Boundary shear stress is a significant parameter employed inriver engineering problems such as sediment transport, pollutantdispersion, river bed and banks protection, and flood controlprojects. The boundary shear stress distribution along the bed andsidewalls of a smooth trapezoidal channel depend on the channelwidth/depth ratio and the structure of secondary flow cells. Theability to differentiate between the boundary shear stress relatedto the bed and sidewall subsections is necessary in almost all stud-ies of open channel flows. Einstein (1942) divided the channel flowcross section into two subsections that correspond to the channelbed and sidewalls, as illustrated in Fig. 1 (channel bed subsection,Ab, and sidewall sub-section, Aw). This division is made by animaginary separation boundary, namely the division curve.Although Einstein did not propose any method for determiningthe exact location of the division curves, his suggestion laid thefoundation for the development of methods used to isolate the flowcross section into bed and sidewall subsections.
Lundgren and Jonsson (1964) proposed a practical method todetermine the division curves for a general cross section, neglectingthe effects of roughness and secondary currents. Their methodol-ogy has been widely used in different studies, including this study.Tominaga et al. (1989) investigated the effects of free surface, chan-nel shape, and the boundary roughness on secondary currents and
three-dimensional (3D) turbulent structures both experimentallyand numerically. They concluded that the secondary currents aregenerated and modified as a result of the anisotropy of turbulence,which is caused by the boundary conditions as well as the channelshape. Yang and Lim (1997, 1998) proposed the concept that thesurplus energy of a unit volume of fluid in a three-dimensionalchannel flow is transported toward and dissipated at the nearestwall boundary. Khodashenas and Paquier (1999) presented an over-view of available methods to compute shear stress distributions.They stated that precise computation is extremely difficult, evenfor simple cases. As an alternative, they proposed a number ofempirical or simplified computational methods. They also usedLundgren and Jonsson’s idea for a general cross section withoutsecondary current effects. Yang and Lim (2005) analyzed the boun-dary shear stress distributions in trapezoidal open channels. Basedon their approach, the flow cross-sectional area in trapezoidal openchannels is divided into various elements according to the shape ofcross section. They presented analytical equations for the local andmean boundary shear stress along the wetted perimeter; howeverthey did not consider the effects of secondary currents. Yang (2005)offered a method for the evaluation of the mean bed and sidewallshear stresses in trapezoidal channels, considering the effect of sec-ondary currents. He obtained the equations of the boundary shearstress and Reynolds shear stress distributions. Zarrati et al. (2008)proposed a semianalytical model for shear stress distribution insimple and compound rectangular and trapezoidal open channels.They applied a simplified streamwise vorticity equation that in-cludes secondary Reynolds stresses and assessed different terms ofthis equation based on experimental data. Ansari et al. (2011) usedcomputational fluid dynamics to determine the distribution of themean bed and sidewall shear stress in trapezoidal channels. Theirresults show a significant contribution of the secondary currentsand internal shear stresses on the overall shear stress at the boun-daries. The idea to use conformal mapping was first suggestedby Leighly (1932). He explained that by neglecting the effects ofsecondary currents, the boundary shear stress acting on the channelbed must be balanced by the component of the weight containedbetween each of the two adjacent orthogonals, but his proposal
1Associate Professor, Dept. of Civil Engineering, Isfahan Univ. ofTechnology, P.O. Box 84156, Isfahan, Iran (corresponding author). E-mail:[email protected]
2M.Sc. Student, Dept. of Civil Engineering, Isfahan Univ. of Technol-ogy, Isfahan, Iran. E-mail: [email protected]
3Associate Professor, Dept. of Civil Engineering, Isfahan Univ. ofTechnology, P.O. Box 84156, Isfahan, Iran. E-mail: [email protected]
did not render any conclusive results (Graf 1971, p. 107). Guo andJulien (2002, 2005) applied Einstein’s consideration and employedconformal mapping to determine the mean shear stress on theboundaries of smooth rectangular open channels. As a first approxi-mation, they obtained the boundary shear stress by neglecting thesecondary currents and assuming a constant eddy viscosity. Thenthey introduced two correction factors for secondary currents andeddy viscosity. Their second approximation agrees well with theexperimental measurements. Their methodology for rectangularchannels is the basis of the approach for trapezoidal open channelsin this paper.
Ample studies have been performed to calculate the open chan-nel boundary shear stresses. However, limited information exists inliterature to describe the actual fraction from the flow field requiredto calculate the sidewall and bed shear stress distributions along thechannel cross section. Based on known continuity and momentumequations for a steady uniform flow, the objective of this study isto formulate a theoretical basis for the boundary shear stress insmooth trapezoidal open channels considering Einstein’s idea andthe effects of secondary currents. Hence, this study mainly focuseson (1) determining a functional relationship for the division curvethat is based on the conformal mapping procedure; (2) generalizinga method to divide the main flow cross section into its various sub-sections considering the effects of secondary currents; (3) assessingthe mean bed and sidewall shear stresses for general trapezoidalchannels, and bed and sidewall shear stress distributions for thebest hydraulic trapezoidal channels; and (4) quantifying the besttrapezoidal hydraulic section aspects based on assessments ofthe analytical approach and the method proposed by Lundgrenand Jonsson (1964). For rectangular channels, this study utilizedthe basis of the methodology presented by Guo and Julien(2002, 2005).
Methodology
Theoretical Consideration
For a steady, uniform, and fully developed turbulent flow in a trap-ezoidal channel, the continuity and momentum equations are com-bined to attain the following equation (Guo and Julien 2005):
τ̄b ¼ρgSoAb
b− 2
b
ZCF
ρuðwdy − vdzÞ þ 2
b
ZCF
ðτ zxdy − τ yxdzÞ ð1Þ
where τ̄b = mean bed shear stress; b = bed width; Ab = area of thebed region; x = streamwise direction; y = distance from the bed; z =distance from the channel centerline; u = longitudinal (streamwise)flow velocity in the x-direction; v and w = velocities in the y and
z-directions, respectively; ρ = water density; g = gravitational ac-celeration; So = channel longitudinal slope; CF = integration path(Fig. 1); and τ yx and τ zx = shear stresses along the x-axis in they-x and z-x planes, respectively (where τ zx ¼ ρðvþ vtÞ∂u=∂z, τ yx ¼ ρðvþ vtÞ∂u=∂y, v is kinematic viscosity, and vt is eddyviscosity). To determine the sidewall shear stress, the overall forcebalance in the flow direction is considered as follows:
2Hffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2
pτ̄w þ bτ̄b ¼ ρgSoHðbþmHÞ ð2Þ
where H = flow depth; and m = channel sidewall lateral slope.Eqs. (1) and (2) are combined to obtain the mean sidewall shearstress, which is given as
τ̄w ¼ ρgSo2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2
p�bþmH − Ab
H
�þ 1
Hffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2
pZCF
ρuðwdy− vdzÞ
− 1
Hffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2
pZCF
ðτ zxdy− τ yxdzÞ ð3Þ
Eqs. (1) and (3) were used through a comprehensive analyticalattempt identified as follows. The existence of secondary currentsand Reynolds shear stress terms complicates the previous equa-tions. As an initial approximation, the effects of secondary currentsand fluid shear stresses are neglected. This approximation is validwhen v and w are almost zero. In addition, the eddy viscosity vt isassumed to be constant. Consequently, the last two terms on theright-hand side of Eqs. (1) and (3) vanish and the only unknownparameter is Ab.
Application of Conformal Mapping
The Poisson equation is generated using the momentum equation,which can be solved by a conformal mapping method as follows(Guo and Julien 2005):
∂2u∂y2 þ
∂2u∂z2 ¼ − gSo
vþ vt¼ C ð4Þ
where C is a constant. With regards to the subject of conformalmapping, a physical plane (ω ¼ zþ iy) is transformed to a map-ping plane (ζ ¼ ξ þ iη). Coordinates ξ and η in the mapping planerepresent orthogonals and isovels, respectively. In any channelcross section, the shear stress on the interface normal to the isovelsis zero, i.e., τξx ¼ 0 (Chiu and Chiou 1986). Hence, orthogonalsare assumed to be surfaces of zero shear stress. Consequently,the component of the weight of water confined between eachof two orthogonals and a boundary must be balanced by the boun-dary shear stress. Division curves are special orthogonals thatstart from the section corners. Based on the Schwarz-Christoffel
Fig. 1. Trapezoidal cross section properties, vector directions, and control volume
transformation (Valentine 1961, p. 183), θ1 and θ4 are on thevertices G and H in infinity (Fig. 2). Hence, these two anglescan be set to zero. For a channel with angles θ2 and θ3 that areequal to π=3, ξ2 and ξ3 are assigned as �b=ð2θ2;3Þ. Hence, theSchwarz-Christoffel transformation in the differential form can bewritten as follows:
Zdωdζ
¼ C1
Z �ζ þ 3b
2π
�−1=3�ζ − 3b
2π
�−1=3dζ þ C2 ð5Þ
where C1 and C2 are constants. By integrating Eq. (5), an equation isobtained that consists of a complicated form, namely, Hyper-geometric2F1, which is a particular case of the general hypergeomet-ric function HypergeometricpFq (a1; : : : ; ap; b1; : : : ; bp; x), wherean and bn are coefficients and x is a variable (Wolfram MathWorld2011). Because of the complication, the previously mentionedequation cannot be solved analytically and numerical integrationwas therefore employed. Then the alternative complexes ω andζ are arranged according to their coordinates (z, y) and (ξ, η),respectively, in four matrices to obtain the relationship between ξ(or η) with the z- and y-coordinates. For all trapezoidal sections,C1 andC2 are b=2 and complex variable αþ iβ (where β=α is equalto 1=m), respectively. Fig. 3 shows the transformed shape of atrapezoidal section, as in Fig. 2 on the mapping plane.
Points G to H on the physical plane correspond to G 0 to H 0 onthe mapping plane, respectively. The results of the previous numeri-cal integration are sketched in Fig. 4 for a typical trapezoidalsection of θ ¼ π=3. Fig. 4 illustrates the contours of constant ξ(orthogonals) of the physical plane. The dashed thick curve(ξ=b ¼ 0.48) in Fig. 4 defines the division curve separating the bedand sidewall regions.
Eq. (4) is not used to estimate the turbulent flow longitu-dinal velocity distribution because the secondary currents and theReynolds shear stresses are neglected. Indeed, these two simplify-ing assumptions result in a laminar solution that does not agree withvelocity measurements in turbulent flows (Guo and Julien 2002).Isovel pattern (constant η, which is perpendicular to constant ξ) is
mutually used to obtain a first estimate for the orthogonal patternthat provides a first approximation of the boundary shear stress.Then, as a second approximation, this methodology is improvedto include the secondary currents and Reynolds shear stresses ofthe turbulent flow (the next section). If the division curve is known,Ab can be obtained using the following equation:
Ab ¼ 2
ZH0
zdy ð6Þ
Consequently, one needs y-z coordinates of each point along thedivision curve of ξ3 ¼ þb=ð2θ3Þ. By employing the previouslymentioned matrices, z-y coordinates of this orthogonal and the re-lationship between its z- and y-coordinates are obtained. The au-thors’ analysis showed that for θ ¼ π=3, the following equation,with R2 ≅ 1, can be introduced for z ¼ fðyÞ along the divisioncurve:
ξ ¼ 3b2π
; z ¼ b2exp
�−1.261 y
b
�ð7Þ
Hence, Ab is written as follows:
Ab ¼ 2
ZH0
zdy ¼ZH0
b exp
�−1.261 y
b
�dy
¼ 0.793b2½1 − expð−1.261H=bÞ� ð8Þ
Improved Mean Bed and Sidewall Shear Stresses
With respect to the dip phenomenon whereby the location of themaximum streamwise velocity appears below the free surface,the actual Ab occupies an area that is less than what is calculatedusing Eq. (8). This is partly due to the action of the secondarycurrents and consequences of Reynolds shear stresses in an open-channel flow (Yang et al. 2004). Tominaga et al. (1989) noted thatfor a typical trapezoidal open channel, the maximum velocity atthe channel centerline never occurs below the water free surface.They stated that this is due to the contrary rotation of the twodominant vortices in the flow cross-sectional area. One of thevortices belongs to the sidewall and the other rotates near the watersurface, so that the division curve CJF deforms to curve CJI (Fig. 5).According to Fig. 5, due to the effects of secondary currents, thedivision curve is changed in a direction from the unknown point J,and point F on the free surface is shifted to point I.
Fig. 2. Schwarz-Christoffel transformation for a trapezoidal section
0
1
2
-3 -1 1 3/b
/b
G' A' B' C' D' H'
E' F'
Fig. 3. Transformed shape of a typical trapezoidal section on themapping plane
0
0.5
1
0 0.2 0.4 0.6 0.8
z /b
y/b
Channel Sidewall0.0
0.13
0.034
/b= 0.96
0.670.48,
Boundary
0.27
CL
Fig. 4. Typical orthogonals in half right side of a trapezoidal openchannel with θ ¼ π=3
Using Eqs. (6) and (7), employing integration by parts, andapplying the mean value theorem for integrals, Eq. (1) yieldsthe following expression:
τ̄bρgSoH
¼ exp
�−1.261H
b
�− λH
�−1.261b
�exp
�−1.261λH
b
�
− 2
ρgSoHb
ZCF
ρuðwdy − vdzÞ
þ 2
ρgSoHb
ZCF
ðτ zxdy − τ yxdzÞ ð9Þ
where λH is a point such that 0 < λ < 1 and also satisfies themean value theorem conditions. Numerical evaluation shows thatthe first term on the right-hand side is the primary term and theothers are only small fractions of the first one [analogous toGuo and Julien (2002)]. Hence, Eq. (9) can be amended to thefollowing form:
τ̄bρgSoH
¼ exp
�−1.261H
b
�þ λ1
�Hb
�exp
�−1.261 λ2H
b
�ð10Þ
where λ1 and λ2 are lumped correction coefficients referred to asthe effects of secondary currents and Reynolds shear stresses.Likewise, Eq. (3) is converted to Eq. (11), as follows:
τ̄wρgSoH
¼ bþmH
2Hffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2
p − 0.5b
Hffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2
p exp
�−1.261H
b
�
− λ1
�1
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2
p�exp
�−1.261 λ2H
b
�ð11Þ
Using boundary conditions of a trapezoidal section with θ ¼π=3, for a given H=b, m, τ̄b, and τ̄w, both λ1 and λ2 are calculatedby solving Eqs. (10) and (11). For the other trapezoidal sections(e.g., θ ¼ π=4, 5π=12, and π=2), modified forms of Eqs. (10)
and (11) are also solved. For this purpose, τ̄b and τ̄w were estimatedaccording to Yang (2005), who offered a method to evaluate theseparameters considering the effects of secondary currents andReynolds shear stresses. For m ¼ ffiffiffi
3p
=3 (θ ¼ π=3), it was con-cluded that λ1 and λ2 are equivalent to 0.43 and 0.722 ðH=bÞ−0.311,respectively. Thus, for θ ¼ π=3, Eqs. (10) and (11) yield
τ̄bρgSoH
¼ exp
�−1.261H
b
�þ 0.43
�Hb
�exp
�−0.91
�Hb
�0.689
�ð12Þ
τ̄wρgSoH
¼ffiffiffi3
p bþ ð ffiffiffi3
p=3ÞH
4H−
ffiffiffi3
pb
4Hexp
�−1.261H
b
�
− 0.186 exp
�−0.91
�Hb
�0.689
�ð13Þ
The proposed analyses were applied for different trapezoidalchannels with θ ¼ π=4, 5π=12, and π=2. Accordingly, the resultsof τ̄b and τ̄w were plotted against H=b in Figs. 6(a and b). Theexperimental data of Knight et al. (1984) for rectangular channelsand Yuen (1989) for trapezoidal channels (θ ¼ π=4) were alsoincluded in Fig. 6. Good agreement between the results of thepresent semianalytical model and those of former investigatorsis clearly seen.
For a specific H=b, Fig. 6 shows that the mean bed shear stressincreases with reducing θ. Tominaga et al. (1989) concluded that ina rectangular cross section, there are two main vortices correspond-ing to the free surface and channel bottom. While the channel crosssection changes from rectangular to trapezoidal, a new longitudinalvortex is generated, the free surface vortex becomes weaker, andthe bottom vortex develops into the depth-scale vortex. Therefore,the area of bed subsection increases, which results in a greater meanbed shear stress.
Another remarkable point is that due to the general form ofSchwarz-Christoffel transformation, for a rectangular cross section(θ ¼ π=2) the power −1=3 in Eq. (5) (for θ ¼ π=3) changes to−1=2. Therefore, Eq. (5) tends to an equation that introduces ζas a function of sinðπω=bÞ (Guo and Julien 2002). Accordingly,the imaginary and real parts are simply separated. However, fortrapezoidal cross sections, the complicated hypergeometric func-tion cannot be solved analytically. To verify the present semiana-lytical model, the results for θ ¼ π=2 were obtained numerically(not shown in this paper). The present semianalytical resultscompared very well with the analytical results of Guo and Julien(2005).
As stated previously, Ansari et al. (2011) proposed a numer-ical model to estimate mean bed and sidewall shear stresses of
Fig. 5. Actual division curve (CJI) compared with the ideal one (CJF)
Fig. 6. Variation of normalized mean: (a) bed and (b) sidewalls shear stresses versus H=b
trapezoidal channels. Table 1 compares the results of this study andthose of Ansari et al. (2011). It can be concluded that for highervalues of b=H, the results of both investigations are very close.However, for lower values of b=H, a maximum discrepancy of15.8% for θ ¼ π=3 and b=H ¼ 4 is evident. For lower values ofb=H, the discrepancies are due to the effects of secondary currents.
Bed Shear Stress Distribution for the Best HydraulicTrapezoidal Sections
By substituting H=b ¼ 1=f2½ð1þm2Þ0.5 −m�g for the besthydraulic trapezoidal sections in Eqs. (12) and (13) with m ¼ð1=3Þ0.5, an identical mean bed and sidewall shear stresses areobtained as follows:
τ̄wρgSoH
¼ τ̄wρgSoH
¼ 0.50 ð14Þ
These analyses were applied for other trapezoidal channels withthe best hydraulic sections. The mean bed shear stress is determinedusing the following integral:
τ̄b ¼0@ Zb=2
−b=2τbdz
1A,
b ð15Þ
Knowing the concrete values of τ̄b=ðρgS0HÞ for the besthydraulic section of different lateral slopes, the actual variation ofthe division curve (CJI) can be determined. Because the divisioncurve ends at the location of maximum velocity, a very smallsegment of the division curve is followed by the free surface.As explained previously, the definite positions of points J and I andthe functional form of curve JI are still unknown. These additionalunknowns were then deduced based on a long trial-and-error pro-cedure using the Wolfram Mathematica 8.0.1 (2011) software tool.The previous integral is broken down as follows:
τ̄b ¼2
b
ZzðIÞ0
ρgSoð0.866bÞdzþ2
b
ZzðJÞzðIÞ
ρgSoy2ðzÞdz
þ 2
b
Zb2zðJÞ
ρgSoy1ðzÞdz ð16Þ
where zðIÞ and zðJÞ ¼ z-coordinates of points I and J; and y1 andy2 = functions in the form of y ¼ fðzÞ that establish curves CJ andJI, respectively. The value of y1 is estimated using Eq. (7). To findthe unknown parameters and the function y2, a computer programwas written in the Mathematica environment. The values of zðIÞand zðJÞ were varied and two boundary conditions for themat point zðJÞ, y1 ¼ y2 and dy1=dz ¼ dy2=dz, were also used as
the constraints of the trial and error procedure. Primarily, y2 isproposed with a polynomial function. The results demonstrated thata third-order polynomial function satisfies the constraints andboundary conditions of the problem. In this situation, it is foundthat zðIÞ decreases and zðJÞ increases until the right- and left-handsides of Eq. (16) become equal. Based on the previous procedure,y2 was determined for different trapezoidal sections. For instance,in a smooth trapezoidal channel with the best hydraulic section andθ ¼ π=3, y2 is
y2ðzÞ ¼ − 90.9b2
z3 þ 42.3b
z2 − 6.22zþ 0.873b ð17Þ
Consequently, zðIÞ=b and zðJÞ=b are calculated as 0.001 and0.263, respectively, for a channel with the best hydraulic sectionand θ ¼ π=3. To calculate the boundary shear stress at any pointon the boundary, ρgSo is multiplied by the height of the elementthat is perpendicular to the boundary up to the division curve.Hence, for this special section, to determine the bed shear stressthe following equations, which were fitted from semianalyticalresults, are introduced:
τb¼0.866ρgSob 0≤ z≤0.001bτb¼ρgSo
�−90.9b2 z
3þ 42.3b z2−6.22zþ0.873b
�0.001b< z≤0.263b
τb¼−0.793ρgSob lnð2zb Þ 0.263b< z≤ b2
ð18Þ
All these procedures have been applied for the other consideredangles, e.g., π=4, 5π=12, and π=2. The only difference betweenrectangular (θ ¼ π=2) and trapezoidal open channels is that thedip phenomenon merely occurs in rectangular channels. When thedip phenomenon arises, point I is created on the channel centerlinewhere the maximum streamwise velocity occurs. To calculate theexact location of this point, the method proposed by Yang et al.(2004) was applied. They offered the following equation for deter-mining the location of the maximum streamwise velocity below thewater free surface:
δH
¼ 1
1þ αð19Þ
where δ ¼ y-coordinate of the location of the maximum stream-wise velocity; and α = factor equal to 1.3 expð−z=HÞ (Yang et al.2004). For a rectangular channel with the best hydraulic section andz ¼ b=2, δ is equivalent to 0.676H or 0.338b. Another importantresult is that as the angle θ decreases, the near-wall vortex becomesstronger and shifts point J closer to the corner of the section. Thisphenomenon differs somewhat from rectangular cross sections.This is partly due to the fact that the location of the maximumvelocity on the centerline is under the free surface. Hence, unlikethe trapezoidal channels, in rectangular channels the locations ofthe maximum velocity and maximum shear stress are not identical.This result has been proven based on experimental data fromTominaga et al. (1989).
Sidewall Shear Stress Distribution for the BestHydraulic Trapezoidal Sections
To calculate sidewall shear stress, one needs to transform the y-axiswith the distance b=2 in the þz-direction, and the coordinate sys-tem must then be rotated by the angle θ (θ is the angle of the chan-nel sidewall). Based on the previously mentioned transformationand rotation, and according to Einstein’s suggestion, the sidewallshear stress can be obtained. For this purpose, the elements that are
Table 1. Comparison between the Numerical Results of Ansari et al.(2011) and Those of the Present Study
perpendicular to the walls and confined to the division curve were used. Consequently, sidewall shear stress equations fitted fromsemianalytical results for θ ¼ π=3 are obtained as
τw ¼ ρgSo�0.703b z 021 þ 1.173z 01
0 ≤ z 01 ≤ 0.325b
τw ¼ ρgSo�165.5b2 z 031 − 196.67
b z 021 þ 78.977z 01 − 10.118b
0.325b < z 01 ≤ 0.5b
τw ¼ ρgSo½ffiffiffi3
p ðb − z 01Þ� 0.5b < z 01 ≤ b
ð20Þ
where z 01 = lateral coordinate of the sidewall.
Results and Discussions
To verify the semianalytical results, two experimental models ofopen channels with θ ¼ π=3 and π=4 were constructed. The chan-nel bed and sidewalls were made of glass. These models satisfiedthe conditions of the best hydraulic section, with bed widths (b) of95 and 70 mm. Models were installed in an 8 m long, 0.3 m wide,and 0.3 m deep, rectangular glass-walled flume. Water entered theflume through a head tank containing perforated internal walls.Over the flume sidewalls, a rolling point gauge of ±0.1 mm read-ing accuracy was mounted to measure the flow depth. Dischargemeasurements were performed using a standard sharp-crestedtriangular weir with precision of �0.1 L=s. The flow dischargewas set as 6.0 L=s for both models. The channel slopes were ad-justed as 0.001 and 0.0009 for θ ¼ π=3 and π=4, respectively. Thewater normal depths were 82 and 84 mm for θ ¼ π=3 and π=4,respectively. Tests were conducted for subcritical approach flowconditions. The velocity profiles were measured at differentsections from x ¼ 4.3 to 5.3 m, where x was measured from thechannel entrance. Velocity measurements were performed usinga static pitot tube. By examining the velocity profiles, it is foundthat the vertical distribution of the velocities remains almostunchanged, and consequently a fully developed, uniform flowwas established at the test section of x ¼ 4.8 m. The bed andsidewall shear velocities in smooth channels were determined byfitting the log-law to velocity profiles as follows:
uu�
¼ 5.75 log
�u�ywν
�þ 5.3 ð21Þ
where u is the longitudinal velocity; u� ¼ ðτo=ρÞ0.5 (τo ¼ τbor τw) is the shear velocity; and yw is the height perpen-dicular to the boundary (e.g., bed or sidewall) in the inner region
(Nezu and Nakagawa 1993, p. 19). This law is traditionally em-ployed for both rectangular and trapezoidal channels (Yang 2005;Nezu and Nakagawa 1993; Tominaga et al. 1989; Ghosh and Roy1970). Ghosh and Roy (1970) used the constant 5.3 for smoothchannels with H=b > 0.5 that covers our model experiments.The present study is limited to smooth channels in which u�ks=ν ≈0.001 < 5 [where ks is the Nikuradse equivalent sand roughness(Nezu and Nakagawa 1993, p. 26)].
For the sidewall shear stresses, coordinates have been trans-formed. The new coordinate system was tangential and perpendic-ular to the sidewall. Due to this transformation and limitation ofthe pitot tube diameter, the experimental data for a small regionclose to the corner of the sidewall was not included. As mentionedpreviously, longitudinal and free surface vortices meet beside thechannel corner. Because pitot tube measurement is limited to one-dimensional flow, these vortices interfere with the application of thepitot tube at the corner. Therefore, accurate experimental measure-ments are not possible in this region. Hence, the results of Ghoshand Roy (1970) for θ ¼ π=4 and π=3 were employed to verify thepresent semianalytical results beside the section corners.
On the basis of the results of present study, it is concludedthat the effects of secondary currents and Reynolds shear stressesin turbulent flows diverge the isovels by the conformal mappingprocedure. Figs. 7 and 8 compare the present analytical and exper-imental results with previous investigations. The results are illus-trated for the right-hand side of the sections. In Figs. 7 and 8, τ̄ isthe mean shear stress equal to ρgSoRh (where Rh is the hydraulicradius of the cross section) and lw is the length of the channelsidewall. According to Figs. 7(a) and 8(a), it is concluded that theanalytical results of bed shear stress distribution are in close agree-ment with the present experimental results, except at the sectioncorners. The reason for these differences was discussed previously.
Also, the results of the study in this paper are very close to thoseof Yang and Lim (2005). This is partly due to the fact that bothmodels are based on Einstein’s suggestion. Again, close agreement
Fig. 7. Comparison of the present analytical results, present experimental data, and those of previous researches for the best hydraulic section withθ ¼ π=3: (a) bed shear stress and (b) sidewall shear stress
is observed among the results of the present analytical approach,present experimental data, and those of Yang and Lim (2005)for the sidewall shear stress [Figs. 7(b) and 8(b)]. However, fora channel with θ ¼ π=4, the free surface becomes wider and moreaccurate results are achieved.
Hence, it is concluded that in sections with wider free surfaces,Einstein’s idea leads to more appropriate results. It is deduced fromthese figures that for approximately 30% of the last parts of the bedshear stress curves, this research’s analytical results are identicalto those of Yang and Lim (2005), but for approximately 70% ofthe first sets of the results, there are small discrepancies. This isbelieved to be due to the influences of vortices that have been con-sidered in this study that were not considered by Yang and Lim(2005). Considering the results of the proposed model and those ofYang and Lim (2005), the error functions’ normalized root-mean-square error (NRMSE) and weighted quadratic deviation(WQD) were determined. Results indicated that for θ ¼ π=3, thepair of (NRMSE, WQD) for bed and sidewall shear stresses are(0.314, 0.078) and (0.417, 0.091), respectively. The results ofZarrati et al. (2008) show significant errors, i.e., negative resultsin the bed subsection. This would be partly due to the fact that theyhave not considered Einstein’s suggestion. To calculate sidewallshear stresses, coordinates have been transformed and sidewallshear stresses on a small portion of the sidewall could not beobtained [the first 20% of sidewall length in Figs. 7(b) and8(b)]. As mentioned previously, an important result of the presentanalytical model is that τ̄b ¼ τ̄w for the channels with the besthydraulic section. This result has been proven according to theexperimental data for the mean shear stress. Based on the boundaryshear stress distributions, it is inferred that maximum bedand sidewall shear stresses are also identical, and that τmax ¼2τ̄b. According to the present analytical results and experimentaldata, (NRMSE, WQD, R2) for θ ¼ π=3 and π=4 are (0.132,0.014, 0.987) and (0.052, 0.009, 0.997), respectively. As statedpreviously, boundary shear stress has been evaluated indirectly byfitting the log-law distribution to experimental data. Hence, whenthe experimental errors are factored, errors result from obtainingthe boundary shear stress in this way. In practice, the average valuesof boundary shear stresses are determined using Fig. 6, and the shearstresses distributions are obtained using Eqs. (18) and (20) forθ ¼ π=3. For other angles, a similar procedure should be applied.
Conclusions
In this study, a conformal mapping procedure is used to obtain aninitial approximation of the mean bed and sidewall shear stresses in
trapezoidal open channels. Then the effects of secondary currentsand Reynolds shear stresses were incorporated into the derivedequations based on two lumped correction coefficients. Next,the boundary shear stress distribution and division curves oftrapezoidal channels with the best hydraulic section were obtainedusing mean bed and sidewall shear stresses. The determineddivision curves are significantly deformed due to the generationof secondary currents. For trapezoidal channels with the besthydraulic section, the mean bed and sidewall shear stresses areidentical. Also, for these kinds of channels, maximum bed andsidewall shear stresses are approximately equivalent to two timesthe mean bed and sidewall shear stresses. The analytical resultsagreed well with the experimental measurements and the resultsof former investigations.
Notation
The following symbols are used in this paper:Ab, Aw = areas corresponding to the bed and sidewall shear
stresses, respectively (m2);B = bed width of channel (m);
C, C1, C2 = constants (−);g = gravitational acceleration (m=s2);H = flow depth (m);ks = Nikuradse equivalent sand roughness (m);L = channel length (m);l = integration length (m);lt = transitional length (m);lw = length of the channel sidewall (m);m = channel side slope (−);n = normal unit vector (−);
u, v, w = velocity components (m=s);umax = maximum longitudinal flow velocity (m=s);u� = shear velocity (m=s);
x, y, z = coordinates;yw = perpendicular height from the boundary (m);
y1, y2 = functions (m);zðIÞ, zðJÞ = z-coordinates of points I and J (m);
z 01 = coordinate of lateral direction on wall;α = factor (−);δ = y-coordinate of the location of maximum streamwise
velocity (m);θ = angle of sidewall with horizontal direction (rad);
λ, λ1, λ2 = correction factors (−);
Fig. 8. Comparison among the present analytical results, present experimental data, and those of previous researches for the best hydraulic sectionwith θ ¼ π=4: (a) bed shear stress and (b) sidewall shear stress
v, vt = kinematic viscosity of water and eddy viscosity,respectively (m2=s);
ρ = density of water (kg=m3);τb, τw = bed and sidewall shear stress, respectively (N=m2);
τo = boundary shear stress (N=m2);τ yx, τ zx = shear stresses in flow direction applied on z-x and
y-x planes, respectively (N=m2);τ̄ = mean shear stress (N=m2); and
τ̄b, τ̄w = mean bed and sidewall shear stresses, respectively(N=m2).
References
Ansari, K., Morvan, H. P., and Hargreaves, D. M. (2011). “Numericalinvestigation into secondary currents and wall shear in trapezoidalchannels.” J. Hydraul. Eng., 137(4), 432–440.
Chiu, C. L., and Chiou, J. D. (1986). “Structure of 3-D flow in rectangularopen-channels.” J. Hydraul. Eng., 112(11), 1050–1068.
Einstein, H. A. (1942). “Formulas for the transportation of bed-load.”Trans. ASCE, 107(2140), 561–597.
Ghosh, S. N., and Roy, N. (1970). “Boundary shear distribution in openchannel flow.” J. Hydraul. Div., 96(4), 967–994.
Graf, W. (1971). Hydraulics of sediment transport, McGraw-Hill, New York.Guo, J., and Julien, P. Y. (2002). “Boundary shear stress in smooth rectan-
gular open-channels.” Advances in Hydraulic Water Engineering, Proc.13th IAHR-APD Congress, World Scientific Publishing Co. Pte. Ltd.,Singapore, 1, 76–86.
Guo, J., and Julien, P. Y. (2005). “Shear stress in smooth rectangularopen-channel flows.” J. Hydraul. Eng., 131(1), 30–37.
Khodashenas, S. R., and Paquier, A. (1999). “A geometrical method forcomputing the distribution of boundary shear stress across irregularstraight open channels.” J. Hydraul. Res., 37(3), 381–388.
Knight, D. W., Demetriou, J. D., and Hamed, M. E. (1984). “Boundary shearin smooth rectangular channels.” J. Hydraul. Eng., 110(4), 405–422.
Leighly, J. B. (1932). “Toward a theory of the morphologic significance ofturbulence in the flow of water in streams.” Univ. California Publ.Geography, 6(1), 1–22.
Lundgren, H., and Jonsson, I. G. (1964). “Shear and velocity distribution inshallow channels.” J. Hydraul. Div., 90(1), 1–21.
Nezu, I., and Nakagawa, H. (1993). Turbulence in open-channel flows,IAHR Monograph Series, A. A. Balkema, Rotterdam, The Netherlands.
Tominaga, A., Nezu, I., Ezaki, K., and Nakagawa, H. (1989). “Three-dimensional turbulent structure in straight open channel flows.”J. Hydraul. Res., 27(1), 149–173.
Valentine, H. R. (1961). Applied hydrodynamics, Butter Worth’s ScientificPublishing, London.
2011).Yang, S. Q. (2005). “Interaction of boundary shear stress, secondary
currents and velocity.” J. Fluid Dyn. Res., 36(3), 121–136.Yang, S. Q., and Lim, S. Y. (1997). “Mechanism of energy transpor-
tation and turbulent flow in a 3D channel.” J. Hydraul. Eng., 123(8),684–692.
Yang, S. Q., and Lim, S. Y. (1998). “Boundary shear stress distributions insmooth rectangular channels.” Proc. ICE Water Marit. Eng., 130(3),163–173.
Yang, S. Q., and Lim, S. Y. (2005). “Boundary shear stress distributions intrapezoidal channels.” J. Hydraul. Res., 43(1), 98–102.
Yang, S. Q., Tan, S. K., and Lim, S. Y. (2004). “Velocity distribution anddip-phenomenon in smooth uniform open channel flows.” J. Hydraul.Eng., 130(12), 1179–1186.
Yuen, K. W. H. (1989). “A study of boundary shear stress, flow resistanceand momentum transfer in open channels with simple and compoundtrapezoidal cross section.” Ph.D. thesis, The Univ. of Birmingham,Birmingham, UK.
Zarrati, A. R., Jin, Y. C., and Karimpour, S. (2008). “Semianalytical modelfor shear stress distribution in simple and compound open channels.”J. Hydraul. Eng., 134(2), 205–215.
