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Experimental study on mean velocity characteristics

of flow over vertical drop
a a a b
C. Lin , W.-Y. Hwung , S.-C. Hsieh & K.-A. Chang
a
Department of Civil Engineering , National Chung-Hsing University , Taichung, 402,
Taiwan
b
Department of Civil Engineering , Texas A&M University , College Station, TX, 77843,
USA
Published online: 26 Apr 2010.

To cite this article: C. Lin , W.-Y. Hwung , S.-C. Hsieh & K.-A. Chang (2007) Experimental study on mean
velocity characteristics of flow over vertical drop, Journal of Hydraulic Research, 45:1, 33-42, DOI:
10.1080/00221686.2007.9521741

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 Journal of Hydraulic Research Vol. 45, No. 1 (2007), pp. 33–42
© 2007 International Association of Hydraulic Engineering and Research

Experimental study on mean velocity characteristics of flow over vertical drop

Etude expérimentale des caractéristiques des vitesses moyennes d’écoulement
en chute verticale
C. LIN, Department of Civil Engineering, National Chung-Hsing University, Taichung 402, Taiwan

W.-Y. HWUNG, Department of Civil Engineering, National Chung-Hsing University, Taichung 402, Taiwan

S.-C. HSIEH, Department of Civil Engineering, National Chung-Hsing University, Taichung 402, Taiwan

K.-A. CHANG, Department of Civil Engineering, Texas A&M University, College Station, TX 77843, USA
Downloaded by [Northeastern University] at 21:33 09 October 2014

ABSTRACT
The characteristics of flows over a vertical drop were investigated experimentally using laser Doppler velocimetry for detailed quantitative velocity
measurements, and a flow visualization technique for qualitative study of flow pattern. A range of velocity and depth of the subcritical approaching
flows was tested to understand the flow structure at the regions of the falling jet, the sliding jet, and the energy dissipating pool. Using the measured
velocity, four similarity profiles of the mean velocity at different locations were obtained: the jet velocity at the intersection of the falling jet and the
sliding jet, the jet velocity along the free surface of the sliding jet, the maximum negative velocity and the mean horizontal velocity of the deflected
wall jet in the pool. Variation trends of several important characteristic velocity and length scales of the deflected wall jet in the pool are also discussed.

RÉSUMÉ
Les caractéristiques des écoulements en chute verticale ont été étudiées expérimentalement en utilisant le vélocimètre laser Doppler pour des mesures
quantitatives détaillées de vitesse, et une technique de visualisation d’écoulement pour l’étude qualitative de la configuration de l’écoulement. Une
gamme de vitesses et de profondeurs des écoulements d’approche sous-critiques a été examinée pour appréhender la structure de l’écoulement dans
les régions de chute du jet, du jet glissant, et la dissipation d’énergie dans le bassin. En utilisant la vitesse mesurée, quatre profils de similitude de la
vitesse moyenne à différents endroits ont été obtenus: la vitesse du jet à l’intersection du jet en chute et du jet glissant, la vitesse du jet à la surface
libre du jet glissant, la vitesse négative maximum et la vitesse horizontale moyenne du jet défléchi par le mur du bassin. Les tendances de variation
de plusieurs échelles caractéristiques importantes de vitesse et de longueur du jet défléchi dans le bassin sont également discutées.

Keywords: Falling jet, sliding jet, impact velocity, deflected wall jet, similarity profile.

1 Introduction the first fundamental study of the overflow was carried out by
Rouse (1936). Experimental investigation on the hydraulic char-
Free overfalls are a very traditional problem in hydraulic engi- acteristics of a free overfall with a subcritical flow approaching
neering dealing with open channels. The meaning of a “free the drop structure was conducted by Moore (1943). Moore indi-
overfall” over a vertical drop is a stream flow encounters a sharp cated that energy loss at the drop depends on the relative height
drop structure in an open channel. Free overfalls exist in both of the drop. White (1943) developed a theoretical solution to
natural and man-made channels. For instance, the flow running predict the energy loss based on several assumptions, which
over a weir constructed across a river is the most common exam- were subsequently questioned and modified by Gill (1979) and
ple. Free overfalls are also formed by riverbed erosion at the Rajaratnam and Chamani (1995). Rand (1955) proposed empiri-
downstream side of the bed protection works that were built to cal equations to describe the characteristics of free overfalls using
control the slope or elevation of the riverbed. Therefore, under- a dimensionless parameter known as the “drop number”. More-
standing the flow characteristics at the drop structure is important over, Marchi (1993) presented an analytic model to predict the
in designing a downstream pool for energy dissipation through falling jet profile of an overflow that is applicable to both the
the use of the structure. upstream subcritical and supercritical flow conditions. Wu and
In the past, the external flow profiles of an overfall, such as the Rajaratnam (1998) investigated experimentally the flow regimes
upper and lower free surface profiles of the falling jet, the length at a rectangular drop when the tailwater depth approaches the
and depth of the pool, and the energy loss of the free overfall, height of the drop or exceeds it with a subcritical approaching
were studied extensively. To the best of the authors’ knowledge, flow.

Revision received May 28, 2004/Open for discussion until August 31, 2007.

33
 34 Lin et al.

Recently, the velocity profiles of both the falling jet and the arc length along the upper surface of the falling and sliding jet
sliding jet of a free overfall were investigated by Rajaratnam and with S = 0 being the upper jet surface at X = 0. S0 is the length
Chamani (1995) using Prandtl probes with external diameters from S = 0 to the intersection of the upper falling jet surface
of 2.6 and 3.2 mm. However, more works with quantitative mea- and the horizontal line of Y = Yp which is the height of the
surements are still needed to better characterize the flow. Detailed pool behind the falling water. In addition, Sa is the arc length at
flow structures such as the velocity fields of the falling and slid- the jet upper surface from S = 0 to the line normal to the jet
ing jet, the circulating vortical flow pattern and corresponding upper surface, passing the jet centerline, and intersecting with
velocity characteristics in the pool, and the energy dissipation the horizontal line Y = Yp . In general, S0 ≈ Sa for all of the
mechanism in the pool have not yet been fully understood. cases tested in the present study.
The purpose of the present study is to investigate the character- The falling jet is defined as the portion of flow after it passes
istics of mean velocity fields in the falling and sliding jet and the the drop (at X = 0) and forms a jet and before the jet intercepts
pool of a two-dimensional free overfall, using a flow visualization with the downstream pool (at Y = Yp ). The sketch is illustrated
technique and laser Doppler velocimetry (LDV) for non-intrusive in Fig. 1. The falling jet becomes a sliding jet after it in contact
qualitative and quantitative measurements. with the pool (at Y = Yp ) and sliding downstream. In the figure
q is the volume flow rate per unit width, Uc and Yc the mean
critical horizontal velocity and critical depth of the upstream flow,
2 Experimental apparatus and conditions H the height of the vertical drop, U1 and Y1 the downstream
mean horizontal velocity and depth, and Vi the impact velocity
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Experiments were conducted in a re-circulating water flume or entering velocity of the falling jet (although the jet does not
located at the Hydraulic Laboratory of the Department of Civil impact but sliding down the pool). The use of the term “impact
Engineering, National Chung-Hsing University. The flume is of velocity” will be explained later. Note that on the vertical walls
8.85 m long with a glass-walled and glass-bottomed test section of the two drop structures (at Y = 9.0 cm for H = 11.0 cm, and
of 3.06 m long, 0.50 m wide and 0.54 m deep. The test section at Y = 18.0 cm for H = 20 cm), five 1-cm diameter ventilating
locates behind a contraction section to increase the flow uniform- holes were drilled to maintain atmospheric pressure in the air
ness and suppress boundary layer development. The flow speed pocket behind the falling water.
is controlled and maintained at a constant-speed condition by a An LDV system was used to measure the flow velocity in the
feedback circuit. Turbulence and disturbance in the flow at the test study. The LDV system is a two-component colorburst-based
section were minimized by three layers of perforated steel plates four-beam fiber-optic system (TSI system 90-3). The system
followed by one honeycomb, four meshes of different sizes, and consists of a 5 W argon-ion laser, a fiber-optic probe, two photo-
the carefully designed contraction section followed by the test multipliers, a 3D traverse, and associated optical lenses and two
section. signal processors. Due to the small dimension of the jet in thick-
Two vertical drops made of acrylic plates with heights of 11.0 ness, only one velocity component was measured in the study
and 20.0 cm were installed, one at a time, in the test section of through the use of two green laser beams. The LDV fiber-optic
the water channel. A schematic diagram on the setup of the test probe was mounted on an angle vise, which not only features
channel and the free overflow is shown in Fig. 1. Two coordinate accurate adjustment on setting obliqueness but also provides an
systems were used in the study. The Cartesian coordinate system extra degree of freedom in rotation (continuous 90◦ adjustment
(X, Y) represents the coordinates with the origin at the lower from horizontal to vertical) in addition to the 3D displacement
corner of the drop. The local coordinate system (S, yr ) presents provided by the traverse. Photo 1 shows the mounting of the LDV
the local coordinates along the upper surface of the falling and probe. Using the device, Vi was obtained by rotating the probe
sliding jet with S tangential to the upper jet surface and yr pointing until the maximum velocity was achieved. Titanium dioxide pow-
outward normal to the surface. In addition, S also represents the der with a mean diameter of 1.0 µm was used as the LDV seeding

Figure 1 Sketch of the overflow.

 Characteristics of flow over vertical drop 35
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Photo 1 Apparatus of the fiber-optic probe and the LDV system. (a) The fiber-optic probe was mounted on the angle vise, which was fixed on the
3D traverse. (b) A magnified view of the fiber-optic probe and the angle vise. (c) A view of the overfall as well as the measuring point of the two
converged LDV laser beams.

Figure 2 Comparison of the free surface profile for the case of Yc /H = 0.146: line, theory from Marchi (1993); dots, laboratory measurements.

particles. In addition to the use of LDV for point velocity mea- As a result, the LDV measurements as well as the vertical light
surements, a second 5 W argon-ion laser with light sheet optics sheet plane for flow visualization throughout the experiments
and a photographic camera (Nikon F5) was used for qualitative were made at approximately 8 cm from the sidewall. Since the
flow visualization. Following the technique employed by Lin flow is quasi-steady, time averaging was used to extract the mean
et al. (1994, 2002), aluminum powder with a 10 µm mean diam- quantity from the LDV velocity data. In addition, the free surface
eter was introduced into the water as the tracing particle and illu- of the overflow was measured using a point gauge mounted on
minated by the thin sheet of laser light. Particle trajectories were the rail at the top of the flume.
then captured using the camera with a controlled exposure time. The free surface measurement using the point gauge for the
A series of experiments with a range of velocity and depth of case of Yc /H = 0.146 was compared with the theoretical results
the subcritical approaching flow was conducted with the two ver- of Marchi (1993) and shown in Fig. 2. From the comparison,
tical drops. In these cases the incoming flow was varied from it is found that the upstream water level and both the upper
Yc /H = 0.068 to 0.490 while Yc was varied from 0.75 to and lower surfaces of the falling jet agree very well with the
5.35 cm. The corresponding Froude number of the approaching theory. Other measured parameters expressed as functions of
flow varies from 0.455 to 0.870. The streamwise velocity distri- Yc /H, such as
E/E0 = 0.115(Yc /H)−0.642 (the relative energy
bution in the transverse direction of the approach flow and the loss at the drop), Yp /H = 1.135(Yc /H)0.713 (the relative pool
pool was first measured using LDV to check whether the flows depth), and Lp /H = 2.403(Yc /H)0.609 (the relative pool length)
are two-dimensional and repeatable. Measurements were made were all compared with previous studies (Moore, 1943; Rand,
at 10 positions along the transverse direction (i.e. the Z direc- 1955; Rajaratnam and Chamani, 1995; Chanson, 1996) and in
tion) with an increment of 1 cm and repeated for three times. We very good agreement (not shown here). This indicates that the
concluded that the mean flows are highly repeatable and two- setup and control conditions are appropriate in the laboratory
dimensional except within the region 4 cm from the sidewalls. and consistent with others.
 36 Lin et al.

3 Flow visualization of free overfall 4 Velocity characteristics of falling and sliding jet

A sample image taken using the qualitative flow visualization The mean velocities of the falling and sliding jet at five cross-
technique for the case of Yc /H = 0.383 is shown in Photo 2(a). In sections were measured using LDV for the following four cases:
the picture, a free falling jet is formed right after the upstream flow Yc /H = 0.218, 0.242, 0.300, and 0.383. The locations of the five
passing the drop. The falling jet then encounters the downstream cross-sections, marked as “a” to “e”, start from the arc length of
flow and forms a pool at the drop. One can easily find that the S = Sa (a very short distance upstream of the intersection of the
jet indeed never fully enters the pool but slides through the pool, falling jet and the pool at Y = Yp as shown in Fig. 1) to near the
although such jet behavior has been described as “impacting” bottom of the downstream flow. Figure 3 shows the mean veloc-
or “entering” the pool by other researchers. We therefore fol- ity profile for the case of Yc /H = 0.300. It clearly demonstrates
lowed the tradition definition of Vi and defined it as the impact or that the large velocity gradient near the sliding jet–pool inter-
entering velocity for consistency and easy comparison. For the section decreases as the jet moving farther downstream. This is
same reason, we used the word impact point to describe the loca- mainly due to the entrainment effect caused by turbulent mixing
tion where the falling jet intercepts with the pool. Photo 2(b,c) along the shear layer. This diffusion-alike mechanism decreases
shows two other cases with different values of Yc /H (0.146 and the velocity gradient as well as momentum exchange (caused by
0.068). From these pictures, it is clear that there exists a very mean flow) as the jet moving downstream, and results in energy
large velocity gradient between the sliding jet and the pool. The exchange between the jet and the pool as well as energy dissipa-
large velocity gradient forms a shear layer that in turn creates a tion due to turbulence. This is indeed the mechanism of how the
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large entrainment effect with energy exchange. The circulating energy dissipating vertical-drop overflow works in practical river
flow pattern and chaotic turbulence in the pool are also clearly protection structures. The jet velocity near the upper surface con-
evident. One can also find from Photo 2 that the primary circu- tinues to increase as the jet moving downstream (i.e. S increases)
lation in the pool does not occupy the full length of the pool. A due to the gravitational acceleration. The coordinates in the mea-
secondary cell of less organized circulation exists near the corner surement points were obtained first through carefully rotating and
of the drop. Such flow pattern was also observed by Rajaratnam matching the measuring volume and the two LDV green beams
and Chamani (1995) through the use of dye injection technique to the upper free surface of the sliding jet and then position-
and video images. ing the coordinates of the measuring points in the yr direction
accordingly. The mean velocity of the sliding jet in the direction
parallel to the jet surface, us , was measured every 0.25 mm along
the cross-sections in the yr direction until us = 0 was reached.
The impact velocity Vi of the falling jet is obviously an impor-
tant parameter in this study. Gill (1979) proposed an equation to
quantify Vi by considering it as a function of both the gravitational
acceleration and the depth of the pool as


Vi = 2g(H + 1.5Yc − Yp ) (1)

in which Yc can be calculated as Yc = (q2 /g)1/3 . Using the

mean impact velocity from our measurement (averaged over the
cross-section in the yr direction), Eq. (1) is compared with
the experimental data. The comparison is shown in Fig. 4 with a
wide range of Yc /H varying from 0.068 to 0.49. From the figure
we found the equation works very well with a maximum discrep-
ancy between the measurements and the calculation being less
than 1.6%. Since Vi is an important parameter and can be accu-
rately predicted, Vi is used as the velocity scale in characterizing
the problem in this study.
From Fig. 3 we have found that the maximum mean velocity
(us )max increases as the sliding jet moving downstream after hit-
ting the pool, i.e. (us )max > Vi for S > S0 . Figure 3 also shows
that the maximum mean velocity (us )max occurs at the upper free
surface of the sliding jet. After an attempt to characterize (us )max
versus the velocity scale Vi with different length scale, including
the upstream critical depth Yc , we found (us )max /Vi can be char-
acterized very well using the length scale S0 . Figure 5 plots the
Photo 2 Pictures taken at the region of sliding jet and pool for three dif- results of (us )max /Vi against S/S0 with four different flow con-
ferent cases. (a) Yc /H = 0.383; (b) Yc /H = 0.146; (c) Yc /H = 0.068. ditions: Yc /H = 0.218, 0.242, 0.300 and 0.383. The maximum
 Characteristics of flow over vertical drop 37
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Figure 3 Jet mean velocity profiles of the case Yc /H = 0.300. Measurements taken at (a) S = Sa = 12.91 cm; (b) S = 13.87 cm; (c) S = 14.82 cm;
(d) S = 15.34 cm; (e) S = 16.19 cm.

Figure 5 Similarity plot of (us )max /Vi against S/S0 for the cases of
Figure 4 Comparison of Gill’s (1979) theory and the measurements for Yc /H = 0.218, 0.242, 0.300, and 0.383.
determining Vi .

mean velocity continuously increases as the jet sliding farther with a correlation coefficient R2 = 0.98 was fitted to the data in
down at the drop. A good linear correlation line of the figure. This implies that the maximum mean velocity of the
(ur )max S sliding jet is influenced mainly by the gravitational effect, not the
= 0.397 + 0.639 (2) upstream critical water depth.
Vi S0
 38 Lin et al.

(a) to a “half-jet”. Although no similarity was found for velocity

us /(us )max versus yr /(S − S0 ) as that of a turbulent jet, there
does exist a similarity between us /(us )max versus yr /br with br
being the half-width of the falling jet right before the intersection
at the “a” cross-section. Figure 6(a) plots us versus yr for the four
cases of Yc /H = 0.218, 0.242, 0.300, 0.383 and at the impact
point while Fig. 6(b) depicts us /(us )max versus yr /br for the same
cases. Using br as a characteristic length scale is the analogy to
the “half-width” of a turbulent jet in which br is equal to the
value of yr at us /(us )max = 0.5. The similarity is clearly seen in
Fig. 6(b) with the fitting line
    
ur (yr /br ) + 0.988
= 0.45 1 + tanh + 0.1
(ur )max 0.1
 
yr
× 0.025 + 1 (3)
br
with an R2 value of 0.99. It should be mentioned that in Fig. 6
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the transition region between the high us region and the low us
region is only about 2–5 mm thick. Such a thin shear layer with a
high velocity gradient could cause large measuring error if a mea-
surement tool with a large measurement volume or an intrusive
probe is used. From the figure we can see that, at the intersecting
(b) point, the maximum mean velocity always occurs at the upper
jet surface. The relatively uniform region having a high velocity
close to (us )max extends to about −0.8yr /br , and the high shear
occurs in the region between −0.8yr /br and −1.2yr /br .

5 Velocity characteristics of pool and deflected wall jet

After the free overflow becoming a falling jet at the drop, the
jet subsequently contacts with the energy dissipating pool and
continues to slide down. The sliding jet eventually touches the
bottom of the lower channel with most of its flow going down-
stream while the rest entraining into the pool. In the vicinity of
the entrainment area the shear flow provides continuous momen-
tum and energy supply to the pool. Following Rajaratnam and
Chamani (1995), who first measured the velocity profile in the
vicinity of the sliding jet, the ratio of the circulating flow, qc /q,
was calculated and showed in Fig. 7. The value of qc /q decreases
from about 0.6 for the case of Yc /H = 0.075 to about 0.17 for

Figure 6 Similarity plot of the jet mean velocity at cross section “a”
(corresponding to the arc length S = Sa ) for the cases of Yc /H = 0.218,
0.242, 0.300, and 0.383. (a) us versus yr ; (b) us /(us )max versus yr /br .

The overflow above the drop structure is similar to a gravity-

controlled jet flow. The difference may be that the water body is
limited by the boundary conditions at the drop, and the incident
angle is limited to a typical and practical drop structure of Yc /H =
O(0.1) to O(1). Through observing the mean velocity profiles of
the falling jet and sliding jet, there seems to exist certain similarity Figure 7 Ratio of circulating flow qc /q versus Yc /H.
 Characteristics of flow over vertical drop 39

Yc /H = 0.48. A regression curve with an R2 value of 0.99 was 0.12 mm) was used in this study. The shear layer, as shown in
fitted to the measurement points as follows Photo 2, is very thin (2–5 mm thick in this study) and could be
 −0.616 very sensitive to the intrusive instrument used.
qc Yc
= 0.108 (4) The horizontal velocity in the pool was measured for the cases
q H of Yc /H = 0.146, 0.218, and 0.383. Figure 8 shows the mea-
The fitting curve is close to what Rajaratnam and Chamani (1995) sured mean velocity profiles for the case of Yc /H = 0.383 at 20
found although certain discrepancy exists. The difference could cross-sections from X = 8.5 to 142 mm (with Lp = 140 mm and
be due to the use of Prandtl probes (external diameters 2.6 Yp = 62 mm). The solid lines in the plot of each profile are for
and 3.2 mm) in Rajaratnam and Chamani’s experiments while visual aid only. Interestingly, the near-bottom mean horizontal
non-intrusive LDV (diameter of measuring volume less than velocity profiles at the inner side (X < Lp ) and outer side

(a)
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(b)

Figure 8 Vertical distributions of the mean horizontal velocity measured at different sections for the case of Yc /H = 0.383: (a) 8.5 mm ≤ X ≤ 80 mm;
(b) 84 mm ≤ X ≤ 142 mm.
 40 Lin et al.

(at X > Lp ) of the pool are very different. The mean velocity U
at the inner side is negative while U becomes positive at the outer
side. Moreover, U decreases when X decreases (closer to the ver-
tical wall), and the maximum negative velocity (hereafter defined
as Umax ) occurs near the bottom of the pool. The mean velocity
distribution in the pool with negative values near the bottom and
positive values near the free surface confirms the circulation pat-
tern shown in Photo 2. Another remarkable feature is that the
mean velocity profiles in the region of 16 mm < X < 92 mm
show similar trend. The mean velocity increases from U = 0
at the bottom to the maximum negative value at approximately
Y = 0.04Yp to Y = 0.05Yp , and then decreases to U = 0 again
at approximately Y = 0.4Yp to Y = 0.45Yp . Above this level, U
increases almost all the way to the free surface. Note that in the Figure 10 Dimensional plot of −Umax against (Lp − X).
region of 92 mm < X < 142 mm, U becomes comparably large
as Y increases over a certain level. This is due to that at this height
the influence of the sliding jet becomes prominent. Similar flow
pattern was also observed in the other two cases, Yc /H = 0.146
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and 0.218 (not shown here).

Based on the measured mean horizontal velocity profiles in the
pool and the flow visualization shown in Photo 2, it is found that
the return flow near the bottom of the pool (hereinafter referred
as a “deflected wall jet”) is seem to be similar to that of a turbu-
lent plane wall jet (Rajaratnam, 1973). In the present case, the
deflected wall jet indeed stems from the bifurcation of the sliding
jet at about X = Lp and issues toward the vertical wall of the
drop structure. The circulation flow in the pool, extending from
the bottom to the free surface, has a finite depth of Yp . On the
other hand, for the case of traditional turbulent plane wall jet, Figure 11 Similarity plot of the non-dimensional maximum negative
the jet issues from a nozzle tangent to a flat smooth wall of infi- velocity −Umax /Vi versus non-dimensional distance (Lp − X)/Yp .
nite length and submerged in a fluid of infinite depth. Following
Rajaratnam (1973), a definition sketch for the deflected wall jet corresponds to a greater value of (Lp − X) because of higher
is shown in Fig. 9. In the figure −Umax denotes the maximum fluid momentum. Figure 11 is the non-dimensional plot of Fig. 10
velocity of return flow, b is the value of Y at U = 1/2Umax , and normalized by the impact velocity Vi and the pool height Yp . It is
b0 is the value of Y at U = 0, which can be considered as the edge interesting that all of the data points collapse onto a single line,
of the deflected wall jet. Curve fittings to the measured points (as indicating the existence of similarity. Regression results in the
shown in Fig. 8) were performed to determine Umax , b and b0 for following linear relationship:
better accuracy. Figure 10 plots −Umax against (Lp − X) for the
Umax (Lp − X)
cases of Yc /H = 0.146, 0.218, and 0.383. It is found that −Umax − = −0.1759 + 0.4675
Vi Yp
decreases linearly with the increase of (Lp − X) within a cer-
tain range for all the three cases. In addition, for the same value (Lp − X)
for 1.0 < < 21.15 (5)
of −Umax , the higher approaching flow (with a greater Yc /H) Yp
Similarly, the growth of the two length scales, b and b0 , indicating
two representative heights of the deflected wall jet, with respect
to (Lp − X) is plotted in Fig. 12. We can find that both b and b0
increase linearly with the increase of (Lp − X), and the tendency
agrees well with that of the traditional turbulent plane wall jet
(Rajaratnam, 1973). Figure 13 shows the non-dimensional plot
of Fig. 12 with the x and y axes scaled with Yp and (Lp − X),
respectively. Interestingly, each of the dimensionless b and b0
collapses into a fitted line:
b Lp − X
= −0.0698 + 0.276 (6)
Lp − X Yp
b0 Lp − X
Figure 9 Definition sketch of the deflected wall jet. = −0.146 + 0.514 (7)
Lp − X Yp
 Characteristics of flow over vertical drop 41

Figure 12 Growth of b and b0 with respect to (Lp − X).

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Figure 14 Similarity plot of the mean horizontal velocity profile for the
deflected wall jet in the pool for the case of Yc /H = 0.146.

Figure 13 Variations of non-dimensional width b/(Lp − X) and

b0 /(Lp − X) against non-dimensional distance (Lp − X)/Yp .

Note that the above two fitted equations are valid for
1.0 < (Lp − X)/Yp < 2.15, meaning that similarities do not
exist when it is to close to the jet deflecting point (at X = Lp ) or
the vertical wall where the vertical velocity component is quite
significant. Moreover, these two lines with different slopes also
illustrate different growth rates for b and b0 . Note that the fit for
b0 /(Lp − X) is far better than that of b/(Lp − X).
Based on the analysis above for Umax , b and b0 , we conclude
that Umax and b0 are important characteristic velocity and length
scales. Accordingly, similarities on the variation of the mean
horizontal velocity along the pool depth (i.e. U vs. Y) for the
deflected wall jet may be obtained by scaling with respect to these
two scales. It is found that there exists a similarity profile within
a certain region for an approaching flow with a specific value of
Yc /H. Surprisingly, a unique similarity profile exists regardless
the approaching flow conditions. Figures 14–16 show the similar-
ity plots for the three cases of Yc /H = 0.146, 0.218, and 0.383,
respectively, within the region of 1.0 < (Lp − X)/Yp < 2.15
Figure 15 Similarity plot of the mean horizontal velocity profile for the
or (0.23−0.32) < X/Lp < 0.66. Regression analysis shows
deflected wall jet in the pool for the case of Yc /H = 0.218.
that the unique similarity profile for the non-dimensional mean
horizontal velocity of the deflected wall jet can be expressed as in which erf( ) is the error function. The curve fits well in all three

2.31(Y/b
1/12
for Y > 0 cases with an R2 value being greater than 0.96 in all the cases. It
U  0) 
= × 1 − erf(0.69Y/b0 ) − 0.76 (8) should be pointed out that the region that fits well with the sim-
Umax 
0 for Y = 0 ilarity curve is the region that neither vertically intersects with
 42 Lin et al.

in Eq. (2) and Fig. 5; (ii) the cross-sectional mean velocity profile
us /(us )max versus yr /br near S/S0 ∼ = 1 of the falling jet as in
Eq. (3) and Fig. 6(b); (iii) the non-dimensional maximum velocity
−Umax /Vi versus (Lp − X)/Yp in the pool as in Eq. (5) and
Fig. 11; and (iv) the non-dimensional mean horizontal velocity
U/Umax versus Y/b0 in the energy dissipating pool in the region
of 1.0 ≤ (Lp − X)/Yp ≤ 2.15 (or 0.23−0.32 < X/Lp < 0.66)
as in Eq. (8) and Figs 14–16.

Acknowledgments

The authors gratefully acknowledge the 2-year financial support

of the Directorate General of the Highways Bureau, Ministry
of Transportation and Communication of Taiwan. The authors
would also like to thank the anonymous reviewer, who made
valuable comments and suggestions on the flow similarities in
the pool, and the correlation between the return flow in the pool
Downloaded by [Northeastern University] at 21:33 09 October 2014

(the deflected wall jet) and the turbulent plane wall jet.

References

1. Chanson, H. (1996). “Discussion on Energy Loss at

Figure 16 Similarity plot of the mean horizontal velocity profile for the
deflected wall jet in the pool for the case of Yc /H = 0.383. Drops”. J. Hydraul. Res. IAHR 34(2), 273–278.
2. Gill, M.A. (1979). “Hydraulics of Rectangular Vertical
Drop Structures”. J. Hydraul. Res. IAHR 17(4), 289–302.
the sliding jet nor closes the vertical wall of the drop structure. 3. Lin, C., Chen, J.Y. and Kuo, C.H. (1994). “Flow Visualiza-
It should also be pointed out that the form of Eq. (8) for the tion on the Near Wake Structure of a Rectangular Cylinder”.
deflected wall jet is very similar to Verhoff’s velocity distribu- Proceedings of the Third Asian Symposium on Visualization,
tion equation for the traditional turbulent plane wall jet (Verhoff, pp. 80–85.
1963; Rajaratnam, 1973). 4. Lin, C., Chiu, P.H. and Shieh, S.J. (2002). “Characteris-
tics of Horseshoe Vortex System Near a Vertical Plate-Base
Plate Juncture”. Exp. Thermal Fluid Sci. 27(1), 25–46.
6 Conclusions
5. Marchi, E. (1993). “On the Free Overfall”. J. Hydraul. Res.
IAHR 31(6), 777–790.
The characteristics of flows over a vertical drop have been studied
6. Moore, W.L. (1943). “Energy Loss at the Base of Free
experimentally using both the qualitative flow visualization tech-
Overfall”. Trans. ASCE 108, 1343–1360.
nique and the quantitative LDV velocity measurement technique.
7. Rajaratnam, N. (1973). Turbulent Jets. Elsevier Scientific
A half-jet or shear-layer-alike structure with a large velocity gra-
Publishing Company, Amsterdam.
dient was observed in the measurement of the falling and sliding
8. Rajaratnam, N. and Chamani, M.R. (1995). “Energy Loss
jet using the techniques. The flow field of the return flow near
at Drops”. J. Hydraul. Res. IAHR 33(3), 373–384.
the bottom of the pool (i.e. the deflected wall jet) was found to
9. Rand, W. (1955). “Flow Geometry at Straight Drop
be comparatively similar to the traditional turbulent plane wall
Spillways”. Proc. ASCE 81, 1–13.
jet. The geometric parameters of the free surface profile, the
10. Rouse, H. (1936). “Discharge Characteristics of the Free
water depth of the pool Yp , the length of the pool Lp , the rel-
Overfall”. Civil Eng. 6, 257.
ative energy loss
E/E0 , and the impact velocity Vi were all
11. Verhoff, A. (1963). The Two-Dimensional Turbulent Wall
compared with previous studies with good agreements. It was
Jet without an External Free Stream. Report No. 626,
found that Yc /H is the appropriate parameter in describing the
Princeton University, Princeton, NJ, USA.
characteristics mentioned above.
12. White, M.P. (1943). “Discussion of Moore”. Trans. ASCE
Four unique similarity profiles were obtained from the LDV
108, 1361–1364.
velocity measurements taken at different locations under different
13. Wu, S. and Rajaratnam, N. (1998). “Impinging Jet and
flow conditions with different values of Yc /H. They are (i) the
Surface Flow Regimes at Drop”. J. Hydraul. Res. IAHR
non-dimensional maximum mean velocity (us )max /Vi versus the
36(1), 69–74.
non-dimensional jet trajectory S/S0 of the falling and sliding jet as