This document describes a 3-D numerical model used to study supercritical flow in gradual channel expansions. The model was first verified against experimental data from a modified Rouse expansion, showing satisfactory agreement, particularly at lower Froude numbers. The model was then applied to optimize the design of a complex expansion structure with variable bottom elevation, piers, and walls. Calculations indicated flow conditions would be satisfactory if bottom ramps ensured uniform bottom variation.

1. Journal of Hydraulic Research Vol. 46, No. 3 (2008), pp. 402–409
doi:10.3826/jhr.2008.3162
© 2008 International Association of Hydraulic Engineering and Research
3-D numerical modeling of supercritical flow in gradual expansions
Modèle numérique tridimensionnel de l’écoulement supercritique dans les
élargissements progressifs
ANASTASIOS I. STAMOU, (IAHR Member), Associate Professor, School of Civil Engineering, Laboratory of Applied Hydraulics,
National Technical University of Athens, Iroon Polytechniou 5, 15780 Athens, Greece. E-mail: stamou@central.ntua.gr
DEMETRIOS G. CHAPSAS, Ph.D. Candidate, School of Civil Engineering, Laboratory of Applied Hydraulics, National Technical
University of Athens, Iroon Polytechniou 5, 15780 Athens, Greece. E-mail: ps04043@mail.ntua.gr
GEORGE C. CHRISTODOULOU, (IAHR Member), Professor, School of Civil Engineering, Laboratory of Applied Hydraulics,
National Technical University of Athens, Iroon Polytechniou 5, 15780 Athens, Greece. E-mail: christod@hydro.ntua.gr
ABSTRACT
A 3-D CFD model based on the Volume of Fluid method to calculate the free surface was used to study supercritical flow in gradual open channel
expansions. The model was first applied to a modified Rouse expansion, for which extensive experimental data are available. A detailed comparison
between the numerical and the experimental results concerning surface profiles, velocities, free surface uniformity, and shock front location showed
satisfactory agreement, especially for lower supercritical Froude numbers. The model was subsequently applied to optimize and finalize the design of
a compound expansion structure with variable bottom elevation, piers, and inner walls. CFD calculations showed that flow conditions in this structure
are satisfactory, provided that bottom ramps are used to ensure a uniform variation of the bottom elevation.
RÉSUMÉ
Un modèle tridimensionnel de dynamique des fluides (3-D CFD) en volumes finis avec surface libre a été utilisé pour étudier l’écoulement supercritique
dans les élargissements progressifs en canal ouvert. Le modèle a d’abord été appliqué à un élargissement de Rouse modifié, pour lequel on dispose
de nombreuses données expérimentales. Une comparaison détaillée entre les résultats numériques et expérimentaux concernant les profils de surface,
les vitesses, l’uniformité de surface libre, et la localisation du ressaut, a montré un accord satisfaisant, particulièrement pour les nombres de
Froude supercritiques inférieurs. Le modèle a ensuite été appliqué pour optimiser et finaliser la conception d’une structure complexe d’élargissement
comprenant une élévation variable du fond, des quais, et des murs intérieurs. Les calculs de CFD ont montré que les conditions d’écoulement dans
cette structure sont satisfaisantes, à condition d’utiliser des rampes inférieures pour assurer une variation uniforme de la cote du fond.
Keywords: Computational Fluid Dynamics (CFD), expansion flow, mathematical model, modified Rouse curve, supercritical flow,
Volume of Fluid (VOF) method
1 Introduction
Gradual expansions in open channel supercritical flow are fre-
quently encountered in hydraulic structures, such as outlets,
spillways, chutes, flood-relief canals, and storm water drainage
channels. The design of these expansions is usually performed
empirically using information from experimental studies or by
means of 1-D hydraulic calculations, which cannot capture the
large surface super-elevations due to the formation of shock
waves.
There exist both experimental and computational studies on
supercritical gradual expansions. The first experimental study
was performed by Rouse et al. (1951), who used dimensional
analysis to determine the free surface in a channel expansion as a
function of the streamwise coordinate and the characteristics of
Accepted on November 16, 2007/Open for discussion until December 31, 2008.
402
the approach flow. Rouse et al. (1951) proposed a continuous wall
geometry to reduce the shock wave formation in a finite width
expansion. Mazumder and Hager (1993) conducted a wide range
experimental investigation in gradual expansions with modified
and reversed Rouse wall geometries for supercritical approach
flow. Their work provided a quite complete flow description con-
taining extensive data, such as: (i) Axial and wall flow depth
profilesandvelocities, (ii)Geometriesofshockfronts, (iii)Distri-
butions of standard deviation of depth, and (iv) Locations of wave
peaks. The non-dimensional Rouse streamwise coordinate was
used for expressing the results. This work was further extended by
Mazumder and Hager (1995) by testing other geometries involv-
ing a gradual linear expansion, an adversely sloping bottom, a
bed deflector or combinations of the two; design guidelines were
also provided in the paper.

2. Journal of Hydraulic Research Vol. 46, No. 3 (2008) 3-D numerical modeling of supercritical flow in gradual expansions 403
Concerning the previous mathematical models of supercritical
gradual expansions, their main characteristics are the following:
(1) 2-D depth-averaged or shallow-water equations (SWE) are
normally applied in transformed coordinates using explicit or
implicit schemes, and (2) Verification of these models was per-
formed successfully using the experimental data of Rouse et al.
(1951), mainly the axial and the wall flow profiles in a gradual
expansion; in only one study a part of the data of Mazumder and
Hager (1993) was used.
Jimenez and Chaudhry (1988) and Bhallamudi and Chaudhry
(1992) used the space and time second-order accurate explicit
McCormack scheme to calculate successfully the water depth in a
modified Rouse curve for an approach flow Froude number equal
to Fo = uo(gho)−1/2
= 2, where uo is the approach flow velocity,
ho is the approach flow depth, and g is the gravitational accelera-
tion; they showed that the shallow-water theory may reasonably
represent smooth, steady, supercritical flow if Fo was not close to
one and the depth to width ratio was less than 0.1. Tseng (1999)
repeated the calculations of Jimenez and Chaudhry (1988) using
four high-resolution, non-oscillatory, shock-capturing explicit
schemes (ROE1, TVD2, ENO2, and ENO3) and the Strang-type
operator splitting technique to treat the flow with bottom slope
and friction terms; his results for the four schemes were similar
to those of Jimenez and Chaudhry (1988) and compared satis-
factorily with measurements. The quantitative comparison of the
relative error in the L2 norm between the calculations and mea-
surements indicated that the ENO3, ENO2, and TVD2 schemes
seemed to be slightly better than that of the ROE1 scheme.
Soulis (1991) introduced a non-orthogonal, boundary-fitted
coordinate system, transformed the quadrilaterals in the physical
domain onto squares in the computational domain and solved the
2-D free surface flow equations using an explicit finite volume
scheme. He calculated the water depths for a modified Rouse
profile for Fo equal to 2 and 4 (Rouse et al., 1951); his calcula-
tions were close to measurements. Klonidis and Soulis (2002)
repeated the calculations of Soulis (1991) using an implicit,
second-order accurate, fast converging and unconditionally sta-
ble scheme to solve the steady, two-dimensional depth-averaged,
free-surface flow equations using a transformation into a non-
orthogonal, boundary-fitted coordinate system to simulate with
accuracy irregular geometries. The computed results were in sat-
isfactory agreement with measurements and with the predictions
of Soulis (1991).
Recently, Krueger and Rutschmann (2006) extended the clas-
sical SWE to simulate supercritical flow in channel transitions
including a modified Rouse channel expansion with a large
approach flow Froude number (Fo = 8), which was experimen-
tally studied by Mazumder and Hager (1993); their extension
involved higher-order distribution functions for pressure and
horizontal and vertical velocities, thus taking into account
non-hydrostatic pressure distribution and vertical momentum.
Wetting and drying of the computational cells and wave break-
ing due to steep free-surface gradients were solved numerically.
The solutions with the extended approach (ESWE) were com-
pared both with a part of the available experimental data (only
flow depths) and with the standard SWE. Despite the pressure
distribution in the expansion of Mazumder and Hager (1993)
was nearly hydrostatic, both simulations and the measurements
were in agreement along the centerline. However, along the
sidewall, the computed separation zone was too large and the
shockwave extended too far downstream; this behavior can
be attributed to the lack of a turbulence model. Krueger and
Rutschmann (2006) also argued that differences could not be due
to 3-D effects, as they had also been reproduced by the ESWE
computations.
The results of the above-mentioned experimental studies in the
form of simple normalized expressions can up to a certain degree
be used as guidelines for the design of supercritical expansions of
simplegeometry. However, thedesignofsupercriticalexpansions
of complicated geometries requires a different approach, which
involves the use of either physical or mathematical models. For
the calibration and the verification of these mathematical models
the experimental data from the above-mentioned studies can be
used.
In the present work, a 3-D numerical model was employed
to study the features of supercritical flow in gradual expansions.
First, the model was tested and verified against four sets of avail-
able experimental data in a modified Rouse profile (Mazumder
and Hager, 1993). Then, it was applied to a complicated expan-
sion structure being designed as part of the drainage network of
the Diakoniaris River in the city of Patras in Greece.
2 Mathematical model
The 3-D flow field is governed by the continuity and the
momentum equations, which are written as
∂
∂x
(u · Ax) +
∂
∂y
(v · Ay) +
∂
∂z
(w · Az) = 0 (1)
∂u
∂t
+
1
VF
u · Ax ·
∂u
∂x
+ v · Ay ·
∂u
∂y
+ w · Az ·
∂u
∂z
= −
1
ρ
∂p
∂x
+ Gx + fx (2)
∂v
∂t
+
1
VF
u · Ax ·
∂v
∂x
+ v · Ay ·
∂v
∂y
+ w · Az ·
∂v
∂z
= −
1
ρ
∂p
∂y
+ Gy + fy (3)
∂w
∂t
+
1
VF
u · Ax ·
∂w
∂x
+ v · Ay ·
∂w
∂y
+ w · Az ·
∂w
∂z
= −
1
ρ
∂p
∂z
+ Gz + fz, (4)
where t is the time, u, v, and w are the velocity components
along the axes x, y, and z, respectively, of a Cartesian coordinate
system, p is the pressure, ρ is the density, and VF is the fractional
volume open to flow. Ax, Ay, and Az are the fractional areas
open to flow along x, y, and z, respectively, Gx, Gy, and Gz are
body accelerations along x, y, and z, respectively, and fx, fy,
and fz, are viscous accelerations along x, y, and z, respectively
(Flow Science Inc., 2007). The viscous accelerations include the

3. 404 A.I. Stamou et al. Journal of Hydraulic Research Vol. 46, No. 3 (2008)
Reynolds stresses, which are calculated via the k−ε model (Rodi,
1980).
The above set of equations is solved by the finite-volume
based code FLOW-3D (Flow Science Inc., 2007), which employs
a structured, patched multi-block, orthogonal, coordinate grid
with a collocated variable arrangement. In FLOW-3D the “free-
griding” approach is combined with simple rectangular grids.The
geometric regions within rectangular grids are defined using
the Fractional Area/Volume Obstacle Representation method.
ThefreesurfaceismodeledwiththeVolumeofFluidmethod(Hirt
and Nichols, 1981). The equations of the model are integrated
over each control volume, such that the relevant quantity (mass
and momentum) is conserved, in a discrete sense, for each control
volume.A spatial discretization is achieved through the HYBRID
scheme, a second-order upwind scheme, and the third-order
QUICK scheme. The basic solution algorithm is the SIMPLEC
pressure correction scheme (Van Doormal and Raithby, 1984),
which uses a variety of linear equation solvers.
3 Verification of model
3.1 Experimental data
FoursetsofexperimentaldatainamodifiedRousewallexpansion
profile reported by Mazumder and Hager (1993) were used for
the verification of the model. These experiments were conducted
in a 10 m long rectangular, horizontal channel (Fig. 1). The
approach flow channel had a width bo = 0.50 m. The approach
flow depth ho was regulated with a gate on which a cover was
attached. In all experiments the downstream width b2 was equal
to 1.5 m, i.e. the expansion ratio was equal to β = b2/bo = 3.
The characteristics of the four runs are shown in Table 1.
The modified Rouse wall profile of Fig. 1 was determined
via the calculation of the transverse coordinate yb using (Chow,
1959)
yb
bo
=
1
2
1 +
1
4
X3/2
(5)
Figure 1 Scheme of modified Rouse wall expansion profile and the
computation domain; dimensions in [m]
Table 1 Characteristics of runs of Mazumder
and Hager (1993) used in present work
Run ho(m) Fo
27 0.048 2.0
24 0.048 4.0
25 0.048 6.0
26 0.048 8.0
with the normalized streamwise coordinate
X =
x
bo · Fo
, (6)
where x is the streamwise coordinate. The design Froude num-
ber was equal to Fo = FD = 1.0; i.e. the wall profile has the
dimensional equation with (x, yb) in [m]
yb = 0.25 + 0.177x3/2
. (7)
Its end is located at x = Lt = 2.0 m, where yb = 0.5 b2 =
0.75 m (Fig. 1).
The flow depths were measured with a conventional point gage
(±0.5 mm reading accuracy), the flow velocities were recorded
with a miniature current meter and the local streamline direc-
tion was determined with an angle probe. The local velocity was
found to vary only slightly in the vertical direction, except for the
boundary layer; thus, a representative velocity magnitude at half
of the local flow depth was taken. Extended preliminary observa-
tions indicated reliable results for flow depths down to 15 mm and
velocities up to 5.0 m/s. The experimental grid was x = 1.0 m
in the streamwise, and y = 0.05 m in the transverse directions.
3.2 Computational domain, boundary conditions,
and numerical grid
The boundary conditions are defined at the borders of the compu-
tation domain, i.e. −1.20 m ≤ x ≤ +10.00 m, −0.75 m ≤ y ≤
+0.75 m and +0.00 m ≤ z ≤ +0.078 m (Fig. 1).At the upstream
end of the approach flow channel (x = −1.20 m), which is called
“inlet,” a parallel flow was imposed, with uniform horizontal
velocity equal to uo = Q/(boho) and vertical velocity equal to
zero. The values of the turbulent energy and its dissipation ko and
εo were assumed to be uniform and were calculated as follows
(Lyn et al., 1992)
ko = au2
o (8)
εo = c0.75
µ
k1.5
o
lo
, (9)
where α is a constant chosen such that a large eddy viscosity
results at the inlet, cµ is an empirical constant and lo is the char-
acteristic inlet dimension taken equal to 0.1 ho. The values of ko
and εo resulted in values of the eddy viscosity at the inlet equal
to approximately 100 times the molecular viscosity of water. The
above-mentioned “inlet” boundary condition approximates geo-
metrically the real boundary condition in the experimental setup,
which consisted of a gate on which a cover was attached. It is
noted that imposing a uniform inflow velocity is a somewhat
“stiff” condition that may cause numerical instabilities and small
perturbations, as discussed below. However, there was no better
alternative available in the numerical code for the definition of a
more realistic velocity distribution.
At the downstream channel end (x = +10.00 m) an “out-
let” condition was used, which sets the discharge equal to the
inlet discharge and the vertical gradients of k and ε equal to
zero. At the rigid walls, the standard wall function approach was
applied, which relates the shear stress at the wall to the cell node

4. Journal of Hydraulic Research Vol. 46, No. 3 (2008) 3-D numerical modeling of supercritical flow in gradual expansions 405
velocity component parallel to the wall.An orthogonal numerical
grid consisting of 358,400 control volumes and grid dimensions
x = 0.02 m, y = 0.025 m, and z ≈ 0.01 m was used
in the computations; these were selected based on a series of
preliminary calculations to ensure grid independent results.
3.3 Computations and comparison with experimental data
Calculations were performed to determine steady state flow
fields. Initially, it was assumed that there was no flow in the
computational domain (cold start). Small perturbations were
observed in the computations, which are evident in some of the
plots. These disturbances are attributed to minor computational
instabilities associated with the “stiff” inlet boundary conditions.
Computations were performed on a single PC with two parallel
2.79 GHz AMD Opteron processors. A time of approximately
70 s was required to reach the steady state conditions; the cor-
responding CPU time was equal to 24 h. Figure 2 shows the
temporal variation of the flow depth at three locations down-
stream of the expansion, providing an indication on the stability
of the calculations.
The computational results at steady state are compared to
the experimental data in Figs 3–8. The end of the expansion
(x = +2.00 m) causes shockwaves, which are deflected toward
0
0.00
0.02
h(m)
0.04
0.06
20 40
t (sec)
60
X = 3.00 m, y = −0.66m
X = 3.00 m, y = −0.71m
X = 3.00 m, y = 0.00m
80
Figure 2 Temporal variation of flow depth h(t) at X = 3 and three
locations y
Fo = 2
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
h/hoo
Model
Experiment
Expansion
Fo = 4
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
h/h
Model
Experiment
Expansion
Fo = 6
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
h/h
Model
Experiment
Expansion
Fo = 8
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
h
αα
αα/hoo
Model
Experiment
Expansion
Figure 3 Calculated vs. measured axial surface profiles (ha/ho)
Fo = 2
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
hw/ho
Model
Experiment
Fo = 6
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
hw/ho
Model
Experiment
Fo = 8
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
hw/ho
Model
Experiment
Fo = 4
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
hw/ho
Model
Experiment
Figure 4 Calculated vs. measured wall surface profiles (hw/ho)
Fo = 2
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
u/uo Model
Experiment
Fo = 4
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
u/uo
Model
Experiment
Fo = 6
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
u/uo
Model
Experiment
Fo = 8
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
u/uo
Model
Experiment
α
αα
α
Figure 5 Calculated vs. measured longitudinal axial velocities surface
profiles (ua/uo)
Fo = 2
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
uw/uo
Model
Fo = 4
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
uw/uo
Model
Experiment
Fo = 6
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
uw/uo
Model
Experiment
Fo = 8
0.0
0.5
1.0
0.0 1.0 2.0 3.0
X
uw/uo
Model
Experiment
Figure 6 Calculated vs. measured longitudinal wall velocities (uw/uo)
thechannelcenterline. InFigs3and4, thenormalizedflowdepths
at the axis and near the wall, ha/ho and hw/ho, respectively,
are compared with the experimental data as a function of the
dimensionless length coordinate X. The agreement of the com-
putations with the experimental data is generally satisfactory;
however, computed depths are higher than measurements imme-
diately downstream of the expansion. The calculated difference
between the maximum and minimum flow depths near the wall

5. 406 A.I. Stamou et al. Journal of Hydraulic Research Vol. 46, No. 3 (2008)
Fo = 2
0.0
1.0
2.0
3.0
0.0 1.0 2.0
X'
y's
Model
Experiment
Fo = 4
0.0
1.0
2.0
3.0
0.0 1.0 2.0
X'
y's
Model
Experiment
Fo = 6
0.0
1.0
2.0
3.0
0.0 1.0 2.0
X'
y's
Model
Experiment
Fo = 8
0.0
1.0
2.0
3.0
0.0 1.0 2.0
X'
y's
Model
Experiment
Figure 7 Calculated vs. measured location of shock front ys(X )
Fo = 2
0.0
0.5
1.0
-3.2 -1.2 0.8 2.8 4.8 6.8
X'
σ
Model
Experiment
Expansion
Fo = 4
0.0
0.5
1.0
-1.6 -0.6 0.4 1.4 2.4 3.4
X'
σ
Model
Experiment
Expansion
Fo = 6
0.0
0.5
1.0
-1.0 0.0 1.0 2.0
X'
σ
Model
Experiment
Expansion
Fo = 8
0.0
0.5
1.0
-0.8 0.2 1.2
X'
σ
Model
Experiment
Expansion
Figure 8 Calculated vs. measured standard deviation σ (X )
was equal to 0.39 m, 0.66 m, 0.80 m, and 0.89 m, for Fo = 2, 4, 6,
and 8, respectively; i.e. it increases with Fo, and so does the flow
non-uniformity. The relatively small local maximum of ha/ho at
X = 1 for Fo = 2, which was observed by Mazumder and Hager
(1993) and was attributed to the crossing of the first disturbance
with the axis, was not “captured” by the numerical model. It
is noted, however, that numerical calculations by Tseng (1999)
and Jimenez and Chaudhry (1988), and the experimental data of
Rouse et al. (1951) in a similar modified Rouse profile showed
trends similar to those in Figs 3 and 4, without the appearance of
such a local maximum.
In Figs 5 and 6 the normalized average velocities along the axis
and near the wall, ua/uo and uw/uo, respectively, are compared
with experimental data as a function of X. Both distributions
show a satisfactory agreement with the experimental data. Fur-
thermore, the calculated vertical velocity profiles showed that
there are no significant variations of the flow velocities in the
vertical direction, as also observed in the experiments.
In Fig. 7 the location of the shock front is plotted as ys(X )
and compared with the experimental data. The coordinate X =
(x − Lt)/(boFo) = X − Lt/boFo, has its origin at the end of
the modified Rouse curve corresponding to Lt = 2.0 m and ys =
ys/(bo/2). Theagreementbetweencomputationsandexperiments
is satisfactory for Fo ≤ 4, while for higher Froude numbers the
calculated plot ys (X ) shows a more linear behavior than the
experiments.
The standard deviation σ = (h/ho−hi/ho)
n−1
2 1/2
of the nor-
malized free-surface profile may be considered as a measure of
free-surfaceuniformity, wherehi/ho isthenormalizedflowdepth
at a location i, h/ho is the normalized cross-sectional average
depth and n is the number of depth values of the cross-section
used in the calculation of σ. In Fig. 8 calculated σ values are
compared with measurements.
As for the shock front (Fig. 7) the agreement is good at
low Fo values, while at high Fo the calculated σ values in the
approach flow channel are much smaller than the measurements
in the expansion region. This results in a significant discrepancy
between calculations and data in the first half of the expansion;
this discrepancy may be attributed to the simplified “inlet” bound-
ary condition requiring σ = 0. In the second half of the expansion
the agreement is satisfactory. For Fo = 2, σ increases along
the expansion and reaches a maximum value σmax = 0.32 at
Xmax = 0.73 downstream of the expansion; then, it decreases
and approaches zero. A similar behavior is observed for Fo = 4;
the maximum value is equal to σmax = 0.44 at Xmax = 0.54.
For Fo = 6 and Fo = 8 the maximum values occur in the
expansion; for Fo = 6, σmax = 0.48 at Xmax = −0.16 and
for Fo = 8 σmax = 0.65 at Xmax = −0.10. These free-surface
uniformity calculations verify the conclusion of Mazumder and
Hager (1993) that flows with Fo/FD ≥ 3 are unacceptable, while
flows with a smaller Froude number ratio, such as Fo/FD = 2 to
2.5, may be considered acceptable.
From the calculations and their comparison with the data in
Figs 3–8 it can be concluded that the model predicts satisfactorily
the flow field characteristics in gradual expansions, especially for
2 ≤ Fo ≤ 4 or, that is, for small ratios of the flow Froude number
Fo to the Froude number FD used for defining the expansion
geometry.
4 Application of the model
4.1 Expansion structure
Diakoniaris is an ephemeral river, which flows for a distance of
approximately 5 km through the west suburbs of the city of Patras,
Greece. In December 2001, during a strong flood event, the exist-
ing cross section of the regulated part of the river was proven to be
inadequate to carry the peak discharge, resulting in human losses
and drawing public awareness. An updated study was carried out
and certain modifications were proposed to increase the design
discharge. One of the most important components of the study
was the compound expansion structure TE1, shown in Fig. 9.
The existing part of the structure is a gradual expansion from a
single 6 m ×2 m covered channel (U2) to two 6 m ×2 m covered
channels (D2-1 and D2-2) with a discharge capacity of 100 m3
/s.
Accordingtotheupdatedstudy, theexpansionistobemodifiedby
adding channels U1, U3, D1, and D3 to the existing structure, as
shown in Fig. 9, thus increasing its capacity to 195 m3
/s. Figure 9
shows that the modified structure involves a double supercritical

6. Journal of Hydraulic Research Vol. 46, No. 3 (2008) 3-D numerical modeling of supercritical flow in gradual expansions 407
H=+1.30
H=+2.01
D1
D2-1
D3
D2-2
H=+2.22
H=+1.80
H=+1.80
H=+1.30
-30.0 0.0 22.9 82.9 x (m)
y (m)
0.0
+8.1
-8.1
3
2-2
2-1
1
Piers
Bottom
Ramps
-10.5
+10.3
U1
U2
U3 3
2-2
2-1
1
H=+2.22
H=+2.22 H=+2.01
Flow direction
Figure 9 Plan view of expansion structure (not in scale), with bottom
elevations (H) in [m]
expansion, which is approximately symmetrical, with variable
bottom elevation (channels D1 and D3 are deeper than D2-1 and
D2-2), two piers with circular cross section of diameter equal to
0.50 m, two inner (existing) walls upstream and three inner (exist-
ing) walls downstream of the expansion. The Froude number of
the approach flow is Fo = 1.6.
The 3-D model previously described was applied to optimize
the design of the expansion structure, aiming in particular at
(a) investigating the flow conditions in the expansion and (b)
ensuring that in all four channels downstream of the expansion
(b1) the water elevation is approximately the same and (b2) the
flow is distributed in proportion to the channel widths.
A linear expansion geometry was proposed by the Consult-
ing firm Hydrotech Hydraulic Studies Ltd, based on approx-
imate one-dimensional hydraulic calculations with HEC-RAS
(USACE, 2002). The geometry of the outer walls is an almost lin-
ear approximation to the respective modified Rouse wall profile
(Rouse et al., 1951), obtained by assuming a simple symmetrical
expansion with a ratio β = b2/bo = 20.40/16.20 = 1.26.
4.2 Computational details
The computation domain is shown in Fig. 9. Boundary condi-
tions were defined as discussed in Section 3.2. In particular, at
the “inlet” a parallel flow was imposed, with uniform approach
flow depth equal to ho = 2.02 m and velocity equal to uo =
6.90 m/s. An orthogonal numerical grid consisting of 753,300
control volumes and grid dimensions x = z = 0.25 m,
y = 0.20 m ÷ 0.27 m was used in the computations after
a series of preliminary calculations to ensure grid independent
results.
4.3 Results and discussion
Several series of calculations were performed to determine the
characteristics of the flow field and to investigate the effects of:
(i) Presence of piers, which are necessary for structural reasons,
and (ii) Presence of the bottom ramps in the second half of the
expansion, between channels U1 and D1, and channels U3 and
D3. The bottom ramps permit a gradual decrease of the bottom
elevation from mid-way of the expansion (H = +2.01 m) to the
beginning of channels D1 and D3 (H = +1.30 m); Fig. 9.
Figure 10 presents a top view of the calculated flow field for
the proposed final design, showing the generation of four shock
Figure 10 View of water surface
waves. The waves initially strike on the sides of the two inner
and the two outer walls defining the first contact points, which
are denoted with (1). Each of the four waves follows its own
path in each of the four downstream channels, being reflected
on the walls of the channel in locations noted as (2), (3), and
(4); then, the waves are dissipated at a distance of approximately
45 m downstream of the beginning of the expansion. Significant
flow disturbances are also observed immediately downstream of
the piers.
In Fig. 11, the water surface profiles are shown along 4 longi-
tudinal sections in the expansion and adjacent channels, namely
1, 2-1, 2-2, and 3, as defined in Fig. 9. From the calculations and
Figs 10 and 11, it is noted that the flow surface never reaches
the soffit of the channels and the minimum freeboards in the four
downstream channels D1, D2-1, D2-2, and D3 are of the order
of 0.40 m. The average elevations of the free surface in the four
downstream channels at the outlet boundary range from 2.82 m
to 2.92 m.
Also, Fig. 11 compares calculated elevations with HEC-
RAS calculations (USACE, 2002). Upstream (−30.00 m ≤ x ≤
0.00 m) and sufficiently downstream (x ≥ 50.00 m) of the expan-
sion, calculations with HEC-RAS can be considered as satisfac-
tory and conservative. As expected, surface elevations (mainly
superelevations) within the expansion (0.00 m ≤ x ≤ 22.90 m)
and immediately downstream (22.90 m ≤ x ≤ 50.00 m) cannot
be captured with the one-dimensional HEC-RAS; in the expan-
sion, thecalculateddepthsarehigherupto0.40 mthanHEC-RAS
−30 0
0
2
4
6
Expansion
Elevation(m)
30 60
Soffit
HEC-RAS
Section 3
Section 2-2
Section 2-1
Section 1
x (m)
Figure 11 Calculated water surface profiles in longitudinal sections 1,
3, 2-2, and 2-1

7. 408 A.I. Stamou et al. Journal of Hydraulic Research Vol. 46, No. 3 (2008)
Figure 12 Velocity vectors close to flow surface
predictions, while downstream of the expansion differences reach
up to 0.60 m.
In Fig. 12 calculated horizontal velocity vectors are shown
at a horizontal plane close to the free surface (H = 2.63 m).
Calculations indicate that there were no significant variations of
the flow velocities in the vertical direction; such uniformity was
also observed in the calculations and the experimental data of the
simple expansion (Mazumder and Hager, 1993). Maximum flow
velocities near the surface were up to 9.7 m/s. The directions of
the velocity vectors were governed by the geometrical character-
istics of the expansion structure; there were sudden changes of
these directions (a) immediately downstream of the two upstream
inner rounded walls and (b) immediately upstream of the three
downstream inner rounded walls. As expected, due to the small
valueoftheexpansionratio(β ≈ 1.3), nosignificantrecirculation
regions occurred.
The percentages of discharge in the four downstream channels
D1, D2-1, D2-2, and D3, were equal to 19.5%, 30.1%, 29.7% and
20.7%, respectively. This distribution is close to the distribution
of the proportional width 17.3%, 32.2%, 32.2%, and 18.4%,
respectively; therefore, the discharge distribution downstream of
the expansion is considered satisfactory.
Concerning the effect of the piers, calculations showed that
their absence slightly improves the general characteristics of the
flow fields and leads to higher velocities in the four downstream
channels (by approximately 2%) and lower flow depths (also by
about 2%).
The effect of the presence of the bottom ramps was investi-
gated by examining an alternative design where the expansion
was assumed to have a uniform slope (from H = 2.22 m to H =
1.80 m) and a bottom drop equal to 0.50 m (from H = 1.80 m
to 1.30 m) at the entrance of channels D1 and D3. Calculations
showed that in that case recirculation regions formed in the ver-
tical direction downstream of the drop. More importantly, the
average water elevations in the downstream end of the four down-
stream channels ranged from 2.73 m to 3.09 m, i.e. there was no
uniform distribution of the water level in the four channels.
5 Conclusions
A 3-D numerical model was applied to supercritical flow in a
gradual expansion following the modified Rouse wall profile.
Results were obtained for four Froude numbers, namely Fo =
2, 4, 6, 8, and compared to experimental data by Mazumder
and Hager (1993) concerning flow depths, velocities, shock front
location, and free surface uniformity. A satisfactory agreement
was obtained in most cases.
Subsequently, the model was applied to design a complicated
expansion structure, between a triple upstream conduit and a
quadruple downstream channel. The CFD calculations showed
that the flow conditions in the expansion are in general satis-
factory; in particular, in all four channels downstream of the
expansion the water elevation is approximately the same and the
flow is distributed in proportion to the channels width. Bottom
ramps are necessary to minimize the effect of uneven bottom
elevations, while the effect of pier removal is minor. CFD models
can be a powerful tool in engineering design therefore, provided
that they are carefully verified with experimental data prior to
application.
Acknowledgments
The research leading to this paper was conducted in connection
to the Project: “Research by physical and mathematical mod-
els of three transition structures for the storm water protection
of Diakoniaris River,” 2003–2004, funded by Mechaniki S.A.
via the Helenic Ministry of Environment, Planning and Pub-
lic Works. The authors would like to thank Mr. V. Sioutas of
Mechaniki S.A. and Mr. Th. Fornier and Mr. M. Gatopoulos of
Hydrotech Hydraulic Studies Ltd for the provision of data and
information from the Final Design of Diakoniaris River.
Notation
Ax = Fractional area open to flow in x-direction
Ay = Fractional area open to flow in y-direction
Az = Fractional area open to flow in z-direction
bo = Approach channel width (m)
b2 = Downstream channel width (m)
cµ = Empirical constant of k-ε model
Fo = Approach flow Froude number
FD = Design Froude number
fx = Viscous acceleration in x-direction (m/s2
)
fy = Viscous acceleration in y-direction (m/s2
)
fz = Viscous acceleration in z-direction (m/s2
)
Gx = Body acceleration in x-direction (m/s2
)
Gy = Body acceleration in y-direction (m/s2
)
Gz = Body acceleration in z-direction (m/s2
)
g = Gravitational acceleration (m/s2
)
H = Elevation (m)
ho = Approach flow depth (m)
h = Average cross-sectional flow depth (m)
hi = Local flow depth at location i(x, y, z) (m)
ha = Flow depth at axis (m)
hw = Flow depth near wall (m)
k = Turbulent kinetic energy per unit mass (m2
/s2
)
ko = Turbulent kinetic energy at inlet (m2
/s2
)

8. Journal of Hydraulic Research Vol. 46, No. 3 (2008) 3-D numerical modeling of supercritical flow in gradual expansions 409
lo = Characteristic inlet dimension (m)
Lt = Expansion length (m)
n = Manning’s roughness coefficient (m1/3
/s)−1
p = Pressure (Pa)
Q = Discharge (m3
/s)
t = Time (s)
u = Velocity component in x-direction (m/s)
uo = Approach flow velocity (m/sec)
ua = Average velocity at axis (m/s)
uw = Average velocity near the wall (m/s)
v = Velocity component in the y-direction (m/s)
VF = Fractional volume open to flow
w = Velocity component in z-direction (m/s)
X = Normalized streamwise coordinate
X = Transformed normalized coordinate in x-direction
x = Coordinate in x-direction (m)
y = Coordinate in y-direction (m)
yb = Transverse coordinate for modified Rouse curve (m)
ys = Shock transverse coordinate (m)
ys = Normalized shock coordinate in y-direction (m)
z = Coordinate in z-direction (m)
α = Constant
β = Expansion ratio
x = Numerical or experimental grid dimension
in x-direction (m)
y = Numerical or experimental grid dimension
in y-direction (m)
z = Numerical grid dimension in y-direction (m)
ε = Turbulent kinetic energy dissipation rate (m2
/s3
)
εo = Turbulent kinetic energy dissipation
rate at inlet (m2
/s3
)
ρ = Density (kg/m3
)
σ = Standard deviation of free-surface profile from
cross-sectional average h/ho
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