(1)
Yee-Meng Chiew and Xingwei Chen
School
of Civil and Structural Engineering
Nanyang Technological
University
Nanyang Avenue, Singapore
639798
E-mail: cymchiew@ntu.edu.sg and cxwhen@ntu.edu.sg
Abstract: The bed shear stress in open channel flow subjected to bed suction is investigated both theoretically and experimentally. By assuming a boundary condition with a slip velocity, the paper proposes a momentum integral equation that can be used to evaluate bed shear stresses for flows subjected to bed suction. Measurements of the turbulent flow are performed in a laboratory flume using a two-dimensional Acoustic Doppler Velocimeter. Preliminary results show that the slip velocity exists. The measured data also show that the bed shear stress is larger in case of bed suction, which is consistent with the prediction of the theoretical analysis.
Keywords: shear stress, open channel, bed seepage
In many natural conditions, e.g., the permeable boundary in a river, seepage flow occurs and it can have a significant influence on the near bed flow field. The interaction between the turbulent flow and seepage can cause changes to the structural features of the flow, such as velocity distribution, turbulent intensity and boundary shear stress, as compared to those associated with an impermeable boundary. Seepage through the permeable bed can take place in an arbitrary angle relative to the bed surface. One is the normal seepage, in which the flow direction is perpendicular to the boundary and thus, the main open channel flow. Normal seepage can either be in the form of upward or downward seepage, which can also be called injection and suction, respectively.
As bed seepage affects the near bed turbulent velocity and the Reynolds shear stress, the bed shear stress also changes accordingly. The typical method used to evaluate bed shear stresses by fitting the logarithmic law of the wall to the measured velocity profiles is no longer valid due to the presence of bed seepage. Oldenziel and Brink (1974), Willettes and Drossos (1975), Maclean (1991a,b) and Prinos (1995) studied the flow experimentally or numerically. But how the bed shear stress varies with seepage still has not been fully understood. More recently, Cheng and Chiew (1998a,b) thoroughly investigated turbulent open channel flow with upward seepage both experimentally and theoretically. They derived a momentum integral equation that can be used to compute the bed shear stress. Because of the difference in the velocity distributions in the cases of upward seepage and downward one, the equation needs to be modified. The objective of the paper is to explore experimentally and analytically the effect of downward seepage on the bed shear stress in open channel flow.
As described in the above section, interaction at the boundary exists when turbulent open channel flows over a permeable bed. Theoretical and experimental investigations, such as those by Gupta and Paudyal (1985), Mendoza and Zhou (1992), have revealed that a slip velocity exists at the permeable surface. The no-slip boundary condition normally assumed in an impermeable boundary is no longer valid. Oldenziel and Brink (1974) measured the velocity profiles at the downstream end of a permeable bed and found that the horizontal velocity near the bed decreased with injection and increased with suction. The velocity profiles were more uniform for flow with bed suction because of the increasing near bed velocity and decreasing near surface velocity. It means that the slip velocity increases with suction, and it is more obvious than that without suction. Therefore the slip boundary condition should be considered in case of bed suction. With this boundary condition, a momentum integral equation for open channel flow subjected to normal seepage is derived in the following section.
For a steady two dimensional, gradually varied open-channel flow with seepage on a horizontal bed as shown in Figure 1, the governing equations can be reduced to:
(1)
(2)
(3)
where x, y = coordinators tangential and normal to the bed boundary, respectively; u, v = velocities in the x and y direction, respectively; t = shear stress; p = pressure; g = gravitational acceleration.
Eqs. (1) to (3) are subjected to the following boundary conditions:
(4)
(5)
(6)
(7)
(8)
where
= slip velocity in the bed;
= seepage velocity normal to the bed; h = water depth.
First, integrating Eq. (1) with respect to y yields the depth-averaged continuity equation:
(9)
Integrating Eq. (3) with respect to y and then differentiating with x, and substituting the resulting equation into Eq. (2) yields:
(10)
Applying the continuity equation (1), Eq. (10) can be rewritten and then be integrated with respect to y, to become
(11)
where
= bed shear stress. With Eq. (8), the first two terms on the left hand side of
Eq. (11)
can be transformed to the form
(12)
Applying the definition of the momentum correction factor
(13)
and Eq. (9), it leads to
(14)
Substituting Eq. (12) and (14) into Eq. (11), the bed shear stress subjected to seepage is expressed as
(15)
Eq.
(15) can be used to evaluate the bed shear stress in open-channel flow with a
horizontal bed in case of seepage. When the seepage is in the upward direction,
the no-slip condition is assumed, i.e.,
, and Eq. (15) reduces to
(16)
and it is the same as that derived by Cheng and Chiew (1998a) in case of upward seepage.
The experiments were conducted in a 30 m long glass-sided horizontal flume. The flume has a seepage zone situated at 16 m from the upstream end of the flume. This seepage zone is a recess that is 2 m long, 0.7 m wide and 0.4 m deep. It is filled with sand particles of median grain diameter = 0.95 mm. Water seeped through the sand by gravity before flowing into 12 identical seepage pipes which were installed at the lower portion of the recess (see Figure 2). Valves were installed on each of the pipes to regulate the flow rate. The open channel flow rate was controlled using a speed inverter and a valve, and was monitored using an electromagnetic flow meter.
A two-dimensional Acoustics Doppler Velocimeter (ADV) was used to measure the instantaneous velocities of the flow so that the turbulent properties of the flow could be explored. The sampling rate of the ADV is 50Hz. The water depth was measured using a Capacity Type Wave Height Meter (CTWHM).
Figure 3 shows the
comparison of the profiles of the Reynolds shear stress with and without bed
suction. The seepage intensity,
is 0.63% and the profile is measured at x = 100 cm, where U0 = depth-averaged velocity at the leading section of the seepage
zone. Clearly, bed suction causes an increase of the near bed Reynolds shear
stress, where
= 1.2 cm2/s2 is
larger than that without suction (
= 0.82 cm2/s2). Qualitatively, Eq. (15) also predicts the
change in the bed shear stress. The gradient of the water surface,
, is larger in case of bed suction and the negative
makes the second term on the right hand side of Eq. (15) positive. Therefore, Eq.
(15) predicts an increase in bed shear stresses where the flow is subjected to
bed suction.
Figure 4 shows a typical velocity distribution (Cheng and Chiew, 1998a) in case of upward seepage. A curve, calculated using the logarithmic law of wall using the same depth-average velocity U and water depth h as that represented by the data points, is superimposed in the figure, where the law of wall is expressed by:
(17)
The figure shows that the no-slip condition is still valid.
When the flow is
subjected to downward seepage the situation changes appreciably. The data in
Figure 5 show the velocity distributions for this case. Similar to that in
Figure 4, the curves in Figure 5 are plotted using the logarithmic law of the
wall with the same depth-average velocity U
and water depth h as that represented
by the data points. Compared with those without bed suction, the velocities near
the bed with bed suction increase and the velocity profiles are more uniformly
distributed. In this case, the logarithmic law can no longer be fitted with the
data. It indicates that the slip velocity becomes important. The data also show
that it is reasonable to assume a slip condition in the derivation of Eq. (15).
Mendoza and Zhou (1992) have proposed a modified logarithmic velocity profile for the permeable bed where the slip velocity is invoked:
(18)
where
is the origin displacement of the mean velocity profile. However, it is still
difficult to determine the values of
and
because the interaction of the
permeable bed and the turbulent flow is still not fully understood. Similarly,
the determinations of
and
in case of bed suction are equally difficult. The present study aims to provide
an improved understanding of
and
, and an accurate determination of the bed shear stresses using Eq. (15).
This paper presents a theoretical
analysis for the evaluation of the bed shear stress on open channel flow with
bed suction. A momentum integral equation is derived, where the slip boundary
condition is invoked. Preliminary experimental measurements show that the
Reynolds shear stress near the bed in case of bed suction is larger than that
without bed suction, which is also qualitatively predicted by the theoretical
analysis. The measured data also show that the slip velocity exists when the
flow is subjected to bed suction. How
the slip velocity varies will be determined in the next stage of work and the
validation of Eq. (15) will be tested more rigorously when additional
experimental data are obtained.
Cheng N. S. and Chiew Y. M., 1998a, “Turbulent open-channel flow with upward seepage”, J. Hydraulic Research, 36(3), 415-431.
Cheng N. S. and Chiew Y. M., 1998b, “Modified logarithmic law for velocity distribution subjected to upward seepage”, J. Hydraulic Engineering, 128(12), 1235-1241.
Gupta A. D. and Paudyal G. N., 1985, “Characteristics of free surface flow over gravel bed”, J. Irrig. and Drain. Eng., ASCE, 111(4), 299-319.
Maclean, A. G., 1991a, “Open channel velocity profiles over a zone of rapid infiltration”, J. Hydraulic Research, 29(1), 15-27.
Maclean, A. G., 1991b, “Bed shear stress and scour over bed-type river intake.”, J. Hydraulic Eng., ASCE, 117(4), 436-451.
Mendoza C. and Zhou D. 1992, “Effects of porous bed on turbulent stream flow above bed”, J. Hydraulic Eng., ASCE, 118(9), 1222-1240.
Oldenziel and Brink, 1974, “Influence of suction and blowing entrainment of sand particles”, Journal of the Hydraulics Division, ASCE, 100(7), 935-947.
Prinos P., 1995, “Bed-suction effects on structure of turbulent open-channel flow”, J. Hydraulic Eng., ASCE, 121(5), 404-412.
Willetts, B. B. and Drossos, M. E., 1975, “ Local erosion caused by rapid forces infiltration”, Journal of the Hydraulics Division, ASCE, 101(12), 1477-1488.
Fig. 1 Definition sketch for open channel flow with seepage on a horizontal bed
Fig.2 Schematic diagram of test section
Fig.3 Reynolds shear stress distributions with and without bed suction
Fig.4 Velocity profile for flow with bed injection (Cheng and Chiew 1998a)
Fig.5 Velocity profiles for flow with bed suction