Experimental Study and 3D Numerical Simulations
for a Free-Overflow Spillway
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Bijan Dargahi1
Abstract: The main objectives of the present work were to investigate the flow field over a spillway and to simulate the flow by means
of a three-dimensional 共3D兲 numerical model. Depending on the wall curvature, the boundary layer parameters decreased or increased
with increasing distance along the spillway. The growth of the boundary layer along the spillway is better described as a function of
Reynolds number than the normalized streamwise length. A simplified form of the 3D momentum equation can be used to obtain a rough
estimate of the skin friction. The velocity profile in the boundary layer along the spillway is described by a velocity–defect relationship.
Numerical models provide a cost-effective means of simulating spillway flows. In this study, the water surface profiles and the discharge
coefficients for a laboratory spillway were predicted within an accuracy range of 1.5–2.9%. The simulations were sensitive to the choice
of the wall function, grid spacing, and Reynolds number. A nonequilibrium wall function with a grid spacing equal to a distance of 30 wall
units gave good results.
DOI: 10.1061/共ASCE兲0733-9429共2006兲132:9共899兲
CE Database subject headings: Three-dimensional models; Simulation; Overflow; Spillways.
Introduction
The recent developments in computer software has advanced the
use of computational fluid dynamics 共CFD兲 in analyzing flow
over spillways. Some recent works are due to Unami et al. 共1999兲,
Savage and Johnson 共2001兲, and Ho et al. 共2003兲. Unami et al.
共1999兲 developed a two-dimensional numerical simulation for
spillway flows. They found a reasonable agreement with experimental data. Savage and Johnson 共2001兲 did a two-dimensional
simulation of flow over an ogee spillway using a commercial
CFD code 共Flow-3D兲. They found a good agreement with experiments for both pressures and discharge. In comparison with
two-dimensional models, there are fewer applications of threedimensional 共3D兲 models for free overflow spillways. Ho et al.
共2003兲 did two- and three-dimensional CFD modeling of spillway
behavior under rising flood levels and validated the results using
published data. The model was also applied to study several spillways in Australia.
CFD complements experimental and theoretical fluid dynamics by providing an alternative cost-effective means of simulating
real flows. However, the usefulness of a mathematical model
depends on the validity of the governing equations and the numerical methods. In order for CFD to become more reliable and
acceptable as a design tool, numerical studies must be carefully
validated with experimental results. Because hydraulic design
1
Associate Professor, Div. Hydraulic Engineering, The Royal Institute
of Technology, Teknikringen 76–3tr, Stockholm 10044, Sweden. E-mail:
bijan@kth.se
Note. Discussion open until February 1, 2007. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on December 30, 2004; approved on August 23, 2005.
This paper is part of the Journal of Hydraulic Engineering, Vol. 132,
No. 9, September 1, 2006. ©ASCE, ISSN 0733-9429/2006/9-899–907/
$25.00.
of spillways is a new application area for CFD, it requires especially careful validation. In the design of these structures the main
challenges are to estimate accurately the discharge coefficient,
frictional losses, the details of local flow patterns, and the position
of the free-surface profile. Traditionally, physical models are used
to study these issues.
The discharge coefficient 共C␩兲 for an overflow spillway is
given by the following equation:
C␩ =
3Q
2B冑2g共ho + Ua兲1.5
共1兲
in which Q = discharge; B = spillway width; g = acceleration of
gravity; h0 = spillway operating head; and Ua = approach velocity
head. The approach velocity head is negligible if the spillway
height is greater than 1.33 hd 共hd = spillway design head兲. The
design head is the vertical distance between the spillway crest and
the water surface upstream of the structure.
The main objectives of the present work were to investigate
the flow field over an overflow spillway and to compare the
results with 3D flow simulations. A commercial code known as
Fluent was used for the numerical simulations. The numerical model accurately simulated the water surface profile under
various head conditions. The discharge coefficients and wall
shear stresses were also accurately estimated using the numerical
results.
The flow over a spillway is described by the following dimensionless parameters
ho ␦ Ks ␦
RF
hd Xl ␦ R
共2兲
in which ␦ = boundary layer thickness, Xl = streamwise length
along the spillway, Ks = wall roughness; R = radius of curvature; R = Reynolds number 共R = UmY m / ␯; Um = mean velocity,
Y m = mean flow depth; and ␯ = kinematic viscosity兲; and
F = Froude number 共F = Um / 冑gY m兲. The first parameter is an indicator of the pressure distribution along the crest of the spillway.
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J. Hydraul. Eng. 2006.132:899-907.
In comparison with the pressure at the design head, the pressure
increases if the ratio ho / hd 共Hn兲 is less than unity and decreases
if the ratio is higher than unity. A low pressure can cause cavitation to occur, resulting in spillway damage. The remaining
parameters in the group are related to the growth of the boundary
layer, the velocity distribution, and the water surface profile. To
estimate the various boundary layer parameters along a spillway the U.S. Army Corps of Engineers 共1952兲 suggested the use
of Eq. 共3a兲 and 共3b兲
冉 冊
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␦
Xl −0.233
= 0.08
Xl
Ks
␦d = 0.18␦,
␦e = 0.22␦
共3a兲
streamline convergence on the development of turbulent boundary layers. Their experimental findings confirm that an adverse
pressure gradient and streamline convergence result in low values
of skin friction, which can be estimated using a modified form
of the Ludwieg–Tillman equation. In an analytical study, Irwin
and Smith 共1975兲 showed that the influence of streamline curvature on turbulence is accounted for by the curvature terms in the
Reynolds stress equation.
One option in calculating the flow along a spillway is to use
Johnston’s 共1957, 1960兲 three-dimensional momentum integral
Eq. 共6兲, which is valid for boundary flows along a plane of symmetry where ⳵w / ⳵x = 0, ⳵u / ⳵z = 0, and ⳵U / ⳵z = 0
1 ⳵ U ␦␪ ⳵ W
␶0x ⳵ ␦␪ ⳵ ␦␪z
+
+ 共2␦␪ + ␦d兲
+
2 =
␳U
⳵x
⳵z
U ⳵x U ⳵z
共3b兲
In which ␦d = displacement thickness; and ␦e = energy thickness.
Their boundary layer definitions are given by Eq. 共3c兲
␦d =
1−
0
u
dy
U
冕 冉 冊
u
u2
1 − 2 dy
U
0 U
共3c兲
共4兲
in which ␦␪ = momentum thickness; and Rx tr = Reynolds number
at the transition point and x is measured from the vertical face of
the spillway
冕 冉 冊
␦
␦␪ =
u
u
1−
dy
U
U
0
1−
0
in which U = free-stream velocity; and u = velocity at depth y.
Spillway flow consists of three different regions: two curved flow
regions and one steep channel region in between. The main characteristic of curved flow is the existence of a radial pressure gradient and a centrifugal force. Universal velocity laws are difficult
to define for a boundary layer with strong longitudinal curvature
because the speed of the free stream flow varies significantly with
the longitudinal distance along the curve. Flows along convex
walls are usually stable 共i.e., Reynolds shear stresses and turbulence energy levels are decreased relative to a parallel shear layer
flow兲 at all values of ␦ / R, because the mean streamwise velocity
increases substantially as the effect of curvature accumulates.
This increase is significant for values of ␦ / R that are greater
than 0.056 共Sherman 1990兲. Furthermore, both Reynolds shear
stresses and the turbulent kinetic energy across the boundary layer
are reduced in comparison with those for a flat boundary layer
共Bradshaw 1978兲, whereas the opposite effect is observed for
flows along concave walls. Flows along concave walls are unstable at increased external flow velocities. Longitudinal vortices
similar to Görtler vortices can develop in the boundary layer.
During spillway discharge, the flow is initially laminar at the
leading edge of the spillway, then it undergoes a rapid transition
to turbulent flow. The location of the transition point can be estimated from the following equation 共White 1991兲:
U␦␪
0.4
⬇ 2.9Rx,tr
␯
冕冉 冊
␦
␦␪z =
␦
␦e =
I II III IV
冕冉 冊
␦
共5兲
While there have been few detailed studies of boundary layer
growth on spillways, effects on streamline curvature and pressure
gradient have been extensively investigated. Results obtained
for curved flows are to some extent applicable to spillways.
Winter et al. 共1968兲 studied the influence of pressure gradient and
共6兲
u w
dy
U U
共7兲
in which W = free stream velocity in z direction; and ␶0x = wall
shear stress in x direction. Terms II and IV represent the effects
of skewing of the velocity profile and the main flow streamline
convergence, respectively.
Experiments
Experiments were conducted in a 4-m-long, 0.403-m-wide, and
0.6-m-deep recirculating hydraulic flume. An overflow spillway
of 0.2 m crest height was placed at a distance of 2.4 m from
the flume inlet. The spillway was designed according to the
Waterways Experiment Station 共WES兲 standard 共U.S. Army
Corps of Engineers 1952兲 for a design head of hd = 0.1 m. The
downstream flow condition in all of the tests was supercritical.
The discharge was measured by means of a calibrated V-notch
weir. Detailed measurements of water surface profiles and velocity distributions normal to the streamlines were made at three
different operating heads of 0.5hd, 1.0hd, and 1.15hd. The Reynolds number based on operating heads was in the range of
104 – 105. The wall shear stresses were measured indirectly by
means of a pitot tube, using the Preston method 共Preston 1954兲
with the universal constants given by Patel 共1965兲. The measurements were taken along the center line section at the operating
head 1.0hd. To investigate the influence of a low Reynolds
number, additional water surface measurements were taken
for ho = 0.1hd. The flume was operated at the discharges computed from Eq. 共1兲 using the four designated operating heads
共ho = 0.1hd, 0.5hd, 1.0hd, and 1.15hd兲, and the C␩ values were
estimated from hydraulic design charts 共U.S. Army Corps of
Engineers 1964兲. The experimental C␩ values were then determined by measuring the actual spillway heads. At operating heads
greater than or equal to hd, the water surfaces were wavy and
unstable near the flume walls. The instantaneous positions of the
water surfaces could not be measured. Instead, the mean water
surface profiles were measured, and the mean wave heights and
mean wavelengths were recorded. Mean water surface measurements were taken using point gages at eight longitudinal sections
Z = ± 0.15, ±0.1, ±0.05, and ±0.025 m. The measurements were
taken at 2 mm intervals 共⌬x = 2 mm兲 over the flume length. A
hot-film anemometer was used for both velocity and turbulence
measurements. Before and after each experiment, the probe was
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Fig. 2. Boundary conditions and initial air and water flow regions
on the VOF method outlined by Hirt and Nichols 共1981兲. In the
VOF method a special advection technique is used that gives a
sharp definition of the free surface, whereas in Fluent the combined air–water flow is solved 共Bombardelli et al. 2001兲. In the
present study the QUICK differencing 共Leonard 1979兲 scheme
was used. The choice is motivated by the greater accuracy of the
scheme in comparison with central differencing or upwind
schemes.
Fig. 1. Spillway model and measurement sections 共Xn = X / hd兲
calibrated in the submerged jet of a calibration apparatus designed
especially for this purpose. The velocity measurements were
taken at 11 different x positions 共Fig. 1 and Table 1兲 along the
flume’s center line 共Z = 0兲. The x positions are given both in normalized Cartesian 共Xn兲 and curvilinear coordinates 共Xln兲 measured along the spillway. hd = 0.1 m was used to normalize the
x-coordinates. At the design head, additional velocity measurements were taken at the same eight longitudinal sections. For each
velocity profile, the measurements were taken at 1 mm intervals
over the flow depth at each station. Using the velocity signals
共sampling rate 200 Hz兲, mean velocities, boundary layer parameters, and relative turbulence intensities 共Ti兲 were computed. The
relative turbulence intensity was defined as the ratio of RMS of
the fluctuating x-velocity component to the local mean velocity.
To compensate for the short flume length, special devices were
tested to obtain a fully developed turbulent flow and symmetric
spanwise velocity profiles upstream of the spillway. At the flume
inlet various combinations of a flow straightener, grid bars and
roughness strips were tested. A reasonable velocity profile that
followed the 1/7 power law was established 1 m downstream
from the flume inlet. The error in discharge measurements was
2.5%, obtained by calibration of the V-notch weir. The water levels were measured with an accuracy of ±0.1 mm. The maximum
error in hot-film velocity measurements was estimated at 4.5%.
All the measurements could be repeated with good accuracy.
Geometry and Grid
The model geometry is shown in Fig. 2. The grids were created
by using a boundary fitted coordinate system 共a structured grid兲.
Near the spillway wall the adjacent grid lines were at nondimensional wall distances in the range 10⬍ Y + = u*y / ␯ ⬍ 50
共0.1– 0.7 mm兲, where u* = shear velocity and y = distance from
the wall. The numbers of grid in XYZ directions were varied in
three different ranges, i.e., 130⫻ 20⫻ 70–130⫻ 20⫻ 514 共Fig. 3兲,
130⫻ 20⫻ 70–130⫻ 120⫻ 70, and 130⫻ 20⫻ 70–330⫻ 20⫻ 70.
Boundary Conditions
To calculate the position of the free water surface the 共P兲VOF
model was used. The basic procedure is to define a primary phase
共water兲 and a secondary phase 共air兲 as shown in Fig. 2. Two
different inlets were needed to define the water flow 共Inlet I兲 and
air flow 共Inlet II兲 into the domain 共Fig. 2兲. These inlets were
defined as streamwise velocity inlets that require the values of the
velocity and the turbulence parameters, i.e., the turbulent kinetic
energy 共k兲 and the dissipation 共␧兲 were obtained from experimental data 共see Dargahi 2004兲. To estimate the effect of the wall on
the flow, empirical wall functions known as standard equilibrium
共Launder and Spalding’s 1972兲 and nonequilibrium wall functions
were used.
Solution Procedures
Numerical Model
The unsteady free surface calculations require a fine grid spacing
and a small initial time step. The grid spacing used was adequate
for solution convergence and good agreement with the experimental results. A time step equal to 0.001 s was selected. During
the calculations, solution convergence and the water surface
profiles were monitored. Convergence was reached when the normalized residual of each variable was on the order of 1 ⫻ 10−3.
An average of 5,000 time steps were required to reach the
correct water surface. The free surface was defined by a value of
VOF= 0.5, which is a common practice for volume fraction
results 共Fluent 1995兲.
Fluent is a general purpose computer program for modeling fluid
flow, heat transfer, and chemical reactions 共Fluent 1995兲, Kim et
al. 共1997兲, and Engelman et al. 共2001兲. It solves the full threedimensional equations of fluid motion in general orthogonal curvilinear coordinates for both laminar and turbulent flows. The
three turbulence models available in Fluent are the k and ␧ model,
the Reynolds stress transport model 共Wilcox 1988兲, and the renormalization group theory-based models 共RNG兲 共Yakhot and
Orszag 1986兲. The method for treating the free surface is the
partial volume of fluid 共PVOF兲 approach, which is partly based
Table 1. Locations of Measurement Sections along Spillway
Section
Inlet
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
Xn
Xln
−14.2
−14.2
0
0
0.083
0.140
0.138
0.243
0.408
0.749
0.562
1.067
0.770
1.334
0.853
1.708
1.254
2.608
1.637
3.447
1.992
4.1400
3.041
6.073
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Fig. 3. Example of 3D grid 共130⫻ 20⫻ 70兲. Only nine grid lines are
shown in Z-direction.
Grid Convergence
To check for grid convergence, the grid numbers were successively increased in three different ranges. It was found that the
increase in grid spacing in X- and Z-directions did not have a
measurable influence on the position of the water surface profiles.
However, the use of a finer grid size in Y direction gave more
accurate water surface and velocity profiles in comparison with
the experiments. The error estimated 共in comparison with experiments兲 for a grid number 130⫻ 60⫻ 70 was less than
0.15% different from the errors estimated for the grid number
130⫻ 120⫻ 70. Thus the resolution 130⫻ 60⫻ 70 is sufficiently
converged to obtain reliable numerical results.
Results
These results are mainly concerned with the center line section
共Zn = Z / B = 0兲 at the design head. Similar results were obtained for
the operating heads 0.5hd and 1.15hd.
Water Surface Profiles
The shapes of the measured water surface profiles were uneven
and wavy. Small irregular surface waves 共0.1– 0.3 mm兲 were observed near the flume walls. These surface waves were induced
by the side walls. The mean center line surfaces of all the three
operating heads agreed well with the upper nappe profiles given
by U.S. Army Corps of Engineers 共1952兲. Simulated water surface profiles agreed well with the experiments and WES data. An
example is given in Fig. 4. The comparison is better defined by
using a relative error which was obtained by taking the difference
between simulation and experimental values and then dividing the
results by the experimental values 共Fig. 5兲. The main results for
the mean water levels are summarized as follows:
Fig. 5. Relative errors between simulated and measured water levels
along Zn = 0 + Hn = 1.15; 共ⴰ兲 Hn = 1, 共쐌兲 Hn = 0.5
1.
The upstream water surfaces are simulated well by the
numerical model;
2. The operating heads are predicted with the maximum
relative error equal to 0.5%;
3. For all three operating heads the maximum relative errors
共4%兲 occur at Xn = 2.2; and
4. For the case Hn = 0.5, the relative errors are higher. One
possible cause for the higher errors could be due to grid
spacing. However, a grid refinement beyond 60 points in
y direction was not possible since in some flow regions
the normalized wall distances were Y + ⬍ 10. This condition is outside the validity range of wall functions, i.e.,
Y + ⬎ 10. It is also probable that the use of the 共P兲VOF
model instead of VOF creates more problems at low flow
depths.
The numerical results were used to compute the discharge coefficients 共C␩兲 by use of Eq. 共1兲. Table 2 compares the numerical
values of C␩ with the experimental values. The two data sets are
in good agreement. For the design head 共Hn = 1兲 the error is 1.5%,
whereas at Hn = 0.5 the error increases to 2.9%.
Mean Velocity Profiles
A good agreement was found between numerical and experimental results, and some examples are given in Figs. 6共a–d兲. In the
boundary layer near the wall, the grid spacing was much finer
than the spacing used in the experiments. Consequently, the simulations gave a finer resolution than the experiments. Close agreement is shown in Figs. 6共a–d兲 for the upper part of the velocity
profiles, i.e., Y n ⬎ 0.5, whereas the degree of agreement for the
lower part of the velocity profiles 共Y n ⬍ 0.5兲 is dependent upon
the choice of turbulence model, i.e., k–␧, and RNG and wall
functions, i.e., standard and nonequilibrium. The experimental velocity profiles agreed better with the numerical data when the
RNG model was used. Fig. 7 illustrates the results for Section S2.
The difference between the two turbulence models is more sig-
Table 2. Comparison between Simulated and Measured Discharge
Coefficients
Operating head
Fig. 4. Comparison between simulated and measured water surface
profiles 共Zn = 0, Hn = 1.15兲: 共ⴰ兲 measured–simulated
0.5hd
1 hd
1.15 hd
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c␩ measured
c␩ simulated
Error 共%兲
0.68
0.745
0.747
0.7
0.756
0.762
2.9
1.48
2
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Fig. 6. Comparison between simulated 共solid line兲 and measured velocity profiles 共ⴰ兲: 共a兲 Section S3 Hn = 1, Zn = 0; 共b兲 Section S3 Hn = 1,
Zn = 0.375; 共c兲 Section S9 Hn = 1, Zn = 0; and 共d兲 Section S3 Hn = 1.15; Zn = 0
nificant near the wall boundary 共Y n ⬍ 0.1兲. The best agreement
with experiments was obtained when the RNG model was used
with the nonequilibrium wall function.
The velocity profiles varied significantly along the spillway.
The measured profiles could be grouped into three regions:
one near the crest of the spillway 0 ⬍ Xln ⬍ 0.243 共S1–S3兲, another along the sloping part of the spillway 0.749⬍ Xln ⬍ 2.068
共S4–S8兲, and the third at the downstream section 3.447⬍ Xln
⬍ 6.073 共S9–S11兲. The velocity profiles in these regions are
shown in Figs. 8共a–c兲. None of the velocity profiles could be
described by a velocity law with universal constants. However, a
general relationship existed 关Eq. 共8兲兴 in which the coefficients A
and B varied along the spillway 共Table 3兲
U−u
y
= A ln + B
u*
␦
共8兲
Fig. 9 compares the velocity defect law with the experimental
data for Groups I–III velocity profiles. Fig. 9 shows that Eq. 共8兲 is
a good fit to the velocity data.
Turbulent and Boundary Layer Variables
The main turbulent variable that could be compared with experiments was the relative turbulence intensity as the velocity measurements were one dimensional. In the wall regions the values of
relative turbulence intensity varied in the range 5–15%. A good
agreement was found with experiments. An example is given in
Fig. 10 at Section S10.
The boundary layer parameters, i.e., ␦, ␦d, ␦e were computed
using the measured velocity profiles along Zn = 0. To find the
boundary layer thickness 共␦兲, two criteria were considered: first ␦
is reached at the position where u = 0.99U; and second ␦ is
roughly equal to the depth of the region in which velocity is a
logarithmic function of flow depth. The results obtained were then
compared and if the values differed by more than 3%, the latter
method was used. The displacement and energy thicknesses were
computed from Eqs. 共3c兲. The boundary layer was laminar for a
short initial distance 关=1 cm, Eq. 共4兲兴, and thereafter it grew to
70% of the flow depth at outlet Section S11 共Fig. 11兲. A comparison is made between the data and Eq. 共3a兲 共Fig. 12兲. Eq. 共3a兲
underestimates the boundary layer thickness. A better fit to the
data is
冉 冊
␦
Xl −0.5
= 2.07
Xl
Ks
Fig. 7. Influence of wall functions and turbulence models
共k-␥ and RNG兲 on lower part of velocity profile at Section S2
共Zn = 0, Hn = 1兲
共9兲
Fig. 13 shows the variation of the other parameters 共␦d, ␦e兲 with x
distance along the spillway. In the region 2 ⬍ Xln ⬍ 4, the values
of ␦d and ␦e differ significantly from Eq. 共3b兲. Better agreement is
obtained outside this region. The other useful parameter is the
shape factors defined by H12 = ␦d / ␦␪. The data agreed well with
Eq. 共10兲 共Table 4兲 given by Hinze 共1975兲.
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Fig. 9. Comparison between velocity “defect law” 关Eq. 共8兲兴 and
experiments 共symbols兲 for Groups I–III velocity profiles 共Zn = 0,
Hn = 1兲
Fig. 8. 共a兲 Measured velocity profiles along spillway 共Zn = 0, Hn = 1兲:
Group I Region 0 ⬍ Xln ⬍ 0.243; 共b兲 measured velocity profiles along
spillway 共Zn = 0, Hn = 1兲: Group II Region 0.749⬍ Xln ⬍ 2.068; and
共c兲 measured velocity profiles along spillway 共Zn = 0, Hn = 1兲: Group
III Region 3.447⬍ Xln ⬍ 6.073
冉
H12 = 1 −
冊
I2 u* −1
,
I1 U
I2
= 4.88
I1
depended on the choice of turbulence models and wall functions.
Fig. 14 compares the results with experimental values for the
four cases of k – ␧ and RNG models, standard wall function, and
nonequilibrium wall function. The k – ␧ model and standard wall
function underestimate the values of wall shear stresses. The differences are more significant for Xln ⬎ 1. The combination of the
RNG model and the nonequilibrium wall function gives the closest fit to the experimental data 共Fig. 15兲. The highest deviations
were found in the range 4 ⬍ Xln ⬍ 5 that correspond to the concave wall region. In this region the shear stresses are overestimated by 10% in comparison with experiments. Eq. 共6兲 was also
applied to the spillway flow to estimate the values of skin friction
coefficient 共C f = ␶0x / 0.5␳U2兲 along the center line, i.e., Zn = 0 and
the order of magnitude of the various terms. The boundary layer
parameters in the equation were calculated using the measured
velocity profiles. Table 5 shows the result of calculations for the
center section with Hn = 1. For comparison the measured skin friction coefficients are included in the table. The dominant terms in
the momentum Eq. 共6兲 are I, III, and IV. Eq. 共6兲 gives a reasonable estimate of C f values along the spillway. The maximum error
is about 15% compared with the numerical data.
共10兲
Skin Friction
The simulated values of wall shear stresses showed a general
agreement with experiments. However, the degree of agreement
Table 3. Velocity Law Coefficients 关Eq. 共8兲兴 for Each Velocity Group
Velocity group
共Fig. 8兲
A
B
I
II
III
Universal law
−1.98
−2.75
−2.73
−2.44
4.30
1.05
0.31
2.5
Fig. 10. Comparison between simulated and measured relative
turbulence intensities, Section S10
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Fig. 11. Boundary layer growth along spillway 共Zn = 0, Hn = 1兲:
共•兲 experiments
Fig. 13. Variation of boundary layer parameters along spillway
共Zn = 0, Hn = 1兲: experiments: 共ⴰ兲 ␦e / ␦; 共•兲 ␦d / ␦
␦
= 11.63共Rl兲−0.42
Xl
Discussion
Velocity profiles in the crest region and steep channel regions
exhibit both a maximum and a minimum 关Figs. 8共a and b兲兴
primarily because of the curvature effect. Similar types of profiles
are observed for wall jet flows due to insufficient fluid momentum
to entrain the boundary layer completely 共see Dvorak 1973兲. At
the distances Xln ⬎ 4.14 the profiles begin to approach the theoretical distribution given by 1/7 power law 关Fig. 8共c兲兴 as the
external flow becomes nearly constant and independent of the
distance. An interesting feature of all the velocity profiles was that
in the boundary layer the velocities were a logarithmic function of
the flow depths and they could be described by a defect law
similar to Eq. 共8兲 having coefficients A and B given in Table 3.
The boundary layer along the spillway is laminar up to the
point 0.1h0. Beyond this laminar region the pressure gradient becomes steep and the boundary layer thickness increases 共with x2兲
more rapidly than a flat plate 共Fig. 11兲. At the outlet section the
boundary layer becomes almost fully developed and the velocity
profile approaches the 1/7 power law. A significant change in the
character of the boundary layer occurs at the concave part of the
spillway. The change is shown by the jumps in the boundary layer
parameters 共Fig. 13兲, and skins friction coefficient 共Fig. 14兲 at a
point that corresponds to the center line of the concave arc of the
spillway. The change in the boundary layer agrees qualitatively
with the boundary layer measurements along a waisted body of
revolution 共Winter et al. 1968兲. For this body the jumps in the
boundary layer parameters and skin friction also take place along
the center line of the arc that marks the concave part of the body.
Between Xln = 3.5 and Xln = 4 the flow divergence leads to an increase in skin friction. The growth of the boundary layer along the
spillway is better described as a function of Reynolds number
based on the length Xl 关Eq. 共11兲兴 than Eq. 共3a兲 共Fig. 16兲
Fig. 12. Comparison of boundary layer growth 共Zn = 0, Hn = 1兲:
共•兲 experiments; 共ⴰ兲 Eqs. 共3a兲 and 共4兲–共9兲
共11兲
Eq. 共3a兲 will underestimate the losses. As an alternative, Eq. 共11兲
can be used with the advantage of not needing the local velocities.
Regarding the other two parameters 共␦d , ␦e兲, Eq. 共3b兲 gives a reasonable estimate in the range 0.15⬍ Xln ⬍ 2. The other parameter
describing the boundary layer is the shape factors H12. The
distribution of this parameter along the spillway agrees well
with Eq. 共10兲 共Table 4兲. The value of the universal constant
I2 / I1 = 4.88 fits the data. This constant can be applied to the spillway flow if the logarithmic velocity distribution given by Eq. 共8兲
is valid. The agreement with the universal constants A and B is
not necessary.
The approximate three-dimensional momentum Eq. 共6兲 can
also be simplified further by ignoring the term II which is less
significant than the other terms in this equation 共Table 5兲. The
remaining terms in the simpler form of the equation can be used
to estimate the skin-friction coefficient with an error of about
15% 共Table 5兲.
The main flow features predicted by the numerical model
agreed well with the experiments. However, the agreement is
dependent on the choice of the grid size, the turbulence model,
wall function, and the Reynolds number. In spillway flows it is
important that the grid has a high resolution near the walls,
otherwise the pressure distribution and the wall shear stresses
would be incorrect. The grid resolution becomes especially important at lower operating heads. A low operating head requires a
finer grid as the flow depth is much smaller. A good choice is a
grid spacing of Y + ⬀ 30. The water surface profiles are less sensitive to grid size variation than the velocity profiles and boundary
Table 4. Comparison between Measured and Computed Shape Factors
共H12兲
Xln
H12
共experiments兲
H12
关Eq. 共10兲兴
Error 共2-3兲/3
共%兲
0.140
0.243
0.749
1.067
1.334
1.708
2.608
3.447
4.140
6.073
1.305
1.276
1.263
1.234
1.232
1.235
1.268
1.200
1.261
1.206
1.351
1.343
1.288
1.277
1.241
1.236
1.233
1.158
1.281
1.247
−3.37
−4.95
−1.95
−3.35
−0.73
−0.04
2.87
3.6
−1.53
−3.23
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Table 5. Variation of Ratios of Terms I–IV 关Eq. 共6兲兴 along Spillway and
Skin Friction Coefficients
Fig. 14. Influence of wall functions and turbulence models
共k-␥ and RNG兲 on distribution of wall shear stresses along spillway
共Non-Eq. = nonequilibrium兲 共Zn = 0, Hn = 1兲
layer properties. The k – ␧ two-equation turbulence model gave
reasonable results. Compared with this model, the RNG model
gives better velocity results near the boundary. However, the
choice of the wall function has a greater influence on the solution
than the turbulence model. Given the two options of a standard
and a nonequilibrium wall function, the latter model gives better
results 共e.g., Fig. 7兲. The solution procedure works well if the
flow is turbulent. At a lower Reynolds number i.e., h0 = 0.1hd,
difficulties occurred in the calculations. During the initial stage of
the calculations, the flow became unstable in the sense that water
drops were formed instead of a smooth continuous flow region.
One reason could be the existence of low flow velocities, i.e., a
low Reynolds number and a low Froude number during the initial
time. To test this hypothesis, an attempt was made to solve the
problem as a laminar flow. However, the water drops also appeared for the laminar flow tests. Once instability was formed,
eliminating it was not possible. The formation of the water drops
can suggest that the implementation of VOF model in the CFD
code has some problems at low values of operating head when the
flow depth is very small and the Reynolds number is low. There is
a need to further address this issue.
Hydraulic structure models are Froudian type 共gravity force
dominates兲 that ensure geometrical, kinematics, and dynamic
similarity. Scale effects are introduced as the Reynolds number in
the prototype is normally several times larger than the model. The
increase in Reynolds number will affect the velocity distributions
Fig. 15. Comparison between simulated, measured and Eq. 共6兲 of
wall shear stress distribution along spillway 共Zn = 0, Hn = 1兲.
Section
II/I
III/I
IV/I
Cf
关Eq. 共6兲兴
Cf
共measured兲
Error
%
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
0.009
0.009
0.012
0.006
0.006
0.005
0.003
0.001
0.003
0.002
0.270
0.750
0.320
0.320
0.090
0.085
0.320
0.230
0.190
0.350
0.070
0.025
0.030
0.022
0.015
0.017
0.090
0.010
0.060
0.050
0.0074
0.0051
0.0043
0.0037
0.0036
0.0032
0.0036
0.0027
0.0040
0.0038
0.0065
0.0059
0.0049
0.0043
0.004
0.0037
0.0034
0.0023
0.0048
0.0041
13.80
13.60
12.20
14.00
10.00
5.90
13.50
15.50
15.20
7.30
and the boundary layer properties. To get some idea, one can
consider the general power type velocity profile given by the
following equation:
冉冊
u
y 1/n
=
U
␦
共12兲
in which the exponent n varies from 5 to 10 for Reynolds
numbers in the range 4 ⫻ 105 – 3.2⫻ 106 共Schlichting 1979兲. The
velocity profile becomes fuller as the Reynolds number increases.
The boundary layer variables yield from integrating Eq. 共12兲
1
␦d
=
␦ 1+n
␦␪
n
=
␦ 共1 + n兲共2 + n兲
共13兲
Eq. 共13兲 shows that the normalized boundary layer parameters
will decrease with increasing Reynolds number. The same result
yields from Eq. 共11兲. The other issue is the frictional resistant that
decreases with increasing Reynolds number for a smooth surface.
However, by keeping the Reynolds number in the turbulent range,
the influence of the scale effects can be slight without being altogether negligible. In the case of a spillway flow, the viscous
resistance effects are negligible at the crest, where the discharge
relation is determined 关Eq. 共1兲兴. The discussion on scale model
effects indicates the need for further research by studying prototype spillways.
Fig. 16. Comparison between measured and Eqs. 共3a兲, 共11兲 of
boundary layer thickness+ experiments
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Conclusions
Depending on the wall curvature, the boundary layer parameters
decrease or increase with increasing distance along the spillway.
The growth of the boundary layer along the spillway is better
described as a function of Reynolds number than the normalized
length. The boundary layer parameters and the skins friction
coefficients are significantly increased at the concave part of
the spillway. The velocity profile in the boundary layer along
the spillway is described by a velocity-defect relationship. The
three-dimensional momentum equation can be used to obtain a
rough estimate of the values of the skin friction. The velocity
profiles in the boundary layer along the spillway are described by
a velocity–defect relationship valid for both smooth and rough
surfaces.
Computational fluid dynamics is an effective tool for analyzing free-surface flows over spillways. The water surface profiles
and the discharge coefficients can be predicted with an accuracy
range of 1.5–2.9% depending on the spillway’s operating head.
Nonequilibrium wall function with a grid spacing equal to a wall
distance of 30 gives good results. However, the difference between the standard wall function and the nonequilibrium function
decreases with decreasing wall distance.
Acknowledgments
The reviewers and the associate reviewer provided useful advice
and tips to improve the manuscript.
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