Flowing of water channels, a force in flow direction is developed on the channel
bed and walls. The reaction of this force which pulls the flow body is known
as tractive force. The flow structure in an open channel is directly affected
by the shear stress distribution along the wetted perimeter. It is difficult
to determine boundary shear distributions in real life applications. Laboratory
flume studies have still been serving to cope with this difficulty.
The problem of separating the bed shear stress and the side-wall shear stress
is very important in most open-channel flow studies. For example, in order to
estimate the amount of bed load, the wall shear stress should be subtracted
from the total shear stress. Similarly, for estimation of erosion in sea coast
and river levees, wall shear stress must be known. Since, the 1960s, several
experimental studies have been reported for measurement of boundary shear stresses
in rectangular open channels with different aspect ratios (Knight,
1981; Knight et al., 1984). Similar works
were conducted later to evaluate effect of sidewalls on bed shear stress and
to reconfigure previous relationship for estimation of the sidewall corrected
bed shear stress in open channel flow (Cheng and Chua, 2005;
Seckin et al., 2006). For the pipe flow, Berlamont
et al. (2003) studied boundary shear stress in partially filled pipes
with high wetted perimeter and hydraulic radius. An analytical approach by Guo
and Julien (2005) revealed considerable effect of sidewall on generation
of eddy viscosity and secondary currents and their effects on bed shear stress.
Recently, Khodashenas et al. (2008) presented
a comparison between verities of methods developed for boundary shear stress
in open channel flow. Results of these shear studies support the understanding
of turbulent boundary layer flow and computation of velocity profile for open
channel flows (e.g., Tang and Knight, 2009).
In spite of the completed work on open channel flow, more shear stress measurements on different flow and boundary conditions are required to validate results and put them into practice. The purpose of this study is to determine contribution of wall and bed shear force on total boundary shear in a relatively wider rectangular channel. That experiments were conducted in a 25% larger channel than have ever been used with aspect ratios around 6-19. A set of nonlinear regression-based equations were developed for estimation of percentage of wall and bed shear force and shear velocity in wider channels and larger aspect ratios.
MATERIALS AND METHODS
Knight et al. (1984) used 43 data points
for range 0.3<B/H and 12 data points for 6<B/H to correlate following
dimensionless log-relationship for percentage of wall shear force
where, SFw is the percentage of the shear force acting on the channel walls; B is the width of the channel; H is the flow depth.
Using the energy principle for unit width of channel (a two dimensional analysis), the dimensionless mean value of wall and bed shear stresses were expressed as:
are the mean wall and bed shear stresses, respectively; ρ is fluid density;
g is gravitational acceleration, Sf is the energy gradient.
Since, application of bed shear velocity
is more common in practice of open channel flow, using Eq. 2,
the dimensionless bed shear velocity will be.
In addition, the mean wall and bed shear stresses may be presented in dimensionless forms using the mean overall boundary shear stress (=ρgRSf).
in which R is hydraulic radius.
|| (a) Experimental flume and Preston tube, (b) pressure transducer
Experimental Set up
The experiments were conducted in 2009 at the Hydraulic Modeling Laboratory
of the School of Water Sciences and Engineering of Shahid Chamran University
of Ahwaz, Iran. A rectangular flume 8.6 m long, 80 cm wide and 55 cm height
was constructed for the experiments. The boundary shear stresses were measured
all around the wetted perimeter of the Plexiglas flume using a Preston tube
(4-mm outside diameter, Fig. 1). All measurements were conducted
in a constant hydraulic gradient of 1.05x10-3 at mid section of the
flume where flow was uniform and fully turbulent. Dynamic and static pressures
were recorded using a differential pressure transmitter (Model: Dwy-PT616) with
a pressure range of ±76 mm of water, accuracy of 0.25% and stability
of 1%. An analog/digital signal converter (Model: RL-PAXP) was used to produce
compatible pressure signals for computer and data acquisition software. A frequency
of 10 s-1 and period of 2 min was assigned for collection of data
in every run of experiments. The time average of the collected data was incorporated
in analyses of wall and bed shear stresses. Velocities were measured with a
Nixon probe micro-propeller and were checked with a sharp V-notch weir down
stream of the water circulating system. The flow depths varying from 4.3 and
13.3 cm were measured using a precision point gage with accuracy of 0.1 mm.
Shear stresses were measured at intervals of 1 to 5 cm around the wetted perimeter
and shear stress distribution curves were drawn. The area under the curves was
measured for wall and bed mean shear forces. The calibration curves of Patel
(1965) were used to convert pressure readings to boundary shear stresses.
Patel (1965) conducted further experiments than those
by Preston (1954) in order to produce reliable and definitive
calibration curves for converting pressure readings to the shear stress as follow:
in which, ΔP is Preston tube differential pressure (dynamic pressure), d is the outside diameter of the probe, ρ is fluid density, υ is kinematic viscosity, τo is overall boundary shear stress; y* is the logarithm of the dimensionless shear stress; x* is logarithm of the dimensionless pressure difference.
Since, then the calibration equations have widely been used for measurement of boundary shear stresses in smooth and rough open channels.
RESULTS AND DISCUSSION
Using Patels calibration equations for the Preston tube pressure readings
in this study, the local bed and wall shear stresses and forces were calculated
in Table 1 for different discharges through the channel at
constant pressure gradient of 2.00x10-3. The numerically integrated
shear stress distributions were used to calculate the bed and wall shear forces
per unit length of the channel. Percentage of the measured wall shear force
to the total boundary shear force for different ratio of B/H is shown in Table
2. Using the left hand side of Eq. 2 to 6,
dimensionless wall and bed shear stresses and shear velocity are also calculated
in Table 2.
The percentage of wall shear force in Table 2 against aspect
ratio is shown in logarithm and natural scales in Figs. 2
and 3, respectively. Data sets from Seckin
et al. (2006) and other sources (Knight et
al., 1984; Knight, 1981) are also shown on the
figures. The data set in this study falls well among the other data sets in
Fig. 2 and 3. Incorporating experimental
results of this study and the data set of Seckin et al.
(2006), Eq. 1 (developed by Knight
et al., 1984 based on his previous studies) is adjusted using linear
regression of the logarithmic scale of %SFw against B/H+3 in Fig.
2, and the proposed equation is as follow,
|| Wall and bed shear stresses measurements
|| Dimensionless wall and bed shear stresses and bed shear velocity
|| Percentage of wall shear force versus B/H+3
|| Percentage of wall shear force versus B/H
In addition, a nonlinear regression of the percentage of wall shear force (%SFw)
and the channel aspect ratio (B/H +3) for all measurements in Fig.
3 produces the following equation,
To be consistent with previous studies, the dimensionless wall and bed shear
data sets reported in Knight et al. (1984) were
re-correlated incorporating the experimental data set of this study and the
Seckin et al. (2006) data set. The results are
compared with Knights correlation Eq. 2 to 6
in Fig. 4-8. The proposed correlated equations
are as follows:
In general, the data sets were best fit with the Sigmoidal-Hill Eq. 3 Parameter and the Rational-4 Parameter models.
The experimental measurements of wall and bed shear forces in Fig.
2 and 3 reveal considerable effect of aspect ratio (B/H)
on percentage of wall shear force. The contributions of wall shear force to
total boundary shear decreases by 10 fold when the aspect ratio increases from
1 to 20 in Fig. 3. A two dimensional approach for normalization
of the wall and bed shear stresses in Fig. 4 and 5
demonstrates a rapid growth of both wall and bed shear stresses in small aspect
ratios, while the curves have stagnated in aspect ratio around 6 and follow
a decrease with increasing aspect ratio.
|| Dimensionless mean wall shear stress versus B/H, (2D analysis)
|| Dimensionless mean bed shear stress versus B/H, (2D analysis)
|| Dimensionless mean bed shear velocity versus B/H, (2D analysis)
Similar results are reported in Fig. 6 for the bed mean
shear velocity. However, normalization of the wall and bed shear stresses with
hydraulic radius (i.e., a 3-dimensional analysis) granted more correct results
in Fig. 7 and 8. Figure 7
shows a reasonable decrease of the mean wall shear stress with increase of aspect
ratio. For the bed shear stress in Fig. 8, the effect of aspect
ratio ore than 6 does not affect wall shear contribution.
In lower aspect ratios, a two dimensional normalization of the wall and bed
shear stresses and shear velocity with flow depth (H) demonstrates a rapid growth
of the parameters with aspect ratios (Fig. 4-6),
while the effect is insignificant for aspect ratios more than 6. However, normalization
of the wall and bed shear stresses with hydraulic radius (i.e., a 3-dimensional
analysis) improved the results considerably. In this case increase of aspect
ratio causes a moderate reduction of average wall shear stress (Fig.
7), while was ineffective on the average bed shear stress (Fig.
The results of this study for contribution of wall and bed shear stresses and
shear forces are well agreed with previous studies. The statistical results
of correlation of the proposed equations for wall and bed shear stresses and
forces (Eq. 12-18) are compared with the
Knights correlation equations (Eq. 1-6)
in Table 3.
|| Dimensionless mean wall shear stress versus B/H, (3D analysis)
|| Dimensionless mean bed shear stress versus B/H, (3D analysis)
|| Statistical results of correlation for equations proposed
by Kight et al. (1984) and current study
The results are very close with some improvement particularly for wall and
bed shear stresses equations.
Results of a 2-D analysis confirm considerable effect of aspect ratio (B/H)
on percentage of wall shear force. The contribution of wall shear force on total
boundary shear decreases rapidly as the aspect ratio increases particularly
at low aspect ratios. A same track is also observed for wall shear stress at
high aspect ratio. However, in aspect ratio less than 6, a growth in wall shear
is recorded due to the increase of relative flow velocity with increasing the
flow depth. In contrast, a result for application of the hydraulic radius instead
of flow depth in form of a 3-D analysis confirms the ability of hydraulic radius
to cover variations in wall shear contribution with aspect ratio.
The proposed equations in this study (Eq. 12-18)
can estimate wall and bed shear forces and stresses as well as shear velocity
as long as channel geometry (B and H), and slope (Sf) are known.
The equations are valid for subcritical and supercritical flows while a uniform
channel flow can be assumed. Direct correlation of shear stress and shear velocity
with aspect ratio is the advantages of the proposed equations over previous
equations that depend on the percentage of wall shear force (%SFw)
as well as the aspect ratio.
The authors acknowledge Shahid Chamran University of Ahwaz, Iran for financial support of the research (Grant No. 696). The authors are also acknowledging Farzad Shamshiri and Kad Controls Co., Tehran, Iran for their technical assistant.
||Percent shear force
||Mean bed shear velocity
||Section mean velocity obtained from the weir
|| Local shear stress
||Mean shear stress
||Mean boundary shear stress calculate by
|| Measured mean boundary shear stress