field measurements of the Aviemore dam
(Cain 1978;Cain and Wood 1981).
3. The self-aeration laboratory data measured by Straub and
Anderson (1958).
Fields Measurements of Aviemore Dam
Different series of prototype measurements were conducted on the
Aviemore dam chute spillway. The Aviemore chute spillway is ini-
tiated by a radial gate and an ogee profile and extends to a 45° steep
chute. Data analyses for different field measurements are pre-
sented below.
Boundary-Layer Development
Velocity and free-surface profiles were measured upstream of the
inception point in sections 401 to 403, as shown in Fig. 2. These
data were reported by (Keller 1972).
Table 1. Rising velocity of bubbles with different sizes
C(%) db(mm) vr(m=s) Comments
4≤C≤15 0.5≤db≤20.06 ≤vr≤0.25 —
C>15 db>2vr¼0.25 Average of 0.2 and 0.3
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Principally, the thickness of the boundary layer is equal to the
distance from the channel invert up to the point where the velocity
is 0.99 times the maximum velocity (surface velocity). The inter-
section location of the boundary layer and the free surface are pre-
sented in Fig. 2. Furthermore, the measured values are compared
with those calculated by the present model. Because the gradient
of longitudinal velocity is assumed to be zero at the free surface
[Eq. (31)], the boundary layer edge near the water surface had
to be extrapolated from the developed boundary layer thickness line
to distinguish its intersection with the free surface (Fig. 2). Given
the experimental uncertainties that may have affected the accuracy
of the reported inception point, the results predicted by the model
are judged to be acceptable.
The measured free surface upstream of the inception point is
also plotted versus the numerical predicted free surface in Fig. 2.
In the numerical model, the upstream velocity profile is set as
uniform. The predicted free surface is also compatible with the
measurements.
Comparison of the measured velocity profile at station 403 ver-
sus the profile calculated by the present model is shown in Fig. 3
(Keller 1972). As seen, the profiles are fairly compatible. In addi-
tion, conformity of the numerical results with the experimental and
field data was similarly evident in different sections as well.
Self-Aerated Flow over Aviemore Dam
Cain (1978) also carried out another series of prototype measure-
ments on the Aviemore dam chute in regard with the free-surface
longitudinal profile and air concentration distribution in-depth
downstream of the inception point along the spillway chute (Cain
1978;Cain and Wood 1981).
Cain and Wood (1981) examined two 0.3- and 0.45-m gate
openings, as described in Table 2. The flow depth and air concen-
tration profiles for each opening were measured along five succes-
sive longitudinal cross sections, sections 501 to 505, with 6.1-m
spacing downstream of the inception point. According to Cain
and Wood (1981), the flow depth decreased gradually downstream
of the gate, toward a normal depth (inducing S2profile), until the
inception point and increased after that due to air entrainment and
bulking effect. The flow characteristics in these measurements
are presented in Table 2for different gate openings. In this table,
qwis the unit flow discharge, Lið501Þis the distance of the inception
point to cross-section 501, and hiis the flow depth at the inception
point. In Figs. 4(a–e) and 5(a–e), air concentration distributions,
calculated by the present numerical model, are depicted versus
the prototype measurements corresponding to the gate opening of
0.45 and 0.30 m, respectively (Cross sections 501 to 505). The air
Fig. 3. Velocity profile at section 403 for the prototype measurements
and the numerical model.
Table 2. Flow condition on Aviemore dam
Gate opening(m) qw(m2=s) Lið501Þ(m) hi(m)
0.45 3.15 2.3 0.194
0.30 2.23 7.7 0.152
Source: Data from Cain and Wood (1981).
Extrapolated
x
(m)
Fig. 2. Free-surface profile and boundary layer development obtained from the prototype measurements compared to the numerical model.
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(a) (b)
(c) (d)
(e)
Fig. 4. Comparison of the computed air concentration profiles with the prototype measurements data for the gate opening of 0.45 m: (a) through
(e) correspond to cross sections 501 to 505, respectively.
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(a) (b)
(c) (d)
(e)
Fig. 5. Comparison of the computed air concentration profiles with the prototype measurements data for the gate opening of 0.3 m: (a) through
(e) correspond to sections 501 to 505, respectively.
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concentration distribution profiles fit satisfactorily in all cross sec-
tions, implying that the present numerical model is favorably quali-
fied to predict air entrainment.
Figs. 6(a and b) show the free-surface profile along the chute,
plotted for 0.45-and 0.3-m gate openings, respectively. As seen,
flow depth is found to increase downstream of the inception point
due to the air entrainment and bulking effect. The nonaerated flow
depth is also plotted in Figs. 6(a and b), illustrating the flow bulking
associated with air entrainment.
Figs. 4–6show the predicted air concentration profiles and free-
surface development (particularly the flow bulking), corresponding
acceptably to the measured data in the prototype. Nevertheless,
as observed, the prototype measurements show some undulation,
which is not far from expectation due to the difficulty regarding
prototype measurements in high-speed flows.
Self-Aerated Flow over a Physical Model
These tests were conducted in a 15-m long channel with a variable
slope of 22.5° and 45°, with 0.46-m width and 0.31-m sidewall
height. The flow velocity and air concentration profiles were
measured at the equilibrium air–water region (at which air entrain-
ment and detrainment are equal) by a Pitot tube and an electrical
air-concentration measuring instrument, respectively (Straub and
Anderson 1958). The test program included the unit discharges
0.13 m2=s<qw<0.93 m2=s.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 10 15 20 25 30 35 40
y (m)
x (m)
Measured Data (Gate Opening 450 mm) Nonaerated Flow Model Prediction
Observed
inception
Station 501--->
Station 502--->
Station 503--->
Station 504--->
Station 505--->
Bulking
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 10 15 20 25 30 35 40
y (m)
x (m)
Measured Data (Gate Opening 300 mm) Nonaerated Flow Model Prediction
Bulking
Observed
inception
Station 501--->
Station 502--->
Station 503--->
Station 504--->
Station 505--->
(a)
(b)
Fig. 6. Free-surface profile: (a) gate opening of 0.45 m; and (b) gate opening of 0.30 m.
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The experimental results (i.e., equilibrium air concentration
profiles at which entrainment and detrainment into the flow are
equal) are plotted in Fig. 7, along with the numerically predicted
plots for unit discharge of qw¼0.32 m2=s in 22.5° and 0.93 m2=s
in 45° sloping chute, respectively. Even though the experimental
and numerical air concentration profiles coincide well on the
22.5° chute slope [Fig. 7(a)], the numerical model underestimates
the air concentration close to the channel bed for the case of flow
over the 45° chute slope. However, the profile trend is predicted
correctly [Fig. 7(b)].
Summary and Conclusions
An in-depth 2D numerical model was developed to simulate the
air–water flow along steep smooth channels or chutes. Assuming
air and water as a continuum, the parabolic equations of mixture
continuity, air mass, and mixture momentum conservation were
solved using a marching algorithm along with Prandtl’s mixing
length turbulent model. A dynamic equation, together with the
marching technique, were utilized for defining the free-surface
profile.
The developed numerical model is capable of predicting the lo-
cation of the free-surface profile along the chutes and analysis of
the flow domain accordingly, including the boundary layer devel-
opment, velocity, and air concentration profiles. One unique feature
of this model is the simulation of self-aeration phenomena consid-
ering the bulking effect. The presented model requires minimal
computational time and memory because of the marching tech-
nique (in prototype scale, in order of minutes compared to a few
days for a 3D model with millions of mesh). Considering the ap-
plication of the σcoordinate system, the presented model is highly
competent in dealing with the water surface variation.
A comparison of the numerical results with the laboratory data
(Straub and Anderson 1958) as well as results of measurements
on prototypes in different flow conditions (Cain 1978;Cain and
Wood 1981;Keller 1972) reveal the capabilities and accuracy of
this model in prediction of air–water flow behavior for practical
purposes.
Data Availability Statement
The code generated and used during the present study is available in
a repository online (https://github.com/Jalili47/ChuteAer) in accor-
dance with funder data retention policies.
Acknowledgments
The third author would like to acknowledge the Iranian National
Elites Foundation for financial support (Grant INEF-AUT 20/
142/2017).
Notation
The following symbols are used in this paper:
c= volumetric air concentration;
g= gravitation acceleration;
h= flow depth;
ks= roughness height;
Lm= Prandtl’s mixing length;
M= mass;
p = pressure;
t = time;
u = flow velocity in x direction;
uai = air velocity component;
uB= bottom parallel velocity at near-wall;
ui= mixture velocity;
us= slip velocity in stream-wise direction;
usi = slip velocity component;
uwi = water velocity component;
u= shear velocity;
v= flow velocity in depthwise direction;
vr= rising velocity in gravitational direction;
vs= slip velocity in direction;
x= streamwise coordinate;
y= vertical axis coordinate;
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 102030405060708090100
Depth (m)
Air Concentration (%)
Measured Data (Straub and Anderson, 1958)
Model Prediction
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 102030405060708090100
Depth (m)
Air Concentration (%)
Measured Data (Straub and Anderson, 1958)
Model Prediction
(a) (b)
Fig. 7. Air concentration profiles across depth for the laboratory sloping chute. Comparison between the present numerical model and the experi-
mental data for: (a) θ¼22.5°; and (b) θ¼45°.
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yo= bottom elevations of the flow;
ΔV= volume of the control volume;
ΔVa= air volume of the control volume;
ΔM= mass of the control volume;
ΔMa= air mass of the control volume;
η= free-surface elevation of the flow respect to datum;
κ= Von Karman constant coefficient;
φ= air mass fraction;
φair-reservoir = air mass fraction at the free surface;
ρ= mixture density;
ρa= air density;
ρw= water density;
σ= depthwise axis in σ-coordinates;
θ= slope of channel;
υ= kinematic viscosity of water;
υt= kinematic eddy viscosity;
(¯) = depth-averaged;
(˜) = density-averaged;
(′) = fluctuation term in depth averaging; and
(″) = fluctuation term in density averaging.
References
Amador, A., M. Sánchez-Juny, and J. Dolz. 2009. “Developing flow region
and pressure fluctuations on steeply sloping stepped spillways.”
J. Hydraul. Eng. 135 (12): 1092–1100. https://doi.org/10.1061/(ASCE)
HY.1943-7900.0000118.
André, S., B. Dewals, M. Pirotton, and A. Schleiss. 2003. “Quasi
2D-numerical model of aerated flow over stepped chutes.”In Proc.,
30th IAHR Congress, 671–678. Hague, Netherlands: International
Association for Hydro-Environment Engineering and Research.
Bauer, W. J. 1954. “Turbulent boundary layer on steep slopes.”Trans. Am.
Soc. Civ. Eng. 119 (1): 1212–1233. https://doi.org/10.1061/TACEAT
.0006971.
Beattie, D. R. H. 1972. “Two-phase flow structure and mixing length
theory.”Nucl. Eng. Des. 21 (1): 46–64. https://doi.org/10.1016/0029
-5493(72)90085-4.
Boes, R. M., and W. H. Hager. 2003. “Two-phase flow characteristics of
stepped spillways.”J. Hydraul. Eng. 129 (9): 661–670. https://doi.org
/10.1061/(ASCE)0733-9429(2003)129:9(661).
Borges, J. E., N. H. C. Pereira, J. Matos, and K. H. Frizell. 2010. “Perfor-
mance of a combined three-hole conductivity probe for void fraction
and velocity measurement in air–water flows.”Exp. Fluids 48 (1):
17–31. https://doi.org/10.1007/s00348-009-0699-1.
Bung, D. B. 2010. “A comparative study of self-aerated stepped spillway
and smooth invert chute flow: The effect of step-induced macro-
roughness.”In Proc., Chinese-German Joint Symp. on Hydraulic
and Ocean Engineering, 451–456. Stuttgart, Germany: Eigenverlag.
Bung, D. B. 2013. “Non-intrusive detection of air–water surface roughness
in self-aerated chute flows.”J. Hydraul. Res. 51 (3): 322–329. https://
doi.org/10.1080/00221686.2013.777373.
Cain, P. 1978. Measurements within self-aerated flow on a large spillway.
Christchurch, New Zealand: Univ. of Canterbury.
Cain, P., and I. R. Wood. 1981. “Measurements of self-aerated flow on
spillways.”J. Hydraul. Div. 107 (11): 1425–1444. https://doi.org/10
.1061/JYCEAJ.0005761.
Campbell, F. B., R. G. Cox, and M. B. Boyd. 1965. “Boundary layer de-
velopment and spillway energy loss.”J. Hydraul. Div. 91 (3): 149–163.
https://doi.org/10.1061/JYCEAJ.0001235.
Casulli, V., and R. A. Walters. 2000. “An unstructured grid, three-
dimensional model based on the shallow water equations.”Int. J. Nu-
mer. Methods Fluids 32 (3): 331–348. https://doi.org/10.1002/(SICI)
1097-0363(20000215)32:3<331::AID-FLD941>3.0.CO;2-C.
Chanson, H. 1993. “Self-aerated flows on chutes and spillways.”
J. Hydraul. Eng. 119 (2): 220–243. https://doi.org/10.1061/(ASCE)
0733-9429(1993)119:2(220).
Chanson, H. 1994. “Closure to ‘self-aerated flows on chutes and spillways’
by h. Chanson (February 1993, vol. 119, no. 2).”J. Hydraul. Eng.
120 (6): 779–782. https://doi.org/10.1061/(ASCE)0733-9429(1994)
120:6(779).
Chanson, H. 1996. Air bubble entrainment in free surface turbulent shear
flows. New York: Academic Press.
Chanson, H. 1997. “Air bubble entrainment in open channels: Flow
structure and bubble size distributions.”Int. J. Multiphase Flow 23 (1):
193–203. https://doi.org/10.1016/S0301-9322(96)00063-8.
Chanson, H. 2009. “Turbulent air–water flows in hydraulic structures:
Dynamic similarity and scale effects.”Environ. Fluid Mech. 9 (2):
125–142. https://doi.org/10.1007/s10652-008-9078-3.
Chanson, H. 2013a. “
Advective diffusion of air bubbles in turbulent water
flows.”In Fluid mechanics of environmental interfaces, edited by C. D.
GualtieriMihailovic and D. Mihailovic. Leiden, Netherlands: Taylor &
Francis.
Chanson, H. 2013b. “Hydraulics of aerated flows: Qui pro quo?”J. Hy-
draul. Res. 51 (3): 223–243. https://doi.org/10.1080/00221686.2013
.795917.
Chanson, H., and L. Toombes. 2001. “Strong interactions between free-
surface aeration and turbulence down a staircase channel.”In Proc.,
14th Australasian Fluid Mechanics Conf.,10–14. Adelaide, Australia:
Adelaide Univ.
Czernuszenko, W., and A. A. Rylov. 2000. “A generalisation of Prandtl’s
model for 3D open channel flows.”J. Hydraul. Res. 38 (2): 133–139.
https://doi.org/10.1080/00221680009498348.
Dong, Z., J. Wang, D. F. Vetsch, R. M. Boes, and G. Tan. 2019. “Numerical
simulation of air–water two-phase flow on stepped spillways behind x-
shaped flaring gate piers under very high unit discharge.”Water 11 (10):
1956. https://doi.org/10.3390/w11101956.
Elghobashi, S., T. Abou-Arab, M. Rizk, and A. Mostafa. 1984. “Prediction
of the particle-laden jet with a two-equation turbulence model.”Int. J.
Multiphase Flow 10 (6): 697–710. https://doi.org/10.1016/0301-9322
(84)90006-5.
Falvey, H. T. 1980. Air-water flow in hydraulic structures, engineering
monograph no. 41. Washington, DC: US Department of the Interior.
Falvey, H. T. 1990. Cavitation in chutes and spillways, engineering mono-
graph no. 42. Washington, DC: US Department of the Interior.
Felder, S., and H. Chanson. 2017. “Scale effects in microscopic air-water
flow properties in high-velocity free-surface flows.”Exp. Therm. Fluid
Sci. 83 (4): 19–36. https://doi.org/10.1016/j.expthermflusci.2016.12
.009.
Felder, S., and M. Pfister. 2017. “Comparative analyses of phase-detective
intrusive probes in high-velocity air–water flows.”Int. J. Multiphase
Flow 90 (Dec): 88–101. https://doi.org/10.1016/j.ijmultiphaseflow
.2016.12.009.
Gross, E. S., V. Casulli, L. Bonaventura, and J. R. Koseff. 1998. “A semi-
implicit method for vertical transport in multidimensional models.”Int.
J. Numer. Methods Fluids 28 (1): 157–186. https://doi.org/10.1002
/(SICI)1097-0363(19980715)28:1<157::AID-FLD144>3.0.CO;2-U.
Haberman, W. L., and R. K. Morton. 1956. “An experimental study of bub-
bles moving in liquids.”Trans. Am. Soc. Civ. Eng. 121 (1): 227–250.
https://doi.org/10.1061/TACEAT.0007317.
Hager, W. H. 1991. “Uniform aerated chute flow.”J. Hydraul. Eng. 117 (4):
528–533. https://doi.org/10.1061/(ASCE)0733-9429(1991)117:4(528).
Heller, V. 2011. “Scale effects in physical hydraulic engineering models.”
J. Hydraul. Res. 49 (3): 293–306. https://doi.org/10.1080/00221686
.2011.578914.
Hirt, C. W. 2016. Dynamic droplet sizes for drift fluxes. Santa Fe, NM:
Flow Science, Inc.
Hohermuth, B., L. Schmocker, R. M. Boes, and D. F. Vetsch. 2021.
“Numerical simulation of air entrainment in uniform chute flow.”J. Hy-
draul. Res. 59 (3): 378–391. https://doi.org/10.1080/00221686.2020
.1780492.
Hooping, P. N., and J. A. Hoopes. 1988. “Development of a numerical
model to predict the behavior of air-water mixture in open channels.”
© ASCE 04022042-11 J. Hydraul. Eng.
J. Hydraul. Eng., 2023, 149(3): 04022042
Downloaded from ascelibrary.org by Karadeniz Technical University on 12/26/22. Copyright ASCE. For personal use only; all rights reserved.

In Proc., Model-Prototype Correlation of Hydraulic Structures:
Proceedings of the Int. Symp. Reston, VA: ASCE.
Jalili Ghazizadeh, M. 2003. “Experimental investigation and development
of a mathematical model for analysis of air-water flow on chute spill-
ways.”Ph.D. thesis, Dept. of Civil and Environmental Engineering,
Amirkabir Univ. of Technology.
Jalili Ghazizadeh, M., and A. R. Zarrati. 2004. “Development and calibra-
tion of a resistivity probe for measurement of air concentration and
bubble count in high-speed air-water flows.”Scientia Iranica 11 (4):
312–319.
Jha, S. K., and F. A. Bombardelli. 2010. “Toward two-phase flow modeling
of nondilute sediment transport in open channels.”J. Geophys. Res.
115 (3): F03015.
Jones, W. P. 1979. Models for turbulent flows with variable density and
combustion. Sint-Genesius-Rode, Belgium: Von-Karman Institute for
Fluid Dynamics.
Jones, W. P., and J. J. McGuirk. 1979. Mathematical modelling of
gas-turbine combustion chambers AGARD-CP-275. Brussels, Belgium:
North Atlantic Treaty Organization.
Keller, R. J. 1972. Field measurement of self-aerated high speed open
channel flow. Christchurch, New Zealand: Univ. of Canterbury.
Kramer, K., and W. H. Hager. 2005. “Air transport in chute flows.”
Int. J. Multiphase Flow 31 (10): 1181–1197. https://doi.org/10.1016/j
.ijmultiphaseflow.2005.06.006.
Kramer, M., B. Hohermuth, D. Valero, and S. Felder. 2020. “Best practices
for velocity estimations in highly aerated flows with dual-tip phase-
detection probes.”Int. J. Multiphase Flow 126 (4): 103228. https://doi
.org/10.1016/j.ijmultiphaseflow.2020.103228.
Launder, B. E., and D. B. Spalding. 1974. “The numerical computation of
turbulent flows.”Comput. Methods Appl. Mech. Eng. 3 (2): 269–289.
https://doi.org/10.1016/0045-7825(74)90029-2.
Lopes, P., J. Leandro, and R. F. Carvalho. 2017. “Self-aeration modell-
ing using a sub-grid volume-of-fluid model.”Int. J. Nonlinear Sci.
Numer. Simul. 18 (7–8): 559–574. https://doi.org/10.1515/ijnsns
-2017-0015.
Ma, J., A. A. Oberai, D. A. Drew, R. T. Lahey, and M. C. Hyman. 2011.
“A comprehensive sub-grid air entrainment model for rans modeling of
free-surface bubbly flows.”J. Comput. Multiphase Flows 3 (1): 41–56.
https://doi.org/10.1260/1757-482X.3.1.41.
Marofi, N., M. R. Jalili Ghazizadeh, and J. Attari. 2012. “Numerical sim-
ulation of self aeration on a chute spillway.”In Proc., 4th IAHR Int.
Symp. on Hydraulic Structures. Lisbon, Portugal: Associação Portuguesa
dos Recursos Hídricos.
Matos, J., Y. Yasuda, and H. Chanson. 2001. “Interaction between free-
surface aeration and cavity recirculation in skimming flows down
stepped chutes.”In Proc., 29th IAHR Congress, 611–617. Beijing:
Tsinghua University Press.
Meireles, I., and J. Matos. 2009. “Skimming flow in the nonaerated re-
gion of stepped spillways over embankment dams.”J. Hydraul. Eng.
135 (8): 685–689. https://doi.org/10.1061/(ASCE)HY.1943-7900
.0000047.
Meireles, I., F. Renna, J. Matos, and F. Bombardelli. 2012. “Skimming,
nonaerated flow on stepped spillways over roller compacted concrete
dams.”J. Hydraul. Eng. 138 (10): 870–877. https://doi.org/10.1061
/(ASCE)HY.1943-7900.0000591.
Morovvat, M., A. R. Zarrati, and M. R. Jalili Ghazizadeh. 2021. “A study
on the accuracy of finite volume numerical models with non-rectangular
mesh.”Scientia Iranica 28 (4): 1963–1972. https://doi.org/10.24200
/SCI.2021.52017.2484.
Namin, M. M., B. Lin, and R. A. Falconer. 2001. “An implicit numerical
algorithm for solving non-hydrostatic free-surface flow problems.”Int.
J. Numer. Methods Fluids 35 (3): 341–356. https://doi.org/10.1002
/1097-0363(20010215)35:3<341::AID-FLD96>3.0.CO;2-R.
Nezu, I., and H. Nakagawa. 1993. Turbulence in open-channel flows.
Rotterdam, Netherlands: Balkema.
Ostad Mirza, M. J., J. Matos, M. Pfister, and A. J. Schleiss. 2016. “The
effect of an abrupt slope change on air entrainment and flow depths
at stepped spillways.”J. Hydraul. Res. 55 (3): 362–375. https://doi
.org/10.1080/00221686.2016.1255263.
Ostad Mirza, M. J., J. Matos, M. Pfister, A. J. Schleiss, and A. R. Zarrati.
2018. “Effect of flow rate on skimming flow properties over an abrupt
slope change on a stepped spillway.”In Proc., E-proceedings of the 5th
IAHR Europe Congress, 367–368. Trento, Italy: IAHR.
Patankar, S. V. 1980. Numerical heat transfer and fluid flow. Boca Raton,
FL: CRC Press.
Patankar, S. V., and D. B. Spalding. 1972. “A calculation procedure for
heat, mass and momentum transfer in three-dimensional parabolic
flows.”Int. J. Heat Mass Transfer 15 (10): 1787–1806. https://doi
.org/10.1016/0017-9310(72)90054-3.
Pfister, M., and W. H. Hager. 2011. “Self-entrainment of air on stepped
spillways.”Int. J. Multiphase Flow 37 (2): 99–107. https://doi.org/10
.1016/j.ijmultiphaseflow.2010.10.007.
Rao, N. S. L., and T. Gangadharaiah. 1971. “Self-aerated flow character-
istics in wall region.”J. Hydraul. Div. 97 (9): 1285–1303. https://doi
.org/10.1061/JYCEAJ.0003062.
Rao, N. S. L., and H. E. Kobus. 1971. “Characteristics of self-aerated free-
surface flows.”In Water and wastewater/current research and practice.
Berlin: Eric Schmidt Verlag.
Reynolds, O. 1895. “Iv. On the dynamical theory of incompressible viscous
fluids and the determination of the criterion.”Philos. Trans. R. Soc.
London, Ser. A 186 (Apr): 123–164. https://doi.org/10.1098/rsta
.1895.0004.
Sabbagh-Yazdi, S.-R., H. Rezaei-Manizani, and N. Mastorakis. 2008.
“Effects of bottom aerator and self-aeration in steep chute spillway
on cell center finite volume solution of depth-averaged flow.”Int. J.
Math. Models Methods Appl. Sci. 2 (2): 154–161.
Schlichting, H. 1979. Boundary layer theory. New York: McGraw-Hill.
Soo, S. L. 1967. Fluid dynamics of multiphase systems. New York:
McGraw-Hill.
Stansby, P. K., and J. G. Zhou. 1998. “Shallow-water flow solver with non-
hydrostatic pressure: 2D vertical plane problems.”Int. J. Numer. Meth-
ods Fluids 28 (3): 541–563. https://doi.org/10.1002/(SICI)1097-0363
(19980915)28:3<541::AID-FLD738>3.0.CO;2-0.
Straub, L. G., and A. G. Anderson. 1958. “Experiments on self-aerated
flow in open channels.”J. Hydraul. Div. 84 (7): 1–35. https://doi.org
/10.1061/JYCEAJ.0000261.
Tekeli, S., and W. H. C. Maxwell. 1978. Behaviour of air bubble screens.
Urbana, IL: Univ. of Illinois.
Towne, A., and T. Colonius. 2015. “One-way spatial integration of hyper-
bolic equations.”J. Comput. Phys. 300 (Aug): 844–861. https://doi.org
/10.1016/j.jcp.2015.08.015.
Valero, D., and D. B. Bung. 2015. “Hybrid investigation of air transport
processes in moderately sloped stepped spillway flows.”In Proc.,
36th IAHR World Congress. Hague, Netherlands: International Asso-
ciation for Hydro-Environment Engineering and Research.
Valero, D., and D. B. Bung. 2016. “Development of the interfacial air
layer in the non-aerated region of high-velocity spillway flows: Insta-
bilities growth, entrapped air and influence on the self-aeration onset.”
Int. J. Multiphase Flow 84 (4): 66–74. https://doi.org/10.1016/j
.ijmultiphaseflow.2016.04.012.
Valero, D., and D. B. Bung. 2018. “Reformulating self-aeration in
hydraulic structures: Turbulent growth of free surface perturbations
leading to air entrainment.”Int. J. Multiphase Flow 100 (Dec): 127–
142. https://doi.org/10.1016/j.ijmultiphaseflow.2017.12.011.
Valero, D., and R. García-Bartual. 2016. “Calibration of an air entrainment
model for cfd spillway applications.”In Proc., Advances in Hydroin-
formatics: SIMHYDRO 2014, edited by P. Gourbesville, J. A. Cunge,
and G. Caignaert, 571–582. Berlin: Springer.
Von Karman, T. 1930. “Mechanical similarity and turbulence.”In Proc.,
3rd Int. Congress of Applied Mechanics,85–92. Washington, DC:
National Advisory Committee on Aeronautics.
Vreugdenhil, C. B. 1994. Numerical methods for shallow-water flow.
Berlin: Kluwer Academic Publishers.
Wilhelms, S. C., and J. S. Gulliver. 2005. “Bubbles and waves description
of self-aerated spillway flow.”J. Hydraul. Res. 43 (5): 522–531. https://
doi.org/10.1080/00221680509500150.
Wood, I. R. 1991. Air entrainment in free-surface flow: IAHR hydraulic
structures design manuals 4. Rotterdam, Netherlands: Balkema.
© ASCE 04022042-12 J. Hydraul. Eng.
J. Hydraul. Eng., 2023, 149(3): 04022042
Downloaded from ascelibrary.org by Karadeniz Technical University on 12/26/22. Copyright ASCE. For personal use only; all rights reserved.

Wood, I. R., P. Ackers, and J. Loveless. 1983. “General method for critical
point on spillways.”J. Hydraul. Eng. 109 (2): 308–312. https://doi.org
/10.1061/(ASCE)0733-9429(1983)109:2(308).
Zarrati, A. R. 1994. “Mathematical modelling of air-water mixtures in open
channels.”J. Hydraul. Res. 32 (5): 707–720. https://doi.org/10.1080
/00221689409498710.
Zarrati, A. R., and J. D. Hardwick. 1991. “Rising velocity of air bubbles in
high speed free surface flows.”In Proc., Int. Symp. on Environmental
Hydraulics. Hong Kong: Univ. of Hong Kong.
Zhang, G., and H. Chanson.2017. “Self-aeration in the rapidly and gradually
varying flow regions of steep smooth and stepped spillways.”Environ.
Fluid Mech. 17 (1): 27–46. https://doi.org/10.1007/s10652-015-9442-z.
© ASCE 04022042-13 J. Hydraul. Eng.
J. Hydraul. Eng., 2023, 149(3): 04022042
Downloaded from ascelibrary.org by Karadeniz Technical University on 12/26/22. Copyright ASCE. For personal use only; all rights reserved.