Journal of Hydraulic Research Vol. 45, No. 4 (2007), pp. 462–471 © 2007 International Association of Hydraulic Engineering and Research
Reynolds shear stress distributions in a gradually varied flow in a roughened channel Distributions du cisaillement de Reynolds dans un écoulement graduellement varié dans un canal rugueux SHU-QING YANG, Professor, School of Environmental Science and Engineering, South China University of Technology, Wushan, Guangzhou, 510640, PR China. E-mail:
[email protected]; Chair Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, Sichuan 610065, PR China (author for correspondence) JOONG-WOO LEE, Professor, Division of Civil and Environmental Engineering, Korea Maritime University, Busan, Korea 606791. Tel.: 82 51 4104461; fax: 82 51 404 6348; e-mail:
[email protected] ABSTRACT The Reynolds shear stress distribution in non-uniform flows has been investigated. The theoretical results show that the wall-normal velocity causes the deviation of Reynolds shear stress from the standard linear distribution, i.e. −u v /u2∗ = 1 − y/ h, but the sum of Reynolds shear stress and the momentum flux, i.e. −(u v + u v)/u2∗ remains a linear distribution. By connecting the velocity gradient with Reynolds shear stress, the study demonstrates theoretically that the linear distribution of Reynolds shear stress and semi-logarithmic distribution of velocity (i.e., log-law) can be observed when and only when the wall-normal velocity is zero; the concave distribution of Reynolds shear stress and dip-phenomenon can be observed when and only when the wall-normal velocity is downward; the convex distribution of Reynolds shear stress can be observed and the wake-law correction is needed when and only when the upflow occurs. The theoretical results are in good agreement with experimental data. RÉSUMÉ La distribution du cisaillement de Reynolds dans des écoulements non uniformes a été étudiée. Les résultats théoriques montrent que la vitesse normale à la partoi produit une déviation du cisaillement de Reynolds par rapport à la distribution linéaire standard, i.e. −u v /u2∗ = 1 − y/ h, mais la somme du cisaillement de Reynolds et du débit de quantité de mouvement, i.e −(u v + u v)/u2∗ demeure une distribution linéaire. En reliant le gradient de vitesse au cisaillement de Reynolds, l’étude démontre que le phénomène de chute est provoqué par la vitesse normale à la paroi de haut en bas. Les résultats théoriques sont en bon accord avec des données expérimentales.
Keywords: Velocity distribution, gradually accelerating flow, bed shear stress, Reynolds shear stress, wall-normal velocity. 1 Introduction
To understand the mechanism, researchers measure the distribution of Reynolds shear stress because the maximum velocity in a profile always corresponds to zero-Reynolds shear stress. In other words, if Reynolds shear stress can be well predicted, then the dip-phenomenon could be inferred from the location of zero shear stress. If the zero shear stress appears below free surface, then the maximum velocity will occur below the free surface. Therefore, the study of Reynolds shear stress distribution could provide a solid basis for understanding the mechanism of dip-phenomenon. The importance of non-zero wall-normal and transverse velocities, or secondary currents has been noted since Prandtl’s era, it is widely believed that these mean velocities would result in the
Non-uniform flow in open channels and rivers is a ubiquitous phenomenon that requires investigations. Reynolds shear stress becomes boundary shear stress at the bed; both shear stresses play an important role for sediment and contaminant transport and velocity distribution. Therefore, the distribution of Reynolds shear stress in non-uniform flows is of great interest to hydraulic engineers. Over 100 years ago, river engineers such as Francis (1878), Stearns (1883) and Gibson (1909) discovered from their measurements that the maximum velocity did not appear on the free surface as expected, or the maximum velocity was immersed. The mechanism of this dip-phenomenon remains still unknown today.
Revision received October 23, 2006/Open for discussion until February 29, 2008.
462
Reynolds shear stress distributions in a gradually varied flow in a roughened channel
deviation of flow characteristics, such as Reynolds shear stress and streamwise velocity, from that in circular pipe flows (Grass, 1971; Hinze, 1975; Nezu and Nakagawa, 1993; Chien and Wan, 1999; Yang et al., 2004; and others). However, there are no available theoretical expressions in the literature to describe the interaction of secondary currents and Reynolds shear stress. Over the past decades, many experimental measurements were conducted in uniform channels, but only a handful of researchers studied the effect of non-uniformity on the velocity and Reynolds shear stress distributions. Cardoso et al. (1991) conducted laboratory experiments using hot-film anemometer in spatially accelerating flows (the upstream water depth is greater than the downstream depth), and concluded that the velocity distributions cannot be represented entirely by the universal log-law except at regions very close to the wall. Tu and Graf (1993) measured friction factor in unsteady flow over gravel beds. Nezu and Nakagawa (1993) measured the turbulence characteristics for accelerating flows in a smooth channel. Nezu et al. (1994) measured the velocity, turbulence intensity, and Reynolds shear stress profiles by using a two-dimensional laser Doppler anemometer in accelerating/decelerating channel flows. Kironoto and Graf (1995) measured the velocity distributions in accelerating and decelerating flows over a rough bed. Song (1994) measured the velocity and turbulence structure of non-uniform flows using a Acoustic Doppler Velocity Profiler (ADVP). Song’s (1994) experiments were performed in a 16.8 m long, 60 cm wide and 80 cm high flume, in which the bed consisted of nearly uniform gravels. The height of the gravel layer was about 10 cm. The size distribution of the gravel was d50 = 1.23 cm, d16 = 0.9 cm and d84 = 1.65 cm. The calibrated Nikuradse roughness height was = 1.49 cm. Song generated the non-uniform flows by adjusting the bed slope and regulating the tail gate. Song’s measurements show that the wallnormal velocity away from the boundary was nonzero, but was much smaller than the streamwise velocity. Details of Song’s (1994) experimental setup and experimental procedure have been described in Song and Graf (1994). Song and Chiew (2001) measured the velocity profile of nonuniform flows using an Accoustic Doppler Velocitymeter (ADV), their experiment confirmed that the wall-normal velocity was non-zero, again the interaction between the wall-normal velocity and the Reynolds shear stress/streamwise velocity was not discussed in their study. Yang (2002, 2005) highlighted the importance of the momentum flux u v for the determination of velocity and Reynolds shear stress distribution, he pointed out that the direct consequence of the flux is that the streamwise velocity profile cannot be described by the universal log–law and the Reynolds shear stress does not follow the standard linear relationship. Yang et al. (2004a) and Yang et al. (2007) found that the wall-normal velocity caused by sediment settlement results in the reduction of von Karman constant in sediment-laden flows. Yang et al. (2004b) revealed that the nonzero wall-normal velocity is responsible for the dip-phenomenon in a uniform channel flow. As a follow-up on the work by Yang (2002), this study investigates the influence of wall-normal velocity on the Reynolds
463
shear stress distribution in non-uniform flows. The objective of the present study is to present a theoretical model for the determination of Reynolds stress in non-uniform open channel flows; and to discuss the mechanism of dip-phenomenon based on the Reynolds shear stress in accelerating/decelerating flows.
2 Reynolds equations in a non-uniform flow For a steady two-dimensional, gradually varied open-channel flow, the governing equations have been derived (see, Nezu and Nakagawa, 1993, p. 13), which can be rewritten as follows by adding the continuity equation to the momentum equations: ∂u ∂v + =0 ∂x ∂y
(1)
∂ u2 + u2 ∂x
∂ u v + u v +
= g sin θ − ∂ u v + u v ∂x
∂y 1 ∂p + ν∇ 2 u ρ ∂x ∂ v2 + v2
(2)
+
∂y 1 ∂p + ν∇ 2 v = −g cos θ − ρ ∂y
(3)
where u and v are time-averaged velocities in the x and y directions, respectively; x is the streamwise direction; y is the wall-normal direction from the bed; p is pressure, shear stress τ/ρ = ν∂u/∂y−u v , ν is kinematic viscosity, ρ is fluid density, u and v are velocity fluctuations in x and y directions, respectively; ∇ 2 denotes the Laplacian operator; θ is the angle of the channel bed to the horizontal axis as shown in Fig. 1, g is gravitational acceleration. For gradually varied turbulent flows, the viscous effect is negligible, and the first term on the left-hand side (LHS) of Eq. (3) is also negligible relative to the term of g cos θ (≈ g), therefore integration of Eq. (3) with respect to y yields p y v 2 − v2 v 2 − v 2 = cos θ 1 − + h + h (4) ρgh h gh gh where h is water depth, vh is the mean vertical velocity at the free surface, v 2h is the magnitude of the vertical fluctuations of v y
v
w
u
h
θ
Figure 1 Coordinate system in non-uniform flows.
x
464
Yang and Lee
at the free surface. The first term on the right-hand side (RHS) of Eq. (4) is the hydrostatic pressure, the second and third terms indicate the contribution of mean vertical velocity and turbulence to the mean pressure. In the following section, experimental data will be used to verify that the second and third terms on RHS of Eq. (4) can be neglected, or the pressure in gradually varied flows is still hydrostatic, i.e. p = g cos θ(h − y) ρ
(5)
the hydrostatic pressure shown in Eq. (5) is consistent with other researchers, e.g. Graf and Altinakar (1993), Kironoto and Graf (1995), Yang et al. (2006), Song and Chiew (2001), and others. It can be derived from Eq. (5) that the pressure gradient in Eq. (2), i.e. ∂p/∂x, is ρg cos θ dh/dx. The boundary conditions of Eq. (2) include: (1) at y = yo , u = v = 0 and τ = −ρu v = τo = ρu2∗ , in which τo is the boundary shear stress, u∗ is the friction velocity; (2) at the free surface where y is water depth = h, τ = −ρu v = 0. Integration of Eq. (2) with respect to y yields y 2 ∂u dh dy − g cos θ − sin θ y + c (6) u v + u v = − dx yo ∂x −u2∗ ;
Using the lower boundary condition, one obtains c = using the upper boundary condition the friction velocity can be determined by the following equation: h 2 τo dh ∂u = u2∗ = − dy − gh cos θ − sin θ − uh vh ρ dx yo ∂x (7) where uh and vh are mean velocities at the free surface in streamwise and vertical directions, respectively. One can determine vh by integrating Eq. (1) with respect to y from y = 0 to y = h using the Leibnitz’s theorem, d vh = − dx
0
h
u dy + uh
dh dx
(8)
for a steady flow with constant discharge, the first term on RHS of Eq (8) is zero, then the wall-normal velocity at the free surface can be expressed as vh = uh
dh dx
(9)
Equation (9) indicates that vh is zero in uniform flows (dh/dx = 0), but it becomes positive or upward in decelerating flows (dh/dx > 0), it is downward or negative in accelerating flows (dh/dx < 0). Substituting Eq. (9) into Eq. (7), one has h τo d dh 2 =− cos θ − sin θ (10) u dy − gh ρ dx yo dx Equation (10) is similar to the bed-shear stress derived by others, such as Song and Chiew (2001) and Nezu and Nakagawa’s (1993).
3 Reynolds shear stress distribution in a gradually varied flow In a gradually varied flow, the Reynolds shear stress can be derived from Eq. (6) as follows dh 2 cos θ − sin θ y (11) u v + u v + u∗ = −fy − g dx where f is the mean value of ∂u2 /∂x on the interval (0, y), it is obvious that in a gradually varied flow, ∂u2 /∂x is not significant, or f is not very large relative to g(dh/dx cos θ − sin θ), this conclusion will be verified using experimental data in the following section. Therefore, Eq. (11) can be rewritten as follows: y u v uv y − 2 = 1− +b + 2 (12) u∗ h h u∗ where b=1+
f dh gh + cos θ − sin θ 2 g dx u∗
Equation (12) demonstrates that in a non-uniform flow the Reynolds shear stress will be different from that in uniform flows, which reads −
u v y =1− 2 u∗ h
(13)
in other words, the Reynolds shear stress in a non-uniform flow will not be linear with relative height y/ h, because the non-zero wall-normal velocity v will result in an additional momentum or −u v, similar to the Reynolds shear stress. Obviously, the momentum flux is not negligible. Eq. (12) can be rewritten as follows: y u v + u v y − +b (14) = 1− 2 h h u∗ Eq. (14) states that −(u v + u v)/u2∗ , rather than −u v /u2∗ , follows a linear relationship with y/ h, thus it is worthwhile to verify Eq. (14) using experimental data. In the literature, Song (1994) systematically measured the Reynolds shear stress distribution and wall-normal velocity in gradually varied flows. ADVP can measure the streamwise and wall-normal velocities (u and v), as well as the Reynolds shear stress (−u v ). The working principles of ADVP and its application were illustrated by Lhermitte and Lemmin (1990, 1993) and Rolland and Lemmin (1997). In Song’s (1994) experiments, the ADVP was connected to three transducers and three acoustic beams from the transducers intersect in a region slightly below the water surface, moving particles in the region reflected the acoustic wave towards the transducer and yielded the Dopplershift frequency from which the instantaneous velocities could be determined using geometric relations. In the experiment, Song obtained shear velocity u∗ from different ways including (a) polynomial-regression fit to the measured Reynolds-stress profiles; (b) fit the measured velocity profile in the inner region; and (c) based on the energy slope. The difference of shear velocities amongst these methods is about 5%. The details of his experiments and data treatment process can be found in Song and Graf (1994,1996).
Reynolds shear stress distributions in a gradually varied flow in a roughened channel
465
straight lines. This means that (a) b = constant, or the previous treatment about f is reasonable; (b) for the accelerating flow, the slope of the dashed line is positive but it becomes negative in the decelerating flow; and (c) the values of dashed lines at the free surface are not zero, but maximum, which indicates that the additional momentum flux u v reaches its maximum values i.e. uh vh . From Eq. (9), it can be deduced that −uh vh /u2∗ is positive for accelerating flows and it is negative for decelerating flows, hence the unknown parameter b can be directly determined using the upper boundary condition; i.e. y = h, −u v = 0. From Eq. (14), one has
4 Measured Reynolds shear stress and influence of wall-normal velocity Song’s (1994) experimental data are used to test Eqs (12) and (14). The typical results in accelerating (AS00-145) and decelerating (DS75-Q60) flows are shown in Fig. 2, in which “A” means accelerating; “D” denotes decelerating; “S” represents bed slope, “Q” is flow rate. For example AS00-Q145 means that the flow is accelerating, bed slope is zero and discharge is 1451/s. First, the measured −u v /u2∗ versus y/ h is plotted in Fig. 2 using the solid symbols, it can be seen clearly that the measured Reynolds shear stresses tends to zero as y/ h approaches to the free surface, and −u v /u2∗ tends to 1 as y/ h approaches to the bed (y = 0), these indicate that at y = 0 and y = h, the Reynolds shear stress in non-uniform flows becomes identical as a uniform flow. However, between these two extremes, the data points locate on both sides of the solid line [or Eq. (13)], the Reynolds shear stress in the accelerating flow (AS00-Q145) is systematically smaller than the values of Eq. (13), and that in the decelerating flow (DS75-Q60) is systematically larger than the values of Eq. (13). Song and Chiew (2001) had also demonstrated that the Reynolds shear stress distribution in a non-uniform flow is not linear. Obviously, the discrepancies between the solid symbols and Eq. (13) cannot be simply attributed to the measurement errors, but the mean wall-normal velocity v or the additional momentum flux u v as shown in Eq. (12). Equation (12) states that the Reynolds shear stress in nonuniform flows deviates from Eq. (13), but Eq. (14) shows that −(u v + u v)/u2∗ follows a linear relationship with y/ h. To verify this, these two datasets (i.e. AS00-Q145 and DS75-Q60) are then re-plotted in the form of −(u v + uv)/u2∗ against y/ h shown in Fig. 2 using the open symbols. It can be seen clearly that as expected by Eq. (14) the data points collapse into two
b=−
uh vh u2∗
(15)
Substituting Eq. (9) into Eq. (15), one obtains 2 uh dh b=− u∗ dx
(16)
in Eq. (14), there is only one unknown parameter b which can be determined using Eq (16) with surface velocity uh and gradient of water depth, i.e. dh/dx. It can be seen from Eq. (16) that b is always positive in an accelerating flow (dh/dx < 0) and becomes negative in a decelerating flow (dh/dx > 0). These theoretical conclusions are consistent with the experimental results shown in Fig. 2. In order to check the validity of Eq. (14), Song’s other experimental data are plotted in Figs. 3 and 4, the hydraulic conditions of Song’s measurements are shown in Table 1. The solid symbols in Figs. 3 and 4 denote the Reynolds shear stress, i.e. −u v /u2∗ , and the open symbols represent the total shear stress: −(u v +uv)/u2∗ ; the straight solid line is Eq. (13), and the dash lines represent Eq. (14). It can be observed that for all profiles of −(u v +uv)/u2∗ is indeed linear as predicted by Eq. (14).
1
y h
0.9
accelerating decelerating
0.8
accelerating decelerating
0.7 0.6 0.5 0.4 0.3 0.2
−
0.1 0 -1.5
-1
-0.5
0
0.5
1
1.5
2
u 'v' u *2
,
−
2.5
u 'v' + u v u *2 3
3.5
Figure 2 Typical −u v /u2∗ and −(u v + u v)/u2∗ distributions in an accelerating flow (AS00-145) and a decelerating flow (DS75-Q60) based on Song’s experimental data.
466
Yang and Lee
y/h
1
1
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
1
AS00-Q145 standard
0.9
AS-00-Q80 b=1.8
b=3.2
0.1−
u ' v' , u *2
−
0 -0.5
0.5
1.5
u ' v' + u v u*2 2.5
Eq. 13 AS-00-Q100 b=3.5
0.1 0 -0.5
3.5
1.5
3.5
0.1 0 -0.5
5.5
0.5
1.5
2.5
(a)
1
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.9
AS-50-Q80 b=2
0.8
0 -0.5
0.5
1.5
2.5
3.5
0 -0.5
b=3.2 AS-75-Q80 1.5
3.5
5.5
AS-93-Q80 b=3.2
0.1 0 -0.5
0.5
1.5
2.5
3.5
(b)
Figure 3 −u v /u2∗ and −(u v )/u2∗ distributions in accelerating flows.
The parameter b shown in Figs. 3 and 4 is determined from the dashed lines which fit the open symbols best. Equation (16) states that the slope or the coefficient b can be estimated using the surface velocity and dh/dx. Song measured dh/dx, and by extrapolating the measured velocity to free surface, one can determine the surface velocity, then the value of b can be calculated using Eq. (16). The last two columns of Table 1 show the measured v2h /gh and v 2h /gh where the velocities at the free surface are obtained. It can be seen that in the gradually varied flows these two values are in the range of 10−4 –10−7 , this is why the last two terms in
Eq. (4) are negligible, or the pressure in gradually varied flows is hydrostatic.
5 Relation between Reynolds shear stress and dip-phenomenon It is worthwhile to discuss the relationship between the Reynolds shear stress and the velocity gradient in gradually varied flows. Boussinesq (1877) drew an analogy coefficient of molecular viscosity in Stokes’s law and proposed a Reynolds shear
Reynolds shear stress distributions in a gradually varied flow in a roughened channel
y/h
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
−3
u ' v' + u v u*2
DS50-Q70 0.2 b=-1.0
0.2
0.1
DS25-Q90 b=-0.25
−
0 -0.5
0
0.5
1
u ' v' u*2
0.2 DS50-Q55 b=-1.5 0.1
0.1
0
0
1.5
-1.5
467
-0.5
0.5
1.5
-1
0
1
(a)
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
DS75-Q80 0.2 b=-1.3
0.2
DS90-Q80 0.1 b=-2.7
0.1
0
0 -1.5
-0.5
DS90-Q70 b=-3.vvv
0.5
1.5
-2
-1
0.2 0.1 0
0
1
-3
-1
1
(b)
Figure 4 −u v /u2∗ and −(u v + uv)/u2∗ distributions in decelerating flows.
the free surface. Inserting Eq. (17) into Eq. (12), one obtains
stress-strain rate relationship: −u v = vt
du dy
(17)
where vt is the eddy viscosity, which is often called the “apparent” or “virtual” viscosity. Prandtl developed the mixing-length theorem to express the eddy viscosity. It can be deduced from Eq. (17) that the maximum velocity occurs at the position where the Reynolds shear stress is equal to zero, in other words, to understand the mechanism of dip-phenomenon, one needs only to identify whether the zero-Reynolds shear stress appears below
−u v y −uh vh u v νt du y = 1− + = 2 − u2∗ u∗ dy h u2∗ uh vh h
(18)
Eq. (18) states that at the free surface where y = h, the Reynolds shear stress is always equal to zero. For accelerating flows, vh is negative or downward due to dh/dx < 0, thus the second term on the RHS of Eq. (18) is negative near the free surface, but the first term on the RHS is positive, therefore in the region near the free surface there must be a position where the Reynolds
468
Yang and Lee Table 1 Determination of coefficient b using different ways. Run no.
sin θ
Q(1s−1 )
Mean velocity (cm/s)
dh/dx
h(cm)
u∗ (cm s−1 )
Measured v2h /gh
Measured v 2h /gh
ASOO-Q145 ASOO-Q100 ASOO-Q80 AS-25-Q100 AS-25-Q80 AS-25-Q60 AS-50-Q110 AS-50-Q80 AS-75-Q100 AS-75-Q80 AS-93-Q100 AS-93-Q80 DS25-Q90 DS50-Q70 DS50-Q55 DS75-Q60 DS75-Q80 DS90-Q80 DS90-Q70
0 0 0 −0.0025 −0.0025 −0.0025 −0.0050 −0.0050 −0.0075 −0.0075 −0.0093 −0.0093 0.0025 0.0050 0.0050 0.0075 0.0075 0.0090 0.0090
145 100 80 100 80 60 110 80 100 80 100 80 90 70 55 60 80 80 70
120.8 102.9 92.0 98.6 88.3 76.3 102.4 83.9 91.6 80.8 87.7 77.5 75.0 70.7 63.2 58.8 65.0 74.1 64.8
−0.0170 −0.0122 −0.0090 −0.0160 −0.0130 −0.0118 −0.0200 −0.0146 −0.0200 −0.0170 −0.0205 −0.0189 0.0016 0.0038 0.0038 0.0042 0.0079 0.0095 0.0096
20 16.2 14.5 16.9 15.1 13.1 17.9 15.9 18.2 16.5 19.0 17.2 20.0 16.5 14.5 17 20.5 18.0 18.0
9.3 8.05 7.21 7.91 7.36 7.54 7.68 6.86 7.42 6.54 6.85 6.80 5.19 6.20 5.20 4.91 5.04 6.52 5.36
2.04E−04 3.51E−04 4.62E−05 1.96E−04 4.06E−04 5.24E−05 4.31E−04 8.94E−05 4.36E−04 1.25E−04 5.09E−04 1.25E−04 3.27E−07 1.67E−05 3.72E−06 3.00E−06 9.64E−06 1.91E−05 8.16E−05
4.47E−04 4.59E−04 4.98E−04 5.11E−04 5.19E−04 3.94E−04 4.96E−04 3.54E−04 1.42E−04 1.98E−04 2.15E−04 1.44E−04 4.17E−04 3.64E−04 7.16E−04 8.12E−04 5.19E−04 7.47E−04 6.59E−04
shear stress is zero. This means that the immersed maximum velocity or dip-phenomenon must occur, or because of downflow in accelerating flows, there must exist two points of zero shear stress, one locates at the free surface, the other under the surface, thus the velocity-dip must appear. In decelerating flows the 2nd term on RHS of Eq. (18) must be positive due to dh/dx > 0 or vh > 0. This means that the Reynolds shear stress is always positive in the whole region from the free surface to the bed, and the velocity gradient is greater than that in circular pipe flows. This is why the wake-log law is always associated with flows where v > 0. Hence the zero Reynolds shear stress only appears at the free surface, or it is impossible to observe the dip-phenomenon in flows with upward wall-normal velocity. It can be concluded from above discussion that the immersed maximum velocity can be only observed in accelerating flows. This conclusion is consistent with Kironoto and Graf’s (1995) observation. They measured the velocity profiles in accelerating and decelerating flows, and found that for the accelerating flow this point of maximum velocity occurred at y/ h = 0.6; while for the decelerating one, the point of velocity-dip moved up to the free surface.
6 Reynolds shear stress distribution in uniform channel flows Equation (9) states that in uniform flows, vh = 0, thus Eq. (18) becomes uv u v y − 2 = 1− + 2 (19) u∗ h u∗
Similarly, Nezu and Nakagawa (1993) developed the following equation for a two-dimensional (2D) channel flow: −u v
y
= gS(h − y) + + h
h y
∂(−u w ) ∂z
∂u v dy + ∂y dy
y
w h
∂u dy ∂z (20)
Eq. (19) states that in uniform flows the Reynolds shear stress may not follow the linear distribution if v = 0, but the total shear stress, i.e., −(u v + u v)/u2∗ must linearly vary with y/ h. This conclusion could be examined using experimental data by Nezu and Nakagawa (1993), they measured the Reynolds shear stress and secondary currents using X-type hot-film anamometers in a uniform flow (dh/dx = 0) with aspect ratio of 7.5, artificial longitudinal ridge elements were attached to the smooth bed. The measured results are reproduced in Figs. 5 and 6. It can be seen clearly from Fig. 5 that over the ridges there existed upflows (v > 0), whilst over the troughs the downflows (v < 0) presented. Consequently, like the accelerating and decelerating flows, the measured Reynolds shear stress shown in Fig. 6 deviates from the linear distribution represented by the dotted line. The measured Reynolds shear stresses over the trough, as predicted by Eq. (19), are systematically lower than the dotted line, but the Reynolds shear stresses over the ridges are larger than the dotted line that is Eq. (13). It is interesting to note that along the profile of z/ h = −0.7, the secondary flow was parallel to the free surface or horizontal, thus the wall-normal velocity v = 0 in the upper flow region (y/ h > 0.5), subsequently the measured data points in Fig. 6 can be well represented by the dotted line; but in the
Reynolds shear stress distributions in a gradually varied flow in a roughened channel
Figure 5 Secondary currents over artificial ridges in a uniform flow measured by Nezu and Nakagawa (1993).
469
It can be seen from Fig. 5 that from profile of z/ h = 0 to profile of z/ h = −0.7, the secondary current is downward, or the wall-normal velocity is negative; subsequently, Fig. 6 shows that (−u v /u2∗ )m − (−u v /u2∗ )t is negative as stated in Eq. (21), and the maximum deviation corresponds to the highest downward velocity. On the other hand, from profile z/ h = −0.7 to z/ h = −1.0, the wall-normal velocity is upward. Consequently the measured Reynolds shear stress in this domain is higher than the dotted line or (−u v /u2∗ )m − (−u v /u2∗ )t > 0. Particularly, along the profile of z/ h = −1.0, the wall- normal velocity is maximum relative to the rest profiles, or u v/u2∗ is maximum, coincided with Eq. (21), the measured Reynolds shear stresses along this profile locate on the topmost of data points band (see Fig. 6). Thus, it can be concluded that Eq. (21) can well explain the observed phenomena in uniform flows. Although Nezu and Nakagawa did not present the streamwise velocity distribution, it can be inferred from Eq. (19) that it would be impossible to observe the velocity dip over the ridge for v > 0 or upwelling flow; and the velocity-dip should occur over the trough region for downwelling flow (v < 0); and the velocity along profile z/ h = 0.7 should be logarithmic due to v = 0. As Eq. (19) explains the measured Reynolds shear stress very well, this implies that the difference between the measured Reynolds shear stress and the dotted line can be represented by u v/u2∗ . Thus Eq. (19) provides another way to estimate the secondary currents based on measured Reynolds shear stress. 7 Conclusions
Figure 6 Distribution of Reynolds shear stress over ridges and troughs (after Nezu and Nakagawa, 1993).
lower portion of this profile (y/ h < 0.5), the wall-normal velocity becomes downward, correspondingly the measured Reynolds shear stresses are less than the dotted line. Equation (19) can be rewritten as follows: u v u v uv − 2 − − 2 = 2 (21) u∗ m u∗ t u∗ where (−u v /u2∗ )m is the measured Reynolds shear stress, and (−u v /u2∗ )t is the theoretical Reynolds shear stress in circular pipe flows, that is u v y − 2 (22) =1− u∗ t h Equation (21) indicates that the measured Reynolds stress shear will deviate from the straight line, i.e., Eq. (22) if v = 0. The deviation is positive if v > 0, or the measured Reynolds shear stress is greater than the straight line; the deviation becomes negative if v < 0, or the measured Reynolds shear stress is less than the straight line. From Eq. 17, one can conclude that the log-law can be observed when and only when v = 0; the dipphenomenon appears when and only when v < 0; Coles’ wakefunction correction is needed when and only when v > 0.
This study investigates the Reynolds shear stress distribution in non-uniform flows based on the Reynolds equations of 2D open channel flows. Equation of the Reynolds shear stress has been developed, and the relationship between Reynolds shear stress and velocity-dip has been discussed. The theoretical results are in good agreement with experimental data. Based on the achievement, the following conclusions can be drawn (1) Similar to the Reynolds shear stress, the non-zero wallnormal velocity introduces an additional momentum flux, i.e. u v, which causes the deviation of Reynolds shear stress from the standard linear distribution, i.e. −u v /u2∗ = 1 − y/ h. (2) In flows with presence of non-zero wall-normal velocity, though the Reynolds shear stress −u v /u2∗ , no longer follows the “standard” linear distribution, the sum of Reynolds shear stress and the momentum flux, i.e., −(u v + u v)/u2∗ still retains a linear distribution, its slope varies with the non-uniformity. (3) In accelerating flows, this slope is positive because of downward flow; but it becomes negative in decelerating flows due to the upward wall-normal velocity. This slope can be theoretically determined using Eq. (16). (4) This study shows theoretically that the linear distribution of Reynolds shear stress and semi-logarithmic distribution of velocity (i.e., log-law) can be observed when and only when the wall-normal velocity is zero; the concave distribution of Reynolds shear stress and dip-phenomenon can be
470
Yang and Lee
observed when and only when the wall-normal velocity is downward; the convex distribution of Reynolds shear stress can be observed and the wake-law correction is needed when and only when the upflow occurs.
Acknowledgments The writers are grateful to three anonymous reviewers for detailed and constructive suggestions that greatly improved the manuscript. We also appreciate Dr. Song T.C. for his kind permission in using the raw data of his Ph.D. work. The first author would like to express his appreciation to Prof. Xu Weilin at Sichuan University for hosting his academic visits in 2005 and 2006, during which the work was initiated.
Notation b = Factor = 1 − gh(sin θ + S)/u2∗ g = Gravitational acceleration h = Water depth p = Pressure u = Velocities in x directions v = Velocities in y directions S = Energy slope u∗ = Shear velocity −u v = Reynolds shear stress −u w = Transverse Reynolds shear stress w = Velocity in z direction x = Streamwise direction y = Vertically upward direction from the bed z = Transverse direction Greek symbols ζ = y/ h θ = Angle of the channel bed to the horizontal plane κ = Von Karman constant = Boundary roughness height ν = Kinematic viscosity of the fluid νt = Eddy viscosity ρ = Fluid density τo = Bed shear stress
References 1. Boussinesq, J. (1877). “Essai sur la theorie des eaux courantes”. Mem. Pres. Acad. Sci. XXIII, 46, Paris. 2. Cardoso, A.M., Gust, G. and Graf, W.H. (1991) “Steady Gradually Accelerating Flow in a Smooth Open Channel”. J. Hydraul. Res. 29(4), 525–543. 3. Chien, N. and Wan, Z. (1999). “Mechanics of Sediment Transport” ASCE, Reston, Va. 4. Coles, D. (1956) “The Law of the Wake in the Turbulent Boundary Layer” J. Fluid Mech. 1, 191–226.
5. Francis, J.B. (1878). “On the Cause of the MaximumVelocity of Water Flowing in Open Channels Being Below the Surface”. Trans. ASCE. May. 6. Gibson, A.M. (1909). “On the Depression of the Filaments of Maximum Velocity in a Stream Flowing through an Open Channels”. Proc. Royal Society of London, Series A 82, 149–159. 7. Graf, W.H. and Altinaker, M.S. (1993). “Hydraulique Fluvial”; Tome 1. Presses Polytechniques et Universitaires Romandes, Lausanne. 8. Grass, A.J. (1971). “Structural Features of Turbulent Flow over Smooth and Rough Boundaries”. J. Fluid Mech. 50(2), 233–255. 9. Hinze, J.Q. (1975). Turbulence, 2nd edn. McGraw-Hill, New York. 10. Kironoto, B.A. and Graf, W.H. (1995). “Turbulence Characteristics in Rough Non-Uniform Open-Channel Flow”. Proc. Instin Civ. Engrs Wat., Marit & Energy 112, 336–348. 11. Lhermitte, R. and Lemmin, U. (1990). “Open Channel Flow and Turbulence Measurement by High Resolution Doppler Sonar”. Ann. Rep. of LRH. Ecole Polytechnique Fédérale, Lausanne, Switzerland, B24, 1–39. 12. Lhermitte, R. and Lemmin, U. (1993). “Turbulent Flow Microstructures Observed by Sonar”. Geophys. Res. Letter 20(9), 823–826. 13. Nezu I. and Nakagawa H. (1993). “Turbulence in Open Channel Flows”. In: Balkema, A.A. (ed.), Monograph series of IAHR, Rotterdam, Netherland. 14. Nezu, I., Kadota, A. and Nakagawa, H. (1994). “Turbulence Structure in Accelerating and Decelerating Open-Channel Flows with Laser Doppler Anemometers”. Proceeding, 9th Congress APD-IAHR, Singapore, pp. 413–420. 15. Rolland, T. and Lemmin, U. (1997). “A Two-Component Acoustic Velocity Profiler for Use in Turbulent OpenChannel Flow”. J. Hydraul. Res. 35(4), 545–561. 16. Song, T.C. (1994). “Velocity and Turbulence Distribution in Non-Uniform and Unsteady Open-Channel Flow”. Ph.D. Thesis, Ecole Polytechnique Federale De Lausanne. 17. Song, T.C. and Graf, W.H. (1994). “Non-Uniform OpenChannel Flow Over a Rough Bed”. J. Hydroscience and Hydraul. Engrg. 12(1), 1–25. 18. Song, T.C. and Graf, W.H. (1996). “Velocity and Turbulence Distribution in Unsteady Open-Channel Flows”. J. Hydrual. Engrg. ASCE 122(3), 141–154. 19. Song, T. and Chiew Y.M. (2001). “Turbulence Measurement in Non-Uniform Open-Channel Flow Using Acoustic Doppler Velocimeter (ADV)”. J. Engrg. Mech. ASCE 127(3), 219–232. 20. Stearns, P.P. (1883). “On the Current Meter, Together with a Reason Why the Maximum Velocity of Water Flowing in Open Channel is Below the Surface”. Trans. ASCE. 21. Tu, H., and Graf, W.H. (1993). “Velocity Distribution in Unsteady Open-Channel Flow Over Gravel Beds”. J. Hydrosci. Hydraul. Engrg. 10(1), 11–25.
Reynolds shear stress distributions in a gradually varied flow in a roughened channel
22. Yang, S.Q. (2002). “1st and 2nd Order Approximate Solutions of Reynolds Equations in 3-D Channels”. Proceeding of 13th IAHR-APD Congress, Vol. 1, World Scientific, Singapore, pp. 70–75. 23. Yang, S.Q., Tan, S.K. and Lim, S.Y. (2004a). “Velocity and Sediment Concentration Profiles in Sediment-Laden Flows”. China Ocean Engrg. 18(2), 229–244. 24. Yang, S.Q., Tan, S.K. and Lim, S.Y. (2004b). “Velocity Distribution and Dip-Phenomenon in Smooth Uniform Open Channel Flows”, J. Hydrual. Engrg. ASCE 130(12) 1179–1186. 25. Yang, S.Q., Xu, W.L. and Yu, G.L. (2006). “Velocity Distribution in Gradually Accelerating Flow”. 29(12), 1969–1980.
471
26. Yang, S.Q., Yu, J.X. and Wang, Y.Z. (2004). “Estimation of Diffusion Coefficients, Lateral Shear Stress and Velocity in Open Channels with Complex Geometry”. Water Resources Research, American Geophysical Union 40(5), 207–217. 27. Yang, S.Q. (2005). “Interactions of Boundary Shear Stress, Secondary Currents and Velocity”. Fluid Dynamics Research 36(3), 121–136. 28. Yang, S.Q. (2007). “Turbulent transfer mechanism in sediment-laden flow”. J. Geophysical Research, AGU, 112, F01005, doi: 10.1029/2005JF000452.
