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Interactions of boundary shear stress, secondary currents and velocity
Fluid Dynamics Research 36 (2005) 121 – 136
Interactions of boundary shear stress, secondary currents and velocity Shu-Qing Yanga, b a Division of Civil and Environmental Engineering, Korea Maritime University, Busan, Korea 606791 b Maritime Research Center, School of Civil and Environmental Engineering, Nanyang Technological University,
Singapore 639798 Received 26 March 2004; received in revised form 18 December 2004; accepted 10 January 2005 Communicated by Z.-S. She
Abstract This paper deals with the interaction of boundary shear stress, velocity distribution and secondary currents in open channel flows. A method for computing boundary shear stress and velocity distribution in steady, uniform and fully developed turbulent flows is developed by applying an order-of-magnitude analysis to the Reynolds equations. A simplified relationship between the wall-tangential and wall-normal terms in the Reynolds equation is hypothesized, then the Reynolds equations become solvable. This analysis suggests that the energy from the main flow is transported towards the nearest boundary to be dissipated through a minimum relative distance or normal distance of the boundary. The equations governing the boundary shear stress and Reynolds shear stress distributions are obtained, and the influence of wall-normal velocity on the streamwise velocity is assessed. It is found that the classical log-law is valid only when the wall-normal velocity is zero, the non-zero wall-normal velocity results in the derivation of measured streamwise velocity from the classical log-law. The derived equations are in good agreement with existing experimental data available in the literature. © 2005 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Boundary shear stress; Turbulent flows; Reynolds shear stress;Velocity profiles; Dip-phenomenon; Secondary currents
E-mail address: [email protected]. 0169-5983/$30.00 © 2005 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:10.1016/j.fluiddyn.2005.01.002
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S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
Nomenclature A Ab and Aw b b1 c c1 D g h ln lb /Db and lw /Dw m n p pw and pb S u, v and w u , v , z u∗b and u∗w u∗ u∗1 x y yo ymax z xy xz b , w b w b
constant area of bed and sidewall sub-region width of channel bed coefficient constant integration constant characteristic length representing energy dissipation of boundary gravitational acceleration water depth the normal distance to the boundary relative distance to the bed and the sidewall coefficient empirical coefficients coefficient sidewall and bed perimeter energy slope mean velocities in x, y and z directions fluctuating velocity components mean bed and sidewall shear velocities shear velocity global shear velocity [=(gRS)0.5 ] streamwise direction wall-normal direction parameter of boundary characteristics location maximum velocity wall-tangential direction dynamic viscosity ju/jy − u v ju/jz − u w mean bed and sidewall shear stresses u2∗b u2∗w typical boundary roughness height coefficient angle of channel wall Karman constant (≈ 0.4) kinematic viscosity y/ h fluid density local boundary shear stress
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136 iso-velocity contour
123
y
lw
h
ln
secondary cells
θ
z
division lines b
Fig. 1. Relationship between observed primary flow, secondary flow, flow geometry and division lines (after Knight et al., 1993).
1. Introduction Turbulent structures and boundary shear stress in steady and uniform flows are fundamentally important and have been studied extensively by many researchers, such as Tracy (1965), Melling and Whitelaw s (1976), Knight and Patel (1985), Rhodes and Knight (1994), Yang (2002), etc., but it is still difficult to predict them exactly even using sophisticated turbulence models. Empirical equations for the boundary shear on the wall and bed have been proposed by Knight (1981), Zheng and Jin (1998), Guo and Julien (2005), etc. Yang (1993), using the hypothesis that the mechanical energy of 3-D turbulent flow is always transferred towards boundary via shortest relative distance, approximated mean boundary shear stresses on the bed and sidewalls in rectangular channels (Yang, 1996, Yang and Lim, 1997,1998). Many researchers measured the mean velocity profiles in a steady and uniform flow, and it is widely reported that the velocity deviates from the classical log-law near the free surface, i.e., the position whereby the maximum velocity appears below the water surface, occurs if the aspect ratio of the channel width to water depth, b/ h, is less than a certain value, this phenomenon was first observed about a century ago by Stearns (1883) and Murphy (1904). The boundary shear stress and the longitudinal velocity deviation from the classical log-law have charmed and challenged to scientists and engineers. To tackle them, Keulegan (1938) suggested that the bisectors of base angles of a polygonal channel could be used to partition a flow region as shown in Fig. 1, but he did not physically explain this suggestion. He also assumed that the log-law is valid to predict the velocity distribution from a point near the bed to the free surface. Einstein (1942) extended Keulegan’s (1938) concept to a river flow and stated that a river flow can be divided into three sub-sections corresponding to the banks and riverbed, respectively. He had, however, failed to explain why the flow region is divisible and how to divide it. The first question was partially answered by Chien and Wan (1999) who claimed that the energy from a unit element of water transmits only in one direction towards the boundary. Yang (1993) proposed that the surplus energy contained in any arbitrary flow volume will be transferred towards the nearest boundary to be dissipated, or the surplus energy is always transported to the boundary
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124
through the minimum relative distance that is defined as the ratio of geometrical distance from boundary to the boundary roughness or thickness of viscous sub-layer, but the concept was not directly derived from the Reynolds equations. The main objectives of this study are: (1) to solve the Reynolds equations by hypothesizing a relationship between the wall-tangential and wall-normal terms (2) to develop a method for estimating the boundary shear stress from the Reynolds equations; (3) to develop a simplified equation suitable for describing the velocity profile from the bed to water surface. 2. Simplified Reynolds equation For a steady, uniform and fully developed turbulent channel flow as shown in Fig. 1, Reynolds equations can be written as follows by adding the continuity equation to the x-direction momentum equation: j(uv − xy /) jy
+
j(uw − xz /) jz
= gS,
(1)
where x is defined as the streamwise direction; y is the wall-normal direction; z is the wall-tangential direction; u, v and w are mean velocities in x, y and z directions, respectively; g is gravitational acceleration; S is energy slope; and xy = ju/jy − u v and xz = ju/jz − u w , is dynamic viscosity, is fluid density, u , v , z are fluctuating velocity components. Experimental data (Tracy, 1965) show that in a region near a wall, the term in the wall-normal direction is much greater than the wall-tangential gradient. It follows that in the region near the bed the first term on the left-hand side (LHS) of Eq. (1) is dominant and the second term on LHS is negligible because the wall-normal direction is y-direction. Likewise, in the region near the sidewall, the second term is dominant, but the first term on LHS can be neglected. In other words, if the first term on LHS of Eq. (1) is negligible, the second term must be dominant, or vice versa. It is impossible that, in any region, both terms on LHS are negligible at same time because the constant gS, i.e., the gravitational term on the right-hand side of Eq. (1), must be balanced. The relative dominance of the two derivatives on LHS of Eq. (1) can be approximately related by j(uw − xz /) lw n j(uv − xy /) lb = , (2) jz Db Dw jy where n = empirical coefficients; lb /Db and lw /Dw are the relative distance to the bed and the sidewall in Fig. 1, the subscripts b and w denote the bed and wall, respectively; D is characteristic length representing energy dissipation of boundary, for a smooth boundary D =v/u∗ and for a rough boundary D = =typical boundary roughness height (Yang, 1993, 2002); l = the normal distance to the boundary as shown in Fig. 1; u∗ = shear velocity; = kinematic viscosity. Eq. (2) means an order-of-magnitude analysis, it is useful for a practical solution to simplify the Reynolds equation, it shows that if lb >lw , j/jz ≈ 0, then only the term j/jy is retained in Eq. (1). Similarly, in the region of lb ?lw , j/jy ≈ 0, then only the term j/jz is left in Eq. (1). For the region between these two extremes, these two terms co-exist to balance the gravitational component, but the importance of each term depends on the ratio of relative distances to the bed and sidewall defined in Eq. (2). Thus, Eq. (2) mathematically expresses that the wall-normal direction is more important than the wall-tangential gradient if n > 0.
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
125
In this study, only smooth boundary is discussed and the results can be extended to rough channel flows, substituting (2) into (1) yields j(uv − xy /) u∗b y n = gS 1+ , (3a) jy u∗w lw where b = u2∗b , w = u2∗w , u∗b and u∗w are mean bed and sidewall shear velocities, b , w are mean bed and sidewall shear stresses. Eq. (1) can be similarly re-arranged as follows j(uw − xz /) u∗w z n = gS 1+ . (3b) jz u∗b y Thus, analytical solutions can be obtained from Eqs. (3a) and (3b), if the value of n is given. Yang and McCorquodale (2004) have examined the value of n and concluded that n → ∞ can generate reasonable results. A direct and simple integration is possible for n → ∞ and it yields similar boundary shear stress in smooth rectangular channel as Yang and Lim (1997) did. In other words, in the region where u∗b y/u∗w lw < 1, Eq. (3a) is simplified as follows: j(uv − xy /) jy
= gS
(4a)
and in the region u∗b y/u∗w lw 1, Eq. (3a) becomes j(uv − xy /) jy
=0
(4b)
The above derivation can be interpreted as that the surplus energy in any arbitrary flow volume will be transferred in one direction towards the nearest boundary (Yang and Lim, 1997). Thus, the division lines in Fig. 1 can be derived by equating the relative normal distances to the bed and the wall, i.e., u∗b y/u∗w lw = 1. It is obvious that the straight division line is consistent to the bisector of base angle as realized by Keulegan (1938) if Db = Dw . In other words, Keulegan’s discovery of straight division lines can be deduced from the Reynolds equation directly. Yang and Lim (1997) examined the straight division lines using experimental data of rectangular channels.
3. Determination of boundary shear stress distribution In the bed region of Fig. 1 where every point satisfies the geometrical condition: u∗b y/u∗w lw < 1, the integration of Eq. (4a) yields uv − xy / = gSy + c
(5)
in which c is the integration constant, it can be determined using the solid boundary condition, i.e., at y = 0, u = 0, v = 0 and xy = boundary shear stress = b . Thus, Eq. (5) can be rewritten as xy / = b / − gSy + uv
(6)
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126
It is obvious that Eq. (6) can be directly derived from Eq. (1) using Prandtl’s method of order-ofmagnitude analysis, but it ambiguously states that Eq. (6) is suitable only in the region far from the sidewall and there is no quantitative definition about how far away this distance from the sidewall is. Alternatively, Eq. (2) describes clearly the region that Eq. (6) is valid, i.e., the region bounded by the two division lines, free surface and bed shown in Fig. 1. Thus, starting from Eq. (6), one is able to assess the boundary shear stress and the interaction of velocity distribution and secondary currents i.e., v. The boundary shear stress b can be determined using the upper boundary condition in the region where y = normal distance from the bed to the upper boundary of the region, i.e., ln , b / = gSl n + (xy / − uv)y=ln
(7)
in the central portion of the bed region shown in Fig. 1, it is obvious that, at the free surface, ln = h, xy = 0, v = 0, thus one has b = gSh
(8)
in the corner portion of the bed region, ln = normal distance from the bed to the division lines, in order to determine the boundary shear stress, one has to evaluate the value of (xy / − uv) at the division line first. From Eq. (4b), one knows that (xy / − uv) is independent of y, thus from the division line to the free surface, (xy / − uv) is constant, and noticing at the free surface (xy / − uv)h = 0, therefore, at the division line (xy / − uv)ln = 0. From Eq. (7), the boundary shear stress near the corner is written as b = gSl n
(9)
Eqs. (8) and (9) show that generally speaking, the flow region can be primarily divided into three parts corresponding to the banks and the bed as suggested by Einstein (1942), and each sub-region can be divided by infinite strips normal to the boundary. If the area of shaded element in Fig. 1 is A corresponding to the wetted perimeter element p, then boundary shear stress can be simply evaluated by b = lim gS p→0
A P
= gS
P l n P
= gSl n .
(10)
The mean bed and sidewall shear stress shown in Fig. 1 can be determined by two ways, one way is to integrate the local boundary shear stress shown in Eq. (10), i.e., gS pb b = b dp, (11a) pb 0 gS pw w = w dp, (11b) pw 0 where b and w are mean bed and sidewall shear stress corresponding to the bed and sidewall perimeter pw and pb , respectively.
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
127
0.05
0.15
0.05
0.15
0.05
0.05
0.25
0.15
0.25 0.25
τw(kg/m2)
0.15
0.25
0.25
0.25
τb
(a)
0.15
0.15
(kg/m2)
(b)
(c)
Fig. 2. Comparison of computed (—) and measured boundary shear stress in trapezoidal channels, (a) = 63.4◦ , (b) = 45◦ , (c) = 30◦ .
Another way is to assess the mean bed and sidewall shear stress by b = gS
Ab , pb
(12a)
w = gS
Aw , pw
(12b)
where Ab and Aw are areas of bed and sidewall sub-region, respectively. Ab is the area bounded by the bed, two division lines and the free surface; Aw is the area bounded by the sidewall, one division line and the free surface. Eqs. (11a) and (11b) are actually identical to Eqs. (12a) and (12b), respectively. Therefore, the slope of the division line shown in Fig. 1 can be determined y u∗w w = = (13) lw u∗b b substituting Eqs. (11a) and (11b) or (12a) and (12b) into Eq. (13), one gets the slope of division line, i.e., y/ lw as follows: 2 3 y h h y − sin = 0. sin + cos − 2 (14) 2 lw b lw b √ For a very wide channel (h/b ≈ 0), Eq. (14) shows that lw = 3 2y ≈ 1.26y regardless of the angle . Tracy (1965) experimentally verified this theoretical conclusion in a very wide wind tunnel, the measured results show that the distributions of Reynolds shear stress and turbulent velocity in the central region of lw > 1.2h are totally different from those in the corner region of lw < 1.2h. Ghosh and Roy (1970) measured the boundary shear stress distribution in trapezoidal channels, in their experiments the boundary shear stress was determined from a Preston tube and the near-wall velocity profiles, typical results are shown in Fig. 2, in which the solid lines represent the theoretical results from Eq. (10). It can be seen that the measured and computed peak sidewall shear stresses approach to the base
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
128 1.2
1
τb ghS
0.8
Knight & Patel (1985) θ =90˚ 0.6
Knight et al. (1993)
0.4
θ =68˚
Knight et al. (1993)
θ =45˚
knight et al (1984)
θ =90˚
0.2
0 0
5
10
15
20
25 b h
30
35
40
45
50
Fig. 3. Comparison of computed (—-) and measured mean bed shear stress versus aspect ratios with various angles.
0.8 0.7 0.6
τb ρghS
0.5
Knight & Patel (1985) θ=90˚ 0.4
Knight et al. (1993)
0.3
θ=68˚
Knight et al. (1993)
θ=45˚
knight et al (1984)
θ=90˚
0.2 0.1 0 0
5
10
15
20
25
30
35
40
45
50
b h
Fig. 4. Comparison of computed (—) and measured mean sidewall shear stress versus aspect ratios with various angles.
corner as the angle decreases. For channel flows with other types of cross section, Yang (1996) found that the concept of energy transport via minimum relative distance reproduces the boundary shear stress better than other available methods. Knight et al. (1984, 1985, 1993) measured the mean bed and sidewall shear stress using the Preston tube and the results are shown in Figs. 3 and 4, in which data points available in the literature are also included. It can be seen that Eqs. (12a) and (12b) yield good agreement with the measured data. It is
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
129
interesting to note that w /ghS approaches the value of 0.63 (=1.26/2) as b/ h increases, which is in accordance with the critical value of corner region (lw ≈ 1.26h) as reported by Tracy (1965). 4. Velocity distribution in inner region To simplify the expression, the discussion in the following sections will be only limited to a special case—rectangular channel flows, i.e., = 90◦ in Fig. 1. In the inner region near the bed, the Reynolds shear stress is constant and the momentum equation (Eq. (6)) can be simplified as −u v = u2∗b + uv,
(15)
where u2∗b = b /. Prandtl’s mixing-length theory is valid to describe the Reynolds shear stress in the inner region and that gives −u v = (y du/dy)2 . Integrating Eq. (15) with respect to y in the inner region and using the boundary condition y = yo , u = 0, yields u 1 y = ln u∗1 yo
(16)
where u∗1 = u∗ + v[1/ ln(u∗ y/) + A]/4; parameter yo depends locally on the boundary characteristics, for rough walls, yo =30/; for smooth walls, yo should be dependent on the local hydraulic characteristics, or yo should be proportional to v/u∗b , therefore for a smooth channel, Eq. (16) becomes u 1 u∗b y = ln + A. u∗1
(17)
In smooth open channels, several researchers have found through their own experiments that and A in Eq. (17) are different from each other, examples are: Klebanoff (1954; 1/ = 2.44, A = 4.9), Townsend (1956; 2.44, 7), Huffman and Bradshaw (1972; 2.44, 5), Steffler et al. (1985; 2.5, 5.5), Nezu and Rodi (1986; 2.43, 5.29), Kirkgoz (1989; 2.44, 5.5). From these investigations, it is clear that has a range of variation between 0.4 and 0.41, and A varies between 4.9 and 7. Eq. (17) shows that the classic law of the wall function is valid in the inner region of 3-D flows, but the shear velocities on both sides may differ from each other. It is widely accepted that the secondary currents bring the fluid particles from the high momentum region to the corner, subsequently the momentum in non-circular duct flows is well mixed and becomes uniform (Nezu and Nakagawa, 1993; Schlichting, 1979). Thus it is possible that u∗1 in Eq. (17) becomes constant. Tracy and Lester (1961) measured velocity profiles in the central lines for it is supposed to be 2-D flow, they first plotted the velocities in the form of u/u∗ and u∗ y/ where u∗ = (ghS)0.5 , and found that the law of wall cannot reproduce the velocity profiles well as the velocity profiles do not have a common slope. However, all measured points collapse into a single line when they plotted the measured point velocities in the form of u/(gRS)0.5 versus (ghS)0.5 y/ where R is the hydraulic radius and all points can be represented by the classical log-law. Obviously, Tracy and Lester (1961) found that the shear velocities in both sides of Eq. (16) are not the same, and an attempt is made herein to extend their discovery to the region besides the channel center line, i.e., u∗1 = (gRS)0.5 , and u∗ = (gl n S)0.5 in Eq. (17).
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
130 25
u y u 1 ln * + A = ν u *1 0.41
20
u u *1
15
10
5
0 0
1
2
2z/h=0.6 2z/h=0.142 2z/h=0.022
3
4 ln u*y ν
2z/h=0.5 2z/h=0.082 2z/h=0.014
5
2z/h=0.4 2z/h=0.042 2z/h=0.009
6
7
8
2z/h=0.2 2z/h=0.032 2z/h=0.8
Fig. 5. Velocity profiles in a sub-region of wind tunnel where z is the distance from a corner.
Fig. 5 shows a plot of the measured velocity profiles in the form of u/u∗1 versus ln(u∗ y/) based on Tracy’s data (1965), it can be seen that all point velocities in the bed region (bounded by the division line, bed, central line and free surface) tends to be a single line. This conclusion has been confirmed by many experimental results (Yang, 1996). It should be mentioned that the above derivation is also valid for the flow dominated by the sidewall shear stress and lateral secondary velocity, i.e., w in the sidewall region, this discussion is omitted in order to focus on vertical velocity in this paper, the detail discussion about velocity distribution in the sidewall region can be found in Yang’s (1996) dissertation. 5. Velocity distribution in outer region Over the past one hundred years, the velocity distributions in steady and uniform flows have been extensively investigated. It is widely reported that the classical log-law can be applied to the whole region from the bed to the free surface if the channel is wide enough when the Karman constant and integral constant yo are properly adjusted to fit the measured velocity profiles. Modern studies show that the classical log-law is only valid in the inner region, the measured velocity deviates systematically from the log-law in the outer region and this deviation cannot be neglected (Nezu and Nakagawa, 1993). The velocity distribution in steady and uniform open channel flows is characterized by the maximum velocity at a vertical position below the free surface that is called the dip-property. The dip-property can be always observed in the vicinity of the sidewall, no matter how much the ratio of width to depth of flow is (Yang et al., 2004). Up to now, the mechanism of dip-property has not been well understood. Most researchers attribute this property to the influence of secondary currents that cannot be predicted exactly even using high-level
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
131
1 z/h=2.5
0.9
z/h=2.0
z/h=1.5
z/h=1.0
z/h=0.5
0.8 0.7
h
y
0.6 0.5 0.4 0.3 0.2 0.1 0
'
0
0.5
'
1
0
0.5
1 -0.2
0.8
-0.2
0.8
-0.4
0.6
-uv u2*1
Fig. 6. Measured Reynolds shear stress distribution in a rectangular open channel for b/ h = 5 (Imamoto and Ishigaki, 1988).
turbulence models. No attempt has been made in this study to completely consider the effects of secondary currents due to its complexity, only a mathematical explanation about the dip-property is provided herein by taking the component of secondary currents v into account. The momentum equation (Eq. (6)) in the outer region can be rewritten as follows: −
u v y y uv = 1 − + b1 + 2 + c, 2 h h u∗1 u∗1
(18)
where b1 and c are independent of y, b1 = (u2∗1 − ghS)/u2∗1 and c = (u2∗b − u2∗1 )/u2∗1 . Fig. 6 shows the measured profiles of Reynolds shear stress −u v /u2∗1 in an open channel from central line to the sidewall, in which z denotes the transverse distance away from the sidewall. The experiment was carried out by Imamoto and Ishigaki (1988) using Laser Doppler Anemometer, the channel used was 6 m long, 20 cm wide and 15 cm deep, the shear velocity was determined from the slope of channel (=1/500). The measured data points shown in Fig. 7 of Imamoto and Ishigaki’s (1988) paper are reproduced in Fig. 6 of this paper, the aspect ratio, b/ h was 5 and the straight lines are obtained by best fit of the data points. It can be seen that the dimensionless Reynolds shear stress −u v /u2∗1 approaches to 1 as y/ h is close to 0, thus Fig. 6 states that c is negligible. Imamoto and Ishigaki (1988) observed negative Reynolds shear stress in the upper flow layer, this agrees with Eq. (18) because u2∗1 is always less than ghS, then b1 = (u2∗1 − ghS)/u2∗1 < 0 and the secondary currents in the upper layer is always downward or v is negative (Shiono and Feng, 2003; Grass, 1971), thus the second on RHS of Eq. (18) is negative. But their experimental results show that the secondary currents in the lower layer are always upward, thus the third term on RHS of Eq. (18) should be positive, this is why the data points inFig. 6 locate above the straight lines. Therefore, one may conclude that the secondary currents lead to the deviation of Reynolds shear
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132
stress from the standard linear relationship, and it subsequently shifts the maximum velocity in a vertical profile from the water surface as realized by Cardoso et al. (1989). Nezu and Nakagawa (1993, p. 66) measured the Reynolds shear stress using LDA in open channels, their experimental results show that the mixing-length theorem is only valid in the inner region and in the outer region the measured the Reynolds shear stress can be modified as follows (Yalin, 1977, p. 202): −
u v y d(u/u∗1 ) = y 1 − h dy u2∗1
substituting Eq. (19) into (18), one obtains u v d −v d 1 b1 = exp exp d + c1 , + u∗ u∗ (1 − ) u∗ (1 − ) (1 − )
(19)
(20)
where = y/ h, c1 is the integration constant, c1 can be determined by the non-slip boundary condition, i.e., at y = yo , u = 0. The classical log-law can be derived from Eq. (20) if b1 = 0, and v = 0, this indicates that the log-law is valid only when b1 and v are negligible. The numerical integration may be necessary to obtain the streamwise velocity distribution using Eq. (20) if the profile of v is given. This study only provides a simple estimation on the influence of the wall-normal velocity on the streamwise velocity, and v is modeled using the boundary condition: at the bed where y = 0, v = 0, and at the free surface where y = h, v = 0, thus v can be approximated by the following equation using these two boundary conditions: v = m (1 − )p , (21) u∗1 where , m and p are coefficients to be determined. Substituting Eq. (21) into (20), one can obtain a simple velocity distribution for any values of , p and m, but to simplify the results, only the case of p = m = 1 is discussed. By substituting Eq. (21) into (20), one has
− − b e 1 e u 1 = e d + d + c1 . (22) u∗ k 1− The first and second terms in the bracket of Eq. (22) can be approximately calculated using the Taylor series and only the first three terms are used, thus one has 2 − 1 1 2 e d ≈ − + d = ln − + , (23) 2 2 (1− ) − e 1− 2 e − − ln(1 − ) + (1 − ) + . (24) d = −e d(1 − ) ≈ −e 1− 1− 2 Using the non-slip boundary condition at = o , and o >1, u = 0, the integration constant c1 can be determined as follows: 1 b − 2 (25) − − ln o . c1 = e 4
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133
2.2 2 2 1.8
u(m/s)
u(m/s)
1.5
Sarma type 2 log-law Eq. 26
1
1.6 1.4
0.5
0 0.01
Sarma type 1 log-law Eq. 26
0.1
1.2
y/h
(a)
1 0.01
1
0.1
1
y/h
(b) 2
2.2
1.9
Sarma type 3
1.7
Eq. 26
u(m/s)
u(m/s)
2
1.8
log-law
1.8
Sarma type 4 log-law Eq. 26
1.6 1.5 1.4
1.6
1.3 1.2
1.4
1.1 1.2 0.01
(c)
0.1
y/h
1 0.01
1
(d)
0.1
1
y/h
Fig. 7. Typical velocity profiles in steady and uniform open channel flows (after Sarma et al., 2000).
Substituting Eqs. (23)–(25) into Eq. (22), one gets
2 2 2 u y 1 − 1 ln + . = e − + − b1 e− ln(1 − ) + (1 − ) + − u∗ yo 2 2 4 (26) It is obvious that Eq. (26) becomes Eq. (16) if b1 = 0 and = 0. Sarma et al. (2000) analyzed the experimental data collected at the Indian Institute of Science over a long period, and concluded that four types of velocity profile exist which are reproduced in Fig. 7. Type 1: the measured velocity in the outer zone is entirely below the log curve; Type 2: the measured velocity is partly above the log-curve and partially below the log curve; Type 3: the measured velocity touches the log curve at the free surface and a certain depth below; Type 4: the measured velocity in the outer zone entirely lies above the log curve. In Fig. 7, it can be seen that Eq. (26) fits the measured velocity profile very well and the hydraulic parameters shown in Fig. 7 are listed in Table 1 where the shear velocities, aspect ratio b/ h, the location
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
134 Table 1 Basic parameters in Fig. 7
Type 1 Type 2 Type 3 Type 4
u∗(mm/s)
Aspect ratio (b/ h)
2z/b
Froude no.
b1
124.4 130.1 105.37 84.6
4.98 4.98 8.22 9.99
0.1 0.389 1.0 1.0
1.57 1.57 1.94 1.88
−1.1 −0.35 −0.1 0
0 0.1 0.045 0.03
of velocity profiles 2z/b and Froude no. were provided by Sarma et al. (2000), b and are obtained by best fit of the data points. As aforementioned, the obtained parameter b is negative and is positive. When the maximum velocity appears below the free surface, at the point of maximum velocity, the velocity gradient is zero, i.e., du/dy = 0, in other words, the maximum velocity occurs at the point where −u v /u2∗1 = 0. Thus, one gets the following equation from Eq. (18): ymax (uv)y=ymax + u2∗1 (1 + c) = , h u2∗1 (1 − b1 )
(27)
where ymax = the distance from the level of the maximum velocity to the bed. If the wall-normal velocity v = 0, then Eq. (27) can be further simplified as follows in the central region where u∗b = (ghS)0.5 :
u2∗b u2∗b ghS 1+c ymax = 1. (28) = = = ghS h 1 − b1 u2∗1 u2∗1 Eq. (28) states that for a velocity profile, the maximum velocity appears on the free surface if the secondary currents disappear. Hence, Eqs. (27) and (28) clearly indicates that the dip-phenomenon is caused by the secondary currents, i.e., v = 0. Strictly speaking, the wall-normal velocity v could be altered by the effect of buoyancy such as density gradient or temperature gradient (Yang, 2002), etc. Many researchers in the field of sediment transport, such as Einstein and Chien (1955), Vanoni (1946) and Coleman (1986), etc. reported that the velocity profile deviates from the classical log-law and some of them concluded that Karman coefficient is not universal, but variable with the sediment concentration. Chien and Wan (1999, p. 548) reported that the observed wind velocity in meteorology follows the loglaw very well when the temperature gradient is zero, but it deviates from the classical log-law significantly when the temperature at the ground surface is different from that at high altitude. It is obvious the buoyant effect caused by the temperature gradient or sediment concentration gradient will increase the wall-normal velocity, and subsequently the temperature gradient or the density gradient causes the deviation of measured velocity from the classical log-law. It is well known that the wall-normal velocity in boundary layer flows is non-zero, and the streamwise velocity profile deviates from the classical log-law, this deviation might be also caused by the non-zero wall-normal velocity, or the mechanism of Coles’ wake-law is physically caused by the wall-normal velocity. Therefore, it is worthwhile to investigate the impact of wall-normal velocity v on the streamwise velocity u. In other words, the wall-normal velocity should not be neglected as widely assumed, i.e., the
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observed deviation of streamwise velocity from the classical log-law should be linked to the presence of wall-normal velocity. 6. Conclusions The distributions of boundary shear and velocity in open channels are discussed. The Reynolds equation is simplified using an order-of-magnitude treatment. A power relationship of the wall-tangential and wallnormal terms in the Reynolds equation is developed, then the boundary shear stress and Reynolds shear stress are obtained directly from the Reynolds equation. Good agreement is achieved between the measured and computed mean bed and sidewall shear stress in trapezoidal channels without empirical coefficients involved. The local boundary shear stress distribution is also well reproduced. An analytical approach is established to express the connection between the streamwise velocity and wall-normal velocity. The theoretical velocity distribution is obtained by solving the Reynolds equation using the simplified mixing-length theorem. The model is compared with the measured velocity profiles available in literature and a good agreement is obtained. The investigation shows that wall-normal velocity greatly affects the velocity profile and leads to the deviation of the measured velocity from the classical log-law. The conclusions include that: 1. The bed shear velocity in an open channel can be approximately estimated using the method of flow region division as suggested by Keulegan (1938). 2. The non-zero wall-normal velocity v driven by the secondary currents caused the deviation of Reynolds shear stress from the conventional linear-distribution in 2-D flows. This deviation subsequently causes the location of the maximum velocity below the free surface, or the maximum velocity appears on the free surface and the classical log-law is valid only when the secondary currents disappear (v = 0). 3. It is widely reported that the measured velocity deviates from the classical log-law in many cases, e.g. boundary layer flows, non-uniform flows and density stratification flows. This velocity deviation can probably attribute to the non-zero wall-normal velocity. References Cardoso, A.H., Graf, W.H., Gust, G., 1989. Uniform flow in a smooth open channel. J. Hydraul. Res. 27 (5), 603–616. Chien, N., Wan, Z., 1999. Mechanics of Sediment Transport. ASCE, Reston, VA. Coleman, N.L., 1986. Effects of suspended sediment on open-channel distribution. Water Resour. Res. AGU 22 (10), 1377–1384. Einstein, H.A., 1942. Formulas for the transportation of bed load. Trans. Am. Soc. Civ. Eng. 107, 133–169. Einstein, H.A., Chien, N., 1955. Effects of heavy sediment concentration near the bed on velocity and sediment distribution. US Missouri River Division Report No. 8, Army Corps of Engineers. Ghosh, S.N., Roy, N., 1970. Boundary shear distribution in open channel flow. J. Hydraul. Div. ASCE 96 (4), 967–994. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries. J. Fluid Mech. 50 (2), 233–255. Guo, J., Julien, P.Y., 2005. Shear stress in smooth rectangular open-channel flows. J. Hydraul. Eng. ASCE 131 (1), 30–37. Imamoto, H., Ishigaki, T., 1988. Measurement of secondary flow in an open channel. Proceedings of 6th IAHR-APD Congress, Kyoto, Japan, pp. 513–520. Keulegan, C.H., 1938. Laws of turbulent flow in open channels. J. Res. Natl. Bur. Stand. 21, 707–740.
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Knight, D.W., 1981. Boundary shear in smooth and rough channels. J. Hydraul. Div. ASCE 107 (7), 839–851. Knight, D.W., Patel, H.S., 1985. Boundary shear in smooth rectangular ducts. J. Hydraul. Eng. ASCE 111 (1), 29–47. Knight, D.W., Demetriou, J.D., Hamed, M.E., 1984. Boundary shear in smooth rectangular channels. J. Hydraul. Eng. ASCE 110 (4), 405–422. Knight, D.W., Yuen, K.W.H., Alhamid, A.A.I., 1993. Boundary shear stress distributions in open channel flow. In: Beven, K., Chatwin, P., Wiley, J. (Eds.), Physical Mechanisms of Mixing and Transport in the Environment. Melling, A., Whitelaw, J.H., 1976. Turbulent flow in a rectangular duct. J. Fluid Mech. 78 (2), 289–315. Murphy, C., 1904. Accuracy of stream measurements. Water Supply and Irrig. Paper 95, 111–112. Nezu, I., Nakagawa, H., 1993. Turbulence in Open Channel Flows. Balkema Publisher, Rotterdam, Netherlands. Nezu, I., Rodi, W., 1986. Open channel flow measurements with a laser Doppler anemometer. J. Hydraul. Eng. ASCE 112 (5), 335–355. Rhodes, D.G., Knight, D.W., 1994. Distribution of shear force on boundary of smooth rectangular duct. J. Hydraul. Div. ASCE 120 (11), 787–806. Sarma, K.V.N., Prasad, B.V.R., Sarma, A.K., 2000. Detailed study of binary law for open channels. J. Hydraul. Eng. ASCE 126 (3), 210–214. Schlichting, H., 1979. Boundary Layer Theory. seventh ed. McGraw-Hill, New York. Shiono, K., Feng, T., 2003. Turbulence measurements of dye concentration and effects of secondary flow on distribution in open channel flows. J. Hydraul. Eng. ASCE 129 (5), 373–384. Stearns, F.P., 1883. A reason why the maximum velocity of water flowing in open channels is below the surface. Trans. Am. Soc. Civ. Eng. 7, 331–338. Tracy, H.J., 1965. Turbulent flow in a three-dimensional channel. J. Hydraul. Div. ASCE 91 (6), 9–35. Tracy, H.J., Lester, C.M., 1961. Resistance coefficients and velocity distribution in smooth rectangular channel. US Geological Survey, Water-supply Paper 1592, A1–A18. Vanoni, V.A., 1946. Transportation of suspended sediment by running water. Trans. ASCE 111, 67–133. Yalin, M.S., 1977. Mechanism of Sediment Transport. second ed. Pergamon, New York. Yang, S.-Q., 1993. The law of boundary shear in rectangular open channels. J. Sediment. Res. Beijing 3, 95–103 (in Chinese). Yang, S.-Q., 1996. Interactions of boundary shear stress, velocity distribution and flow resistance in 3-D channels. Ph. D. Dissertation, Nanyang Technological University. Yang, S.-Q., 2002. 1st and 2nd order approximate solutions of Reynolds equations in 3-D channels. Proceedings of 13th IAHRAPD Congress, Vol. 1, Singapore, pp. 70–75. Yang, S.-Q., Lim, S.-Y., 1997. Mechanism of energy transportation in a 3-D channel. J. Hydraul. Eng. ASCE 123 (8), 684–692. Yang, S.-Q., Lim, S.-Y., 1998. Boundary shear stress distribution in smooth rectangular open channel flows. Proc. Inst. Civil Eng. Water Maritime and Energy 130 (9), 163–173. Yang, S.-Q., McCorquodale, J.A., 2004. Determination of boundary shear stress and Reynolds shear stress in smooth rectangular channel flows. J. Hydraul. Eng. ASCE 130 (5), 458–462. Yang, S.-Q., Tan, S.-K., Lim, S.-Y., 2004. Velocity distribution and dip-phenomenon in smooth uniform channel flows. J. Hydraul. Eng. ASCE 130 (12), 1179–1186. Zheng, Y., Jin, Y.C., 1998. Boundary shear in rectangular ducts and channels. J. Hydraul. Eng. ASCE 124 (1), 86–89.
Interactions of boundary shear stress, secondary currents and velocity Shu-Qing Yanga, b a Division of Civil and Environmental Engineering, Korea Maritime University, Busan, Korea 606791 b Maritime Research Center, School of Civil and Environmental Engineering, Nanyang Technological University,
Singapore 639798 Received 26 March 2004; received in revised form 18 December 2004; accepted 10 January 2005 Communicated by Z.-S. She
Abstract This paper deals with the interaction of boundary shear stress, velocity distribution and secondary currents in open channel flows. A method for computing boundary shear stress and velocity distribution in steady, uniform and fully developed turbulent flows is developed by applying an order-of-magnitude analysis to the Reynolds equations. A simplified relationship between the wall-tangential and wall-normal terms in the Reynolds equation is hypothesized, then the Reynolds equations become solvable. This analysis suggests that the energy from the main flow is transported towards the nearest boundary to be dissipated through a minimum relative distance or normal distance of the boundary. The equations governing the boundary shear stress and Reynolds shear stress distributions are obtained, and the influence of wall-normal velocity on the streamwise velocity is assessed. It is found that the classical log-law is valid only when the wall-normal velocity is zero, the non-zero wall-normal velocity results in the derivation of measured streamwise velocity from the classical log-law. The derived equations are in good agreement with existing experimental data available in the literature. © 2005 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Boundary shear stress; Turbulent flows; Reynolds shear stress;Velocity profiles; Dip-phenomenon; Secondary currents
E-mail address: [email protected]. 0169-5983/$30.00 © 2005 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:10.1016/j.fluiddyn.2005.01.002
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Nomenclature A Ab and Aw b b1 c c1 D g h ln lb /Db and lw /Dw m n p pw and pb S u, v and w u , v , z u∗b and u∗w u∗ u∗1 x y yo ymax z xy xz b , w b w b
constant area of bed and sidewall sub-region width of channel bed coefficient constant integration constant characteristic length representing energy dissipation of boundary gravitational acceleration water depth the normal distance to the boundary relative distance to the bed and the sidewall coefficient empirical coefficients coefficient sidewall and bed perimeter energy slope mean velocities in x, y and z directions fluctuating velocity components mean bed and sidewall shear velocities shear velocity global shear velocity [=(gRS)0.5 ] streamwise direction wall-normal direction parameter of boundary characteristics location maximum velocity wall-tangential direction dynamic viscosity ju/jy − u v ju/jz − u w mean bed and sidewall shear stresses u2∗b u2∗w typical boundary roughness height coefficient angle of channel wall Karman constant (≈ 0.4) kinematic viscosity y/ h fluid density local boundary shear stress
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136 iso-velocity contour
123
y
lw
h
ln
secondary cells
θ
z
division lines b
Fig. 1. Relationship between observed primary flow, secondary flow, flow geometry and division lines (after Knight et al., 1993).
1. Introduction Turbulent structures and boundary shear stress in steady and uniform flows are fundamentally important and have been studied extensively by many researchers, such as Tracy (1965), Melling and Whitelaw s (1976), Knight and Patel (1985), Rhodes and Knight (1994), Yang (2002), etc., but it is still difficult to predict them exactly even using sophisticated turbulence models. Empirical equations for the boundary shear on the wall and bed have been proposed by Knight (1981), Zheng and Jin (1998), Guo and Julien (2005), etc. Yang (1993), using the hypothesis that the mechanical energy of 3-D turbulent flow is always transferred towards boundary via shortest relative distance, approximated mean boundary shear stresses on the bed and sidewalls in rectangular channels (Yang, 1996, Yang and Lim, 1997,1998). Many researchers measured the mean velocity profiles in a steady and uniform flow, and it is widely reported that the velocity deviates from the classical log-law near the free surface, i.e., the position whereby the maximum velocity appears below the water surface, occurs if the aspect ratio of the channel width to water depth, b/ h, is less than a certain value, this phenomenon was first observed about a century ago by Stearns (1883) and Murphy (1904). The boundary shear stress and the longitudinal velocity deviation from the classical log-law have charmed and challenged to scientists and engineers. To tackle them, Keulegan (1938) suggested that the bisectors of base angles of a polygonal channel could be used to partition a flow region as shown in Fig. 1, but he did not physically explain this suggestion. He also assumed that the log-law is valid to predict the velocity distribution from a point near the bed to the free surface. Einstein (1942) extended Keulegan’s (1938) concept to a river flow and stated that a river flow can be divided into three sub-sections corresponding to the banks and riverbed, respectively. He had, however, failed to explain why the flow region is divisible and how to divide it. The first question was partially answered by Chien and Wan (1999) who claimed that the energy from a unit element of water transmits only in one direction towards the boundary. Yang (1993) proposed that the surplus energy contained in any arbitrary flow volume will be transferred towards the nearest boundary to be dissipated, or the surplus energy is always transported to the boundary
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through the minimum relative distance that is defined as the ratio of geometrical distance from boundary to the boundary roughness or thickness of viscous sub-layer, but the concept was not directly derived from the Reynolds equations. The main objectives of this study are: (1) to solve the Reynolds equations by hypothesizing a relationship between the wall-tangential and wall-normal terms (2) to develop a method for estimating the boundary shear stress from the Reynolds equations; (3) to develop a simplified equation suitable for describing the velocity profile from the bed to water surface. 2. Simplified Reynolds equation For a steady, uniform and fully developed turbulent channel flow as shown in Fig. 1, Reynolds equations can be written as follows by adding the continuity equation to the x-direction momentum equation: j(uv − xy /) jy
+
j(uw − xz /) jz
= gS,
(1)
where x is defined as the streamwise direction; y is the wall-normal direction; z is the wall-tangential direction; u, v and w are mean velocities in x, y and z directions, respectively; g is gravitational acceleration; S is energy slope; and xy = ju/jy − u v and xz = ju/jz − u w , is dynamic viscosity, is fluid density, u , v , z are fluctuating velocity components. Experimental data (Tracy, 1965) show that in a region near a wall, the term in the wall-normal direction is much greater than the wall-tangential gradient. It follows that in the region near the bed the first term on the left-hand side (LHS) of Eq. (1) is dominant and the second term on LHS is negligible because the wall-normal direction is y-direction. Likewise, in the region near the sidewall, the second term is dominant, but the first term on LHS can be neglected. In other words, if the first term on LHS of Eq. (1) is negligible, the second term must be dominant, or vice versa. It is impossible that, in any region, both terms on LHS are negligible at same time because the constant gS, i.e., the gravitational term on the right-hand side of Eq. (1), must be balanced. The relative dominance of the two derivatives on LHS of Eq. (1) can be approximately related by j(uw − xz /) lw n j(uv − xy /) lb = , (2) jz Db Dw jy where n = empirical coefficients; lb /Db and lw /Dw are the relative distance to the bed and the sidewall in Fig. 1, the subscripts b and w denote the bed and wall, respectively; D is characteristic length representing energy dissipation of boundary, for a smooth boundary D =v/u∗ and for a rough boundary D = =typical boundary roughness height (Yang, 1993, 2002); l = the normal distance to the boundary as shown in Fig. 1; u∗ = shear velocity; = kinematic viscosity. Eq. (2) means an order-of-magnitude analysis, it is useful for a practical solution to simplify the Reynolds equation, it shows that if lb >lw , j/jz ≈ 0, then only the term j/jy is retained in Eq. (1). Similarly, in the region of lb ?lw , j/jy ≈ 0, then only the term j/jz is left in Eq. (1). For the region between these two extremes, these two terms co-exist to balance the gravitational component, but the importance of each term depends on the ratio of relative distances to the bed and sidewall defined in Eq. (2). Thus, Eq. (2) mathematically expresses that the wall-normal direction is more important than the wall-tangential gradient if n > 0.
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In this study, only smooth boundary is discussed and the results can be extended to rough channel flows, substituting (2) into (1) yields j(uv − xy /) u∗b y n = gS 1+ , (3a) jy u∗w lw where b = u2∗b , w = u2∗w , u∗b and u∗w are mean bed and sidewall shear velocities, b , w are mean bed and sidewall shear stresses. Eq. (1) can be similarly re-arranged as follows j(uw − xz /) u∗w z n = gS 1+ . (3b) jz u∗b y Thus, analytical solutions can be obtained from Eqs. (3a) and (3b), if the value of n is given. Yang and McCorquodale (2004) have examined the value of n and concluded that n → ∞ can generate reasonable results. A direct and simple integration is possible for n → ∞ and it yields similar boundary shear stress in smooth rectangular channel as Yang and Lim (1997) did. In other words, in the region where u∗b y/u∗w lw < 1, Eq. (3a) is simplified as follows: j(uv − xy /) jy
= gS
(4a)
and in the region u∗b y/u∗w lw 1, Eq. (3a) becomes j(uv − xy /) jy
=0
(4b)
The above derivation can be interpreted as that the surplus energy in any arbitrary flow volume will be transferred in one direction towards the nearest boundary (Yang and Lim, 1997). Thus, the division lines in Fig. 1 can be derived by equating the relative normal distances to the bed and the wall, i.e., u∗b y/u∗w lw = 1. It is obvious that the straight division line is consistent to the bisector of base angle as realized by Keulegan (1938) if Db = Dw . In other words, Keulegan’s discovery of straight division lines can be deduced from the Reynolds equation directly. Yang and Lim (1997) examined the straight division lines using experimental data of rectangular channels.
3. Determination of boundary shear stress distribution In the bed region of Fig. 1 where every point satisfies the geometrical condition: u∗b y/u∗w lw < 1, the integration of Eq. (4a) yields uv − xy / = gSy + c
(5)
in which c is the integration constant, it can be determined using the solid boundary condition, i.e., at y = 0, u = 0, v = 0 and xy = boundary shear stress = b . Thus, Eq. (5) can be rewritten as xy / = b / − gSy + uv
(6)
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It is obvious that Eq. (6) can be directly derived from Eq. (1) using Prandtl’s method of order-ofmagnitude analysis, but it ambiguously states that Eq. (6) is suitable only in the region far from the sidewall and there is no quantitative definition about how far away this distance from the sidewall is. Alternatively, Eq. (2) describes clearly the region that Eq. (6) is valid, i.e., the region bounded by the two division lines, free surface and bed shown in Fig. 1. Thus, starting from Eq. (6), one is able to assess the boundary shear stress and the interaction of velocity distribution and secondary currents i.e., v. The boundary shear stress b can be determined using the upper boundary condition in the region where y = normal distance from the bed to the upper boundary of the region, i.e., ln , b / = gSl n + (xy / − uv)y=ln
(7)
in the central portion of the bed region shown in Fig. 1, it is obvious that, at the free surface, ln = h, xy = 0, v = 0, thus one has b = gSh
(8)
in the corner portion of the bed region, ln = normal distance from the bed to the division lines, in order to determine the boundary shear stress, one has to evaluate the value of (xy / − uv) at the division line first. From Eq. (4b), one knows that (xy / − uv) is independent of y, thus from the division line to the free surface, (xy / − uv) is constant, and noticing at the free surface (xy / − uv)h = 0, therefore, at the division line (xy / − uv)ln = 0. From Eq. (7), the boundary shear stress near the corner is written as b = gSl n
(9)
Eqs. (8) and (9) show that generally speaking, the flow region can be primarily divided into three parts corresponding to the banks and the bed as suggested by Einstein (1942), and each sub-region can be divided by infinite strips normal to the boundary. If the area of shaded element in Fig. 1 is A corresponding to the wetted perimeter element p, then boundary shear stress can be simply evaluated by b = lim gS p→0
A P
= gS
P l n P
= gSl n .
(10)
The mean bed and sidewall shear stress shown in Fig. 1 can be determined by two ways, one way is to integrate the local boundary shear stress shown in Eq. (10), i.e., gS pb b = b dp, (11a) pb 0 gS pw w = w dp, (11b) pw 0 where b and w are mean bed and sidewall shear stress corresponding to the bed and sidewall perimeter pw and pb , respectively.
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
127
0.05
0.15
0.05
0.15
0.05
0.05
0.25
0.15
0.25 0.25
τw(kg/m2)
0.15
0.25
0.25
0.25
τb
(a)
0.15
0.15
(kg/m2)
(b)
(c)
Fig. 2. Comparison of computed (—) and measured boundary shear stress in trapezoidal channels, (a) = 63.4◦ , (b) = 45◦ , (c) = 30◦ .
Another way is to assess the mean bed and sidewall shear stress by b = gS
Ab , pb
(12a)
w = gS
Aw , pw
(12b)
where Ab and Aw are areas of bed and sidewall sub-region, respectively. Ab is the area bounded by the bed, two division lines and the free surface; Aw is the area bounded by the sidewall, one division line and the free surface. Eqs. (11a) and (11b) are actually identical to Eqs. (12a) and (12b), respectively. Therefore, the slope of the division line shown in Fig. 1 can be determined y u∗w w = = (13) lw u∗b b substituting Eqs. (11a) and (11b) or (12a) and (12b) into Eq. (13), one gets the slope of division line, i.e., y/ lw as follows: 2 3 y h h y − sin = 0. sin + cos − 2 (14) 2 lw b lw b √ For a very wide channel (h/b ≈ 0), Eq. (14) shows that lw = 3 2y ≈ 1.26y regardless of the angle . Tracy (1965) experimentally verified this theoretical conclusion in a very wide wind tunnel, the measured results show that the distributions of Reynolds shear stress and turbulent velocity in the central region of lw > 1.2h are totally different from those in the corner region of lw < 1.2h. Ghosh and Roy (1970) measured the boundary shear stress distribution in trapezoidal channels, in their experiments the boundary shear stress was determined from a Preston tube and the near-wall velocity profiles, typical results are shown in Fig. 2, in which the solid lines represent the theoretical results from Eq. (10). It can be seen that the measured and computed peak sidewall shear stresses approach to the base
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
128 1.2
1
τb ghS
0.8
Knight & Patel (1985) θ =90˚ 0.6
Knight et al. (1993)
0.4
θ =68˚
Knight et al. (1993)
θ =45˚
knight et al (1984)
θ =90˚
0.2
0 0
5
10
15
20
25 b h
30
35
40
45
50
Fig. 3. Comparison of computed (—-) and measured mean bed shear stress versus aspect ratios with various angles.
0.8 0.7 0.6
τb ρghS
0.5
Knight & Patel (1985) θ=90˚ 0.4
Knight et al. (1993)
0.3
θ=68˚
Knight et al. (1993)
θ=45˚
knight et al (1984)
θ=90˚
0.2 0.1 0 0
5
10
15
20
25
30
35
40
45
50
b h
Fig. 4. Comparison of computed (—) and measured mean sidewall shear stress versus aspect ratios with various angles.
corner as the angle decreases. For channel flows with other types of cross section, Yang (1996) found that the concept of energy transport via minimum relative distance reproduces the boundary shear stress better than other available methods. Knight et al. (1984, 1985, 1993) measured the mean bed and sidewall shear stress using the Preston tube and the results are shown in Figs. 3 and 4, in which data points available in the literature are also included. It can be seen that Eqs. (12a) and (12b) yield good agreement with the measured data. It is
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129
interesting to note that w /ghS approaches the value of 0.63 (=1.26/2) as b/ h increases, which is in accordance with the critical value of corner region (lw ≈ 1.26h) as reported by Tracy (1965). 4. Velocity distribution in inner region To simplify the expression, the discussion in the following sections will be only limited to a special case—rectangular channel flows, i.e., = 90◦ in Fig. 1. In the inner region near the bed, the Reynolds shear stress is constant and the momentum equation (Eq. (6)) can be simplified as −u v = u2∗b + uv,
(15)
where u2∗b = b /. Prandtl’s mixing-length theory is valid to describe the Reynolds shear stress in the inner region and that gives −u v = (y du/dy)2 . Integrating Eq. (15) with respect to y in the inner region and using the boundary condition y = yo , u = 0, yields u 1 y = ln u∗1 yo
(16)
where u∗1 = u∗ + v[1/ ln(u∗ y/) + A]/4; parameter yo depends locally on the boundary characteristics, for rough walls, yo =30/; for smooth walls, yo should be dependent on the local hydraulic characteristics, or yo should be proportional to v/u∗b , therefore for a smooth channel, Eq. (16) becomes u 1 u∗b y = ln + A. u∗1
(17)
In smooth open channels, several researchers have found through their own experiments that and A in Eq. (17) are different from each other, examples are: Klebanoff (1954; 1/ = 2.44, A = 4.9), Townsend (1956; 2.44, 7), Huffman and Bradshaw (1972; 2.44, 5), Steffler et al. (1985; 2.5, 5.5), Nezu and Rodi (1986; 2.43, 5.29), Kirkgoz (1989; 2.44, 5.5). From these investigations, it is clear that has a range of variation between 0.4 and 0.41, and A varies between 4.9 and 7. Eq. (17) shows that the classic law of the wall function is valid in the inner region of 3-D flows, but the shear velocities on both sides may differ from each other. It is widely accepted that the secondary currents bring the fluid particles from the high momentum region to the corner, subsequently the momentum in non-circular duct flows is well mixed and becomes uniform (Nezu and Nakagawa, 1993; Schlichting, 1979). Thus it is possible that u∗1 in Eq. (17) becomes constant. Tracy and Lester (1961) measured velocity profiles in the central lines for it is supposed to be 2-D flow, they first plotted the velocities in the form of u/u∗ and u∗ y/ where u∗ = (ghS)0.5 , and found that the law of wall cannot reproduce the velocity profiles well as the velocity profiles do not have a common slope. However, all measured points collapse into a single line when they plotted the measured point velocities in the form of u/(gRS)0.5 versus (ghS)0.5 y/ where R is the hydraulic radius and all points can be represented by the classical log-law. Obviously, Tracy and Lester (1961) found that the shear velocities in both sides of Eq. (16) are not the same, and an attempt is made herein to extend their discovery to the region besides the channel center line, i.e., u∗1 = (gRS)0.5 , and u∗ = (gl n S)0.5 in Eq. (17).
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
130 25
u y u 1 ln * + A = ν u *1 0.41
20
u u *1
15
10
5
0 0
1
2
2z/h=0.6 2z/h=0.142 2z/h=0.022
3
4 ln u*y ν
2z/h=0.5 2z/h=0.082 2z/h=0.014
5
2z/h=0.4 2z/h=0.042 2z/h=0.009
6
7
8
2z/h=0.2 2z/h=0.032 2z/h=0.8
Fig. 5. Velocity profiles in a sub-region of wind tunnel where z is the distance from a corner.
Fig. 5 shows a plot of the measured velocity profiles in the form of u/u∗1 versus ln(u∗ y/) based on Tracy’s data (1965), it can be seen that all point velocities in the bed region (bounded by the division line, bed, central line and free surface) tends to be a single line. This conclusion has been confirmed by many experimental results (Yang, 1996). It should be mentioned that the above derivation is also valid for the flow dominated by the sidewall shear stress and lateral secondary velocity, i.e., w in the sidewall region, this discussion is omitted in order to focus on vertical velocity in this paper, the detail discussion about velocity distribution in the sidewall region can be found in Yang’s (1996) dissertation. 5. Velocity distribution in outer region Over the past one hundred years, the velocity distributions in steady and uniform flows have been extensively investigated. It is widely reported that the classical log-law can be applied to the whole region from the bed to the free surface if the channel is wide enough when the Karman constant and integral constant yo are properly adjusted to fit the measured velocity profiles. Modern studies show that the classical log-law is only valid in the inner region, the measured velocity deviates systematically from the log-law in the outer region and this deviation cannot be neglected (Nezu and Nakagawa, 1993). The velocity distribution in steady and uniform open channel flows is characterized by the maximum velocity at a vertical position below the free surface that is called the dip-property. The dip-property can be always observed in the vicinity of the sidewall, no matter how much the ratio of width to depth of flow is (Yang et al., 2004). Up to now, the mechanism of dip-property has not been well understood. Most researchers attribute this property to the influence of secondary currents that cannot be predicted exactly even using high-level
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
131
1 z/h=2.5
0.9
z/h=2.0
z/h=1.5
z/h=1.0
z/h=0.5
0.8 0.7
h
y
0.6 0.5 0.4 0.3 0.2 0.1 0
'
0
0.5
'
1
0
0.5
1 -0.2
0.8
-0.2
0.8
-0.4
0.6
-uv u2*1
Fig. 6. Measured Reynolds shear stress distribution in a rectangular open channel for b/ h = 5 (Imamoto and Ishigaki, 1988).
turbulence models. No attempt has been made in this study to completely consider the effects of secondary currents due to its complexity, only a mathematical explanation about the dip-property is provided herein by taking the component of secondary currents v into account. The momentum equation (Eq. (6)) in the outer region can be rewritten as follows: −
u v y y uv = 1 − + b1 + 2 + c, 2 h h u∗1 u∗1
(18)
where b1 and c are independent of y, b1 = (u2∗1 − ghS)/u2∗1 and c = (u2∗b − u2∗1 )/u2∗1 . Fig. 6 shows the measured profiles of Reynolds shear stress −u v /u2∗1 in an open channel from central line to the sidewall, in which z denotes the transverse distance away from the sidewall. The experiment was carried out by Imamoto and Ishigaki (1988) using Laser Doppler Anemometer, the channel used was 6 m long, 20 cm wide and 15 cm deep, the shear velocity was determined from the slope of channel (=1/500). The measured data points shown in Fig. 7 of Imamoto and Ishigaki’s (1988) paper are reproduced in Fig. 6 of this paper, the aspect ratio, b/ h was 5 and the straight lines are obtained by best fit of the data points. It can be seen that the dimensionless Reynolds shear stress −u v /u2∗1 approaches to 1 as y/ h is close to 0, thus Fig. 6 states that c is negligible. Imamoto and Ishigaki (1988) observed negative Reynolds shear stress in the upper flow layer, this agrees with Eq. (18) because u2∗1 is always less than ghS, then b1 = (u2∗1 − ghS)/u2∗1 < 0 and the secondary currents in the upper layer is always downward or v is negative (Shiono and Feng, 2003; Grass, 1971), thus the second on RHS of Eq. (18) is negative. But their experimental results show that the secondary currents in the lower layer are always upward, thus the third term on RHS of Eq. (18) should be positive, this is why the data points inFig. 6 locate above the straight lines. Therefore, one may conclude that the secondary currents lead to the deviation of Reynolds shear
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stress from the standard linear relationship, and it subsequently shifts the maximum velocity in a vertical profile from the water surface as realized by Cardoso et al. (1989). Nezu and Nakagawa (1993, p. 66) measured the Reynolds shear stress using LDA in open channels, their experimental results show that the mixing-length theorem is only valid in the inner region and in the outer region the measured the Reynolds shear stress can be modified as follows (Yalin, 1977, p. 202): −
u v y d(u/u∗1 ) = y 1 − h dy u2∗1
substituting Eq. (19) into (18), one obtains u v d −v d 1 b1 = exp exp d + c1 , + u∗ u∗ (1 − ) u∗ (1 − ) (1 − )
(19)
(20)
where = y/ h, c1 is the integration constant, c1 can be determined by the non-slip boundary condition, i.e., at y = yo , u = 0. The classical log-law can be derived from Eq. (20) if b1 = 0, and v = 0, this indicates that the log-law is valid only when b1 and v are negligible. The numerical integration may be necessary to obtain the streamwise velocity distribution using Eq. (20) if the profile of v is given. This study only provides a simple estimation on the influence of the wall-normal velocity on the streamwise velocity, and v is modeled using the boundary condition: at the bed where y = 0, v = 0, and at the free surface where y = h, v = 0, thus v can be approximated by the following equation using these two boundary conditions: v = m (1 − )p , (21) u∗1 where , m and p are coefficients to be determined. Substituting Eq. (21) into (20), one can obtain a simple velocity distribution for any values of , p and m, but to simplify the results, only the case of p = m = 1 is discussed. By substituting Eq. (21) into (20), one has
− − b e 1 e u 1 = e d + d + c1 . (22) u∗ k 1− The first and second terms in the bracket of Eq. (22) can be approximately calculated using the Taylor series and only the first three terms are used, thus one has 2 − 1 1 2 e d ≈ − + d = ln − + , (23) 2 2 (1− ) − e 1− 2 e − − ln(1 − ) + (1 − ) + . (24) d = −e d(1 − ) ≈ −e 1− 1− 2 Using the non-slip boundary condition at = o , and o >1, u = 0, the integration constant c1 can be determined as follows: 1 b − 2 (25) − − ln o . c1 = e 4
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133
2.2 2 2 1.8
u(m/s)
u(m/s)
1.5
Sarma type 2 log-law Eq. 26
1
1.6 1.4
0.5
0 0.01
Sarma type 1 log-law Eq. 26
0.1
1.2
y/h
(a)
1 0.01
1
0.1
1
y/h
(b) 2
2.2
1.9
Sarma type 3
1.7
Eq. 26
u(m/s)
u(m/s)
2
1.8
log-law
1.8
Sarma type 4 log-law Eq. 26
1.6 1.5 1.4
1.6
1.3 1.2
1.4
1.1 1.2 0.01
(c)
0.1
y/h
1 0.01
1
(d)
0.1
1
y/h
Fig. 7. Typical velocity profiles in steady and uniform open channel flows (after Sarma et al., 2000).
Substituting Eqs. (23)–(25) into Eq. (22), one gets
2 2 2 u y 1 − 1 ln + . = e − + − b1 e− ln(1 − ) + (1 − ) + − u∗ yo 2 2 4 (26) It is obvious that Eq. (26) becomes Eq. (16) if b1 = 0 and = 0. Sarma et al. (2000) analyzed the experimental data collected at the Indian Institute of Science over a long period, and concluded that four types of velocity profile exist which are reproduced in Fig. 7. Type 1: the measured velocity in the outer zone is entirely below the log curve; Type 2: the measured velocity is partly above the log-curve and partially below the log curve; Type 3: the measured velocity touches the log curve at the free surface and a certain depth below; Type 4: the measured velocity in the outer zone entirely lies above the log curve. In Fig. 7, it can be seen that Eq. (26) fits the measured velocity profile very well and the hydraulic parameters shown in Fig. 7 are listed in Table 1 where the shear velocities, aspect ratio b/ h, the location
S.-Q. Yang / Fluid Dynamics Research 36 (2005) 121 – 136
134 Table 1 Basic parameters in Fig. 7
Type 1 Type 2 Type 3 Type 4
u∗(mm/s)
Aspect ratio (b/ h)
2z/b
Froude no.
b1
124.4 130.1 105.37 84.6
4.98 4.98 8.22 9.99
0.1 0.389 1.0 1.0
1.57 1.57 1.94 1.88
−1.1 −0.35 −0.1 0
0 0.1 0.045 0.03
of velocity profiles 2z/b and Froude no. were provided by Sarma et al. (2000), b and are obtained by best fit of the data points. As aforementioned, the obtained parameter b is negative and is positive. When the maximum velocity appears below the free surface, at the point of maximum velocity, the velocity gradient is zero, i.e., du/dy = 0, in other words, the maximum velocity occurs at the point where −u v /u2∗1 = 0. Thus, one gets the following equation from Eq. (18): ymax (uv)y=ymax + u2∗1 (1 + c) = , h u2∗1 (1 − b1 )
(27)
where ymax = the distance from the level of the maximum velocity to the bed. If the wall-normal velocity v = 0, then Eq. (27) can be further simplified as follows in the central region where u∗b = (ghS)0.5 :
u2∗b u2∗b ghS 1+c ymax = 1. (28) = = = ghS h 1 − b1 u2∗1 u2∗1 Eq. (28) states that for a velocity profile, the maximum velocity appears on the free surface if the secondary currents disappear. Hence, Eqs. (27) and (28) clearly indicates that the dip-phenomenon is caused by the secondary currents, i.e., v = 0. Strictly speaking, the wall-normal velocity v could be altered by the effect of buoyancy such as density gradient or temperature gradient (Yang, 2002), etc. Many researchers in the field of sediment transport, such as Einstein and Chien (1955), Vanoni (1946) and Coleman (1986), etc. reported that the velocity profile deviates from the classical log-law and some of them concluded that Karman coefficient is not universal, but variable with the sediment concentration. Chien and Wan (1999, p. 548) reported that the observed wind velocity in meteorology follows the loglaw very well when the temperature gradient is zero, but it deviates from the classical log-law significantly when the temperature at the ground surface is different from that at high altitude. It is obvious the buoyant effect caused by the temperature gradient or sediment concentration gradient will increase the wall-normal velocity, and subsequently the temperature gradient or the density gradient causes the deviation of measured velocity from the classical log-law. It is well known that the wall-normal velocity in boundary layer flows is non-zero, and the streamwise velocity profile deviates from the classical log-law, this deviation might be also caused by the non-zero wall-normal velocity, or the mechanism of Coles’ wake-law is physically caused by the wall-normal velocity. Therefore, it is worthwhile to investigate the impact of wall-normal velocity v on the streamwise velocity u. In other words, the wall-normal velocity should not be neglected as widely assumed, i.e., the
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observed deviation of streamwise velocity from the classical log-law should be linked to the presence of wall-normal velocity. 6. Conclusions The distributions of boundary shear and velocity in open channels are discussed. The Reynolds equation is simplified using an order-of-magnitude treatment. A power relationship of the wall-tangential and wallnormal terms in the Reynolds equation is developed, then the boundary shear stress and Reynolds shear stress are obtained directly from the Reynolds equation. Good agreement is achieved between the measured and computed mean bed and sidewall shear stress in trapezoidal channels without empirical coefficients involved. The local boundary shear stress distribution is also well reproduced. An analytical approach is established to express the connection between the streamwise velocity and wall-normal velocity. The theoretical velocity distribution is obtained by solving the Reynolds equation using the simplified mixing-length theorem. The model is compared with the measured velocity profiles available in literature and a good agreement is obtained. The investigation shows that wall-normal velocity greatly affects the velocity profile and leads to the deviation of the measured velocity from the classical log-law. The conclusions include that: 1. The bed shear velocity in an open channel can be approximately estimated using the method of flow region division as suggested by Keulegan (1938). 2. The non-zero wall-normal velocity v driven by the secondary currents caused the deviation of Reynolds shear stress from the conventional linear-distribution in 2-D flows. This deviation subsequently causes the location of the maximum velocity below the free surface, or the maximum velocity appears on the free surface and the classical log-law is valid only when the secondary currents disappear (v = 0). 3. It is widely reported that the measured velocity deviates from the classical log-law in many cases, e.g. boundary layer flows, non-uniform flows and density stratification flows. This velocity deviation can probably attribute to the non-zero wall-normal velocity. References Cardoso, A.H., Graf, W.H., Gust, G., 1989. Uniform flow in a smooth open channel. J. Hydraul. Res. 27 (5), 603–616. Chien, N., Wan, Z., 1999. Mechanics of Sediment Transport. ASCE, Reston, VA. Coleman, N.L., 1986. Effects of suspended sediment on open-channel distribution. Water Resour. Res. AGU 22 (10), 1377–1384. Einstein, H.A., 1942. Formulas for the transportation of bed load. Trans. Am. Soc. Civ. Eng. 107, 133–169. Einstein, H.A., Chien, N., 1955. Effects of heavy sediment concentration near the bed on velocity and sediment distribution. US Missouri River Division Report No. 8, Army Corps of Engineers. Ghosh, S.N., Roy, N., 1970. Boundary shear distribution in open channel flow. J. Hydraul. Div. ASCE 96 (4), 967–994. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries. J. Fluid Mech. 50 (2), 233–255. Guo, J., Julien, P.Y., 2005. Shear stress in smooth rectangular open-channel flows. J. Hydraul. Eng. ASCE 131 (1), 30–37. Imamoto, H., Ishigaki, T., 1988. Measurement of secondary flow in an open channel. Proceedings of 6th IAHR-APD Congress, Kyoto, Japan, pp. 513–520. Keulegan, C.H., 1938. Laws of turbulent flow in open channels. J. Res. Natl. Bur. Stand. 21, 707–740.
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