Stoch Environ Res Risk Assess (2015) 29:1–11
DOI 10.1007/s00477-014-0959-3
ORIGINAL PAPER
Comparison between Shannon and Tsallis entropies for prediction
of shear stress distribution in open channels
Hossein Bonakdari • Zohreh Sheikh
Mohtaram Tooshmalani
•
Published online: 21 September 2014
Ó Springer-Verlag Berlin Heidelberg 2014
Abstract The concept of Tsallis entropy was applied to
model the probability distribution functions for the shear
stress magnitudes in circular channels (with filling ratios of
0.506, 0.666, 0.826), circular with flat bed (filling ratios of
0.333, 0.666), rectangular channel (1.34, 2, 3.94, 7.37
aspect ratios) and compound channel (with relative depths
of 0.324, 0.46). The equation for the shear stress distribution was obtained according to the entropy maximization
principle, and is able to estimate the shear stress distribution as much on the walls as the channel bed. The approach
is also compared with the predictions obtained based on the
Shannon entropy concept. By comparing the two prediction
models, this study highlights the application of Tsallis
entropy to estimate the shear stress distribution of open
channels. Although the results of the two models are similar in the circular cross-section, the differences between
them are more significant in circular with flat bed and
rectangular channels. For a wide range of filling ratio
values, experimental data are used to illustrate the accuracy
and reliability of the proposed model.
Keywords
Bed
Entropy Open channel Sediment Wall H. Bonakdari (&) Z. Sheikh M. Tooshmalani
Department of Civil Engineering, Faculty of Engineering, Razi
University, Kermanshah, Iran
e-mail: bonakdari@yahoo.com
H. Bonakdari Z. Sheikh M. Tooshmalani
Water and Wastewater Research Center, Razi University,
Kermanshah, Iran
1 Introduction
Depending on the cross section shape (Knight et al. 1994),
bed roughness (Flintham and Carling 1988) and hydraulics
of flow (Ghosh and Roy 1970), the boundary shear stress in
any channel cross section is one of the main factors to be
considered in different open channel studies such as that
caused by flow resistance (Rhodes and Knight 1994),
velocity distribution (Tominaga et al. 1989; Wilcock
1996), sediment transport rate (Chien and Wan 1999),
channel erosion or deposition bed erosion (Julien 1995),
and morphological and geometrical changes in rivers
(Khodashenas and Paquier 2002). A considerable number
of experimental works carried out in open channel have
demonstrated that it is difficult to determine boundary
shear stress distribution in an open channel (Lane 1953;
Cruff 1965; Myers 1978; Yuen 1989; Olivero et al. 1999;
Galip et al. 2006; Lashkar-Ara and Fathi-Moghadam 2009;
Kabiri-Samani et al. 2013). To cope with this difficulty,
empirical, analytical or simplified computational methods
have been carried out by several researchers such as Yang
and Lim (1997), Khodashenas and Paquier (1999), Berlamont et al. (2003), Galip et al. (2006), Guo and Julien
(2005), Bonakdari et al. (2008), Yang (2010), Bonakdari
and Tooshmalani (2010), Javid and Mohammadi (2012)
and Huai et al. (2013). Still, despite using sophisticated
turbulence models, accurate calculation of the local shear
stress value is a difficult task.
Depending on Shannon entropy (1948) as essential reference, Chiu (1987) proposed a new approach introducing
the maximum entropy and probability concepts in
hydraulics to develop new equations for the distribution of
both velocity and shear stress in open channels. Based on
Chiu’s work, (1991) and other researchers like Cao and
Knight (1996), Araujo and Chaudhry (1998) and Sterling
123
2
and Knight (2002) used the Shannon entropy concept to
predict shear stress distribution in open channels. Sterling
and Knight (2002) developed a new approach to predict the
distribution of boundary shear stress in a circular open
channel. However, their study showed limitations in
reflecting the hydraulic behavior of open channels and was
somewhat unsatisfactory in terms of reliability. This might
be attributed to the complexity of the model parameters
assumption and high sensitivity of the model results to the
parameters estimated.
Besides Shannon entropy, Tsallis (1988) proposed a
generalization of the celebrated Boltzmann–Gibbs entropic
measure, which is one of a family of functions for quantifying the diversity, uncertainty or randomness of a system.
The Tsallis entropy is a generalization of the Shannon
entropy containing an additional parameter which can be
used to make it less susceptible to the shape of the probability distribution (Maszczyk and Dush 2008). Singh and
Luo (2011) make use of the Tsallis entropy to predict
velocity distribution in open channels. The main objective
of the present study is to introduce a new method which
results from maximizing the Tsallis entropy for predicting
the shear stress distribution. A short introduction to the
suggested model is given and this relation is used to predict
the boundary shear stress in circular, circular with flat bed,
rectangular and compound channels. These results are
compared with the experimental results to validate this new
equation. A comprehensible comparison is made between
this suggested equation and the model suggested by Sterling
and Knight (2002) that resulted from the Shannon entropy.
2 Shear stress distribution model
2.1 Shannon entropy
Sterling and Knight (2002) used the Lagrange coefficient to
maximize the Shannon entropy and introduce an equation
to predict shear stress as follows:
y
1
s¼
1 þ eksmax 1
;
ð1Þ
k
L
where s is the local boundary shear stress, smax the maximum value of boundary shear stress, y the transverse
coordinate, L half the wetted perimeter and the Lagrange
multipliers, k, can be estimated as:
1
smax eksmax
k ¼ ks
qgRS
;
ð2Þ
e max 1
where q is the mass density, g the gravitational acceleration, R the hydraulic radius and S the bed slope of channel.
This equation was only used in the circular cross section.
123
Stoch Environ Res Risk Assess (2015) 29:1–11
Fig. 1 Cross section of a circular open channel and its notation
The wall and bed relationships in a circular channel with a
flat bed, respectively, must be defined as follows:
2ðy yw Þ
1
sw ¼
1 þ ekw smaxðwÞ 1
yw \y\Pw =2;
kw
Pw
ð3Þ
ks
2ðy yw Þ
1
b maxðbÞ
1þ e
1
sb ¼
kb
Pb
Pw =2\y\Pb =2 þ yw ;
ð4Þ
where kw and kb are the Lagrange multipliers corresponding to the wall and bed, respectively, Pw and Pb, the wetted
perimeter corresponding to the wall and bed of channel,
respectively. Figure 1 shows the channel cross section and
the notation considered for a circular conduit with a flat
bed, where h is the flow height, t the height of sediment in
channel bed, Pb the bed wet area, Pw the total wet area on
the channel wall and D the channel diameter.
To make use of Eqs. (3), (4) and also Eq. (1) the average and maximum shear stress are to be estimated first.
Knight (1981) had conducted their tests in a 15 m flume,
0.46 m wide with a constant bed slope of 9.58 9 10-4. The
percentage of the shear force carried by the walls or bed
was calculated for different aspect ratio (b/h = 1.5, 2, 3, 5,
7, 7.5, 10 and 15). For each depth and roughness the shear
stress was measured by a Preston tube or by semi-logarithmic plotting of the velocity profiles. Their proposed
empirical method to compute these shear stress values on
the bed and wall is as follows [Eqs. (5)–(8)]:
smeanðwÞ
¼ 0:01 %SFw ð1 þ Pb =Pw Þ;
ð5Þ
qgRS
smeanðbÞ
1
¼ ð1 0:01 %SFw Þ 1 þ
;
ð6Þ
Pb =Pw
qgRS
Stoch Environ Res Risk Assess (2015) 29:1–11
h
i
smaxðwÞ
¼ 0:01 %SFw 2:0372ðPb =Pw Þ0:7108 ;
qgRS
h
i
smaxðbÞ
¼ ð1 0:01 %SFw Þ 2:1697ðPb =Pw Þ0:3287 ;
qgRS
3
ð7Þ
ð8Þ
where smean(w) and smean(b) are the average and smax(w) and
smax(b) the maximum shear stress at the channel wall and bed,
respectively and Pb and Pw the wetted perimeters corresponding to the bed and wall, respectively. Knight et al. (1994) proposed an empirical relationship for the prediction of the shear
force percentage carried by the walls, as follows [Eq. (9)]:
%SFw ¼ Csf expð3:23 logðPb =C2 Pw þ 1Þ þ 4:6052Þ;
ð9Þ
where Csf = 1 for Pb/Pw \ 6.546, otherwise Csf = 0.5857
(Pb/Pw)0.28471 and in a subcritical flow C2 = 1.5. Thereby,
depending on the water depth and the bed slope, the
transverse distribution of shear stress can be evaluated for
any given channel.
2.2 Tsallis entropy
wet area and y is the channel wall path changing from 0 to
L. The two other parameters are k0 and k2, which can be
evaluated by the following equations (see Appendices 1,
2):
h 0
ik
0
k þ k2 smax ½k k ¼ k2 kk ;
ð15Þ
h 0
ik h 0
ikþ1
0
smax ðk þ 1Þk2 k þ k2 smax k þ k2 smax
þ½k kþ1
¼ ðk þ 1Þk22 kk smean ;
ð16Þ
where smax is maximum shear stress on the channel wall.
According to the Eq. (14), the shear stress on the channel
wall and bed can be estimated as follows:
1
0
1 0 k k2w yw k k kw
sw ¼
k kw þ
;
ð17Þ
k2w
Pw =2
k2w
1
0
1 0 k k2b yb k k kb
sb ¼
kb þ
k :
ð18Þ
k2b
Pb
k2b
0
Considering a consistent variable s, and based on the
Tsallis (1988) idea, the entropy equation is defined as:
Z smax
1
HðsÞ ¼
f ðsÞ 1 f ðsÞq1 ds;
ð10Þ
q1
0
where q is a real number, H(s) a function of Tsallis entropy
and f(s) the probability density function. Chiu (1987) and
Cao’s (1995) methods of maximizing Lagrange coefficient
were used to estimate the probability density f(s), as follows:
o
1
f ðsÞ 1 f ðsÞq1
of ðsÞ q 1
ð11Þ
o½f ðsÞ
o½sf ðsÞ
þ k2
¼ 0:
þ k1
of ðsÞ
of ðsÞ
Simplifying the above equation, f(s) is defined as:
1
q1
q 1 0
f ðsÞ ¼
k þ k2 s
;
ð12Þ
q
0
1
þ k1 : The cumulative distribution funcin which k ¼ q1
tion of the shear stress is introduced by Eq. (13):
Z s
FðsÞ ¼
f ðsÞds ¼ y=L:
ð13Þ
0
Inserting Eq. (12) into Eq. (13), thus solving the integral
and simplifying it, the shear stress function becomes:
2
31k
!k
0
0
k 4 k
k2 y 5 k
þ
;
ð14Þ
s¼
k2
L
k
k2
q
and q is an exponential parameter in Tsallis
where k ¼ q1
relation and has real quantities, L is constant and equals the
0
To calculate kw ; kb ; k2w and k2b, Eqs. (15) and (16)
were used. Also, Knight et al. (1994) used Eqs. (5) and (8)
to estimate the maximum and average shear stress, smax
and smean, respectively, in Eqs. (15) and (16). It should be
taken into consideration that the wall average, Eq. (5), and
0
maximum shear stress, Eq. (7), were used to calculate kw
0
and k2w, respectively, and for calculating kb and k2b the bed
average, Eq. (6), and maximum shear stress in bed, Eq. (8),
respectively, were used. The results of the comparison
between the equations [Eqs. (17), (18)] proposed in this
study and the extant previous equations [Eqs. (3), (4)] are
presented herein using the criteria for root mean square of
error (RMSE) and the relative error (RE) as defined below:
n
X
siðpredictedÞ siðmeasuredÞ
;
ð19Þ
RE ¼ 100 siðmeasuredÞ
i¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u n u1 X siðpredictedÞ siðmeasuredÞ 2
RMSE ¼ 100 t
;
n i¼1
siðmeasuredÞ
ð20Þ
where n is the total number of nodes. Considering the error
percent, we concluded that both methods show a good
performance in predicting the shear stress distribution in a
circular channel.
3 Experimental data
Knight and Sterling (2000) used the Preston pipe technique
to measure shear stress distribution in different depths of a
circular channel with a diameter of 244 mm and a wall
123
4
Stoch Environ Res Risk Assess (2015) 29:1–11
Table 1 Summary of main hydraulic parameters in circular channel and circular channel with flat bed, Knight and Sterling (2000)
t/D
(1)
h (mm)
(2)
(h ? t)/D
(3)
Pb/Pw
(4)
S0
(5)
Q (l/s)
(6)
s0 (N/m2)
(8)
F
(9)
0.394
0.4411
0.516
R (9104)
(10)
0
81.3
0.333
0
0.001
0
123.5
0.506
0
0.001
11.7
0.493
0.5971
0.505
11.0
0
162.6
0.666
0
0.001
17.3
0.524
0.6897
0.441
13.5
0
201.5
0.826
0
0.001
22.9
0.554
0.7209
0.375
15.0
0.25
20.3
0.333
4.7
0.00862
0.750
1.4905
1.710
4.9
0.25
101.5
0.666
1
0.00862
1.625
4.8040
1.590
34.2
38.9
6.49
U (m/s)
s0 (N/m2)
Available data for
local shear stress
References
7.58
0.187
0.084
Bed
Tominaga et al. (1989)
15.14
0.192
0.7
Bed and walls
Tominaga et al. (1989)
3.9
7
0.338
0.405
0.36
0.164
Bed
Walls
Knight et al. (1984)
Knight et al. (1984)
3.91
18.38
0.495
0.352
Bed
Knight et al. (1984)
7.73
22.34
0.464
0.456
Bed
Knight et al. (1984)
b (cm)
h (cm)
b/h
40
10.15
3.94
40
19.90
2.01
15.2
15.2
7.6
11.3
2
1.34
38.1
9.75
61.0
7.89
Q (l/s)
thickness of 3 mm. Making a sediment layer with the
thickness of t over the bed of a circular channel resulted in
a circular conduit with a flat bed. This was used for measuring the different ratios of t/D shear stress distribution in
different depths.
A summary of hydraulic parameters in circular channel
and circular channel with flat bed is shown in Table 1,
where S0 is the slope of channel length, Q the discharge in
channel, U the average velocity, F the Froude number and
R the hydraulic radius.
The experimental results acquired from the study of
Tominaga et al. (1989) and Knight et al. (1984) were made
use of in the present study. The most important flow conditions and shear stress measurements are tabulated in
Table 2, namely (b = channel width, h = depth of flow,
b/h = aspect ratio, Q = discharge, U = mean velocity,
smean = average shear stress, smean(w) = average shear
stress at wall, and smean(b) = average shear stress at bed).
4 Analysis of results
According to its description, the Tsallis entropy becomes a
convex function when q [ 0. Thus, q should stay positive
to maintain the entropy function convex, so as to achieve
the maximum entropy. Based on Singh and Luo (2011)
proposition and using both field and experimental data, the
feasible range of q proposed was from 0 to 2. A larger
q value leads to more difficulty in solving the equations,
and the solutions of k values for larger q values are not as
stable as for smaller ones. For different values of q,
123
3.39
1.4
1.2
1
τ/ρgRS
Table 2 Summary of
experimental data of Tominaga
et al. (1989) and Knight et al.
(1984)
5.36
U (m/s)
(7)
0.8
0.6
0.4
0.2
q=1/3
q=2/3
q=3/4
q=4/5
q=5/4
q=3/2
q=2
Experimental results
0
0
0.2
0.4
0.6
0.8
1
y/P
Fig. 2 Variation of shear stress distribution with q in circular channel
for (h ? t)/D = 0.333, t = 0
different values of Lagrange multipliers are found. Also,
the values of these parameters vary significantly with those
of q. The shear stress distribution in a circular channel for
(h ? t)/D = 0.333, t = 0 was calculated for the values of
q = 1/3, 2/3, 3/4, 4/5, 5/4, 3/2, 2, as shown in the Fig. 2.
As can be seen in the figure, for q = 5/4, 3/2, the calculated shear stress distribution is different from the experimental results. For the value of q less than 1, the results
obtained are closer to the experimental outcomes than for
q [ 1. The values of q = 2/3, 3/4, 4/5 deliver the same
results and very good estimation of the shear stress. The
smallest difference between estimated shear stress and
experimental results obtained for these values of q. Table 3
illustrates the RE and RMSE between the introduced model
Stoch Environ Res Risk Assess (2015) 29:1–11
5
Table 3 RE and RMSE between the introduced model and the experimental results for different values of q in circular and circular with flat bed
cross section
q = 1/3
RMSE
4.37
6.46
Circular (h ? t)/D = 0.666, t = 0
Circular (h ? t)/D = 0.826, t = 0
3.96
4.33
Circular (h ? t)/D = 0.333, t = 0.25
Circular (h ? t)/D = 0.666, t = 0.25
RE
6.6
RMSE
q = 3/4
RE
RMSE
q = 4/5
RE
RMSE
q = 5/4
RE
RMSE
q = 3/2
RE
RMSE
q=2
RE
RMSE
RE
2.31
2.8
2.3
2.8
2.3
2.8
4.02
5
15.27
15
8.08
8.8
3.5
4.4
3.5
4.4
3.5
4.4
3.78
4.4
17.96
18.9
8.97
9.4
6.8
6.3
3.22
3.91
5.6
7.1
2.4
2.6
3.2
3.9
2.4
2.6
3.2
3.9
4.2
12.84
5.6
17.3
14.34
22.13
15.6
27
11.14
13.49
12.5
16.7
4.41
6.4
1.5
2.4
1.5
2.4
1.5
2.4
6.31
7.4
17.06
18
10.37
12
21.32
23.1
3.9
4.6
3.9
4.6
3.9
4.6
9.05
10.4
14.49
15.3
13.41
13.9
10
and the experimental results for different values of q in
circular and circular with flat bed cross section in different
cases. Based on Singh and Luo’s (2011) proposition and
the obtained results (Table 3), the value of q = 3/4 (k =
-3) was regarded as constant throughout the modeling in
this study.
By using the suggested model [Eq. (14)] and the equation of Sterling and Knight (2002) in the present study, the
shear stress distribution in the circular channel wall in four
different depths was compared with Knight and Sterling
(2000) experimental results, as shown in Table 1. In order
to compare the results, the local boundary shear stress was
normalized by the total shear stress. s0 = qgRS0 (Table 1,
column 7). Figures 2, 3, 4 and 5 show the comparisons
between the two models and the experimental results,
where the horizontal axis shows the dimensionless wet area
resulting from dividing y (Fig. 1) into P (wetted perimeter
in whole channel), the vertical axis showing the dimensionless shear stress. As seen in Figs. 3, 4, 5 and 6 the
predicted values for 0.2 \ y/P \ 0.8 by two models are
close to experimental data, whereas the measured values
for y/P \ 0.2 and y/P [ 0.8 in (h ? t)/D = 0.333, t = 0,
(h ? t)/D = 0.506, t = 0 and (h ? t)/D = 0.666, t = 0
are less than those predicted. Because the boundary shear
stress at the free surface is always difficult to measure, on
increasing flow depth, the shear stress distribution tended
to be uniform. In every situation, the results of the suggested model [Eq. (14)] and those of the Sterling and
Knight (2002) model produce similar results. Using the
Eqs. (19) and (20), an average percent error of these
models was calculated in comparison with the experimental results displayed in Table 4, which illustrates the
RE and RMSE of the introduced model and the Sterling and
Knight (2002) relation for a circular cross section.
Knight and Sterling (2000) filled the bed of a circular
channel with sediments and, according to Table 1, used the
Preston pipe method to measure the shear stress distribution on a bed slope of S0 = 0.00862 at two different depths,
t = 0.25, (h ? t)/D = 0.333 and t = 0.25, (h ? t)/
D = 0.666. The circular channel wall has a steady
1.2
1
0.8
τ/ρgS
Circular (h ? t)/D = 0.333, t = 0
Circular (h ? t)/D = 0.506, t = 0
q = 2/3
0.6
Present model
0.4
Sterling and Knight (2002)
Experimental results
0.2
0
0
0.2
0.4
0.6
0.8
1
y/P
Fig. 3 Shear stress distribution in circular channel for (h ? t)/
D = 0.333, t = 0
1.2
1
0.8
τ/ρgS
Case
0.6
Present model
0.4
Sterling and Knight (2002)
Experimental results
0.2
0
0
0.2
0.4
0.6
0.8
1
y/P
Fig. 4 Shear stress distribution in circular channel for (h ? t)/
D = 0.506, t = 0
curvature. As the channel bed gets filled, the wall curvature, not being fixed, leads to the wall breaking down,
creating a flat bed shaped cross section. As a result, the
shear stress distribution in a circular channel with flat bed
is estimated by using a rectangular channel pattern. Estimating it in the channel wall and bed, the Eqs. (17) and
123
Stoch Environ Res Risk Assess (2015) 29:1–11
1.2
1.2
1
1
0.8
0.8
τ/ρgS
τ/ρgS
6
0.6
Present model
0.6
Present model
0.4
0.4
Sterling and Knight (2002)
Sterling and Knight (2002)
Experimental results
0.2
0.2
0
Experimental results
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
y/P
Fig. 5 Shear stress distribution in circular channel for (h ? t)/
D = 0.666, t = 0
0.8
1
Fig. 7 Shear stress distribution in circular channel with flat bed for
(h ? t)/D = 0.333, t = 0.25
1.2
1.2
1
1
0.8
τ/ρgS
0.8
τ/ρgS
0.6
y/P
0.6
Present model
0.4
0.6
Present model
0.4
Sterling and Knight (2002)
Sterling and Knight (2002)
0.2
Experimental results
Experimental results
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y/P
0
0
0.2
0.4
0.6
0.8
1
y/P
Fig. 6 Shear stress distribution in circular channel for (h ? t)/
D = 0.826, t = 0
Table 4 Comparison of the experimental results in a circular channel
with the results obtained by the proposed model and the Sterling and
Knight equation
Case
Present
model
Eq. Sterling and Knight
(2002)
RE
RE
RMSE
RMSE
(h ? t)/D = 0.333, t = 0
2.3
2.8
3.6
4.6
(h ? t)/D = 0.506, t = 0
3.5
4.4
2.1
2.5
(h ? t)/D = 0.666, t = 0
(h ? t)/D = 0.826, t = 0
2.4
2.6
3.2
3.9
2.2
3.8
4.3
4.4
(18), respectively, were used. The same was done to esti0
0
mate kw ; kb ; k2w and k2b. To predict shear stress distribution, the Eqs. (3) and (4) were solved by using the Sterling
and Knight (2002) equation. The shear stress distribution at
two different depths in circular channels with flat bed for
123
Fig. 8 Shear stress distribution in circular channel with flat bed for
(h ? t)/D = 0.666, t = 0.25
the suggested model and that of Sterling and Knight (2002)
is shown in Figs. 7 and 8.
As seen in Fig. 7, the results in a channel bed of the
suggested model correspond to the experimental results,
but the Sterling and Knight relationship predicts a larger
shear stress in comparison with the experimental results.
Both methods showed different results from the experimental ones in the channel wall. The discrepancy ought to
be related to the small flow depth (2 cm) in this section.
Figure 8 shows the shear stress distribution in t = 0.25,
(h ? t)/D = 0.666. As shown, the suggested model results
are close to the experimental ones, but the Sterling and
Knight relation predicts a smaller shear stress at the
channel wall and a larger one on the channel bed in comparison with the experimental results. The average RE
percentage and RMSE of both models at two different flow
depths on wall and bed were estimated by the Eq. (19) and
are presented in Table 5. It is seen that the average error
percent for the Sterling and Knight method is twice as large
as that for the suggested method. Based on these results we
can conclude that the method suggested in this study
Stoch Environ Res Risk Assess (2015) 29:1–11
Case
Present model
Wall
RE
Eq. Sterling and Knight (2002)
Bed
RMSE
RE
Wall
Bed
RMSE
RE
RMSE
RE
RMSE
(h ? t)/D = 0.333, t = 0.25
6.6
7.3
1.5
2.4
11.5
12.1
2.4
2.5
(h ? t)/D = 0.666, t = 0.25
3.5
5.1
3.9
4.6
9.2
9.8
6.8
8.1
1.4
1.4
1.2
1.2
1
1
0.8
0.8
τ/ρgRS
τ/ρgRS
Table 5 Comparison of the
experimental results in a
circular channel with flat bed
with the results obtained by the
proposed model and the Sterling
and Knight equation
7
0.6
0.6
Present model
0.4
Sterling and Knight (2002)
Experiment Tominaga et al. (1989)
0.2
0.4
Present model
0.2
Sterling and Knight (2002)
Experiment Knight et al. (1984)
Experiment Knight et al. (1984)
0
0
0
0.2
0.4
0.8
1
0
y/P
(a) b/h=2
0.2
0.4
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.8
1
0.8
1
y/P
(c) b/h=7.37
τ/ρgS
τ/ρgRS
0.6
0.6
Present model
0.4
Experiment Tominaga et al. (1989)
0.2
Present model
0.4
Sterling and Knight (2002)
Sterling and Knight (2002)
0.2
Experiment Knight et al. (1984)
Experiment Knight et al. (1984)
0
0
0
0.2
(b) b/h=3.94
0.4
0.6
0.8
y/P
1
0
0.2
(d) b/h=1.34
0.4
0.6
y/P
Fig. 9 Shear stress distribution in rectangular channel with different aspect ratios
performs better when compared with the Sterling and
Knight (2002) model.
The experimental data of Knight et al. (1984) and
Tominaga et al. (1989) were used to validate the proposed
equation. Figure 9a–d show the shear stress distribution in
a rectangular channel with different b/h ratios. The straight
lines represent the shear stress value obtained by the proposed equation, the dash lines are for Eqs. (3) and (4) and
the points are the experimental data. The Sterling and
Knight model tends to overestimate the bed shear stress;
this overestimation being the most important near the
corners in all cases. The bed shear stress calculated by the
Sterling and Knight model is nearly uniform for b/h = 2
and 1.34. For all aspect ratios the bed shear stress
calculated by the proposed model reaches its maximum
value in the center. The Sterling and Knight model indicates slightly higher peak shear stresses.
As seen in Fig. 9a, the results indicate that the proposed
equation predictions agree with the laboratory measurements, but the Sterling and Knight relationship predicts a
larger shear stress in comparison with the experimental
results for b/h = 2. The average RE at channel bed found
between the proposed equation and Sterling and Knight
relation and the experiments of Tominaga et al. (1989) is
3.6 and 10.3, respectively, and the RMSE is 5.8 and 17.7,
respectively (Table 5). In Fig. 9b, the boundary shear
stress at the bed predicted by the two models are compared
with experimental data for b/h = 3.94. Reasonably good
123
8
Table 6 Comparison of the
experimental results in a
rectangular channel with the
results obtained by the proposed
model and the Sterling and
Knight equation
Stoch Environ Res Risk Assess (2015) 29:1–11
Case
Present model
Wall
RE
Eq. Sterling and Knight (2002)
Bed
RMSE
RE
Wall
RMSE
Bed
RE
RMSE
RE
RMSE
b/h = 2, Tominaga et al. (1989)
6.2
7.3
3.6
5.8
13.4
19.5
10.3
17.7
b/h = 2, Knight et al. (1984)
–
–
5.2
7.0
–
–
9.8
12.8
b/h = 3.94, Tominaga et al. (1989)
–
–
7.1
10.2
–
–
11.9
17.1
b/h = 3.94, Knight et al. (1984)
–
–
7.1
7.6
–
–
11.7
14.3
b/h = 7.37, Knight et al. (1984)
–
–
1.4
1.5
–
–
2.9
3.9
b/h = 1.34, Knight et al. (1984)
4.7
5.5
–
–
21.8
28.6
–
–
1.8
Present model
1.6
Sterling and Knight (2002)
1.4
Experiment Rajaratnam and Ahmadi (1981)
τ/ρgRS
1.2
1
0.8
0.6
0.4
Fig. 10 Compound channel cross section characteristics (Rajaratnam
and Ahmadi 1981)
0.2
0
0
0.2
0.4
(a) h/H=0.324
0.8
1
0.8
1
1.8
Present model
1.6
Sterling and Knight (2002)
1.4
Experiment Rajaratnam and Ahmadi (1981)
1.2
τ/ρgRS
agreement is given by the proposed model at bed corners. It
is also interesting to note that there is no significant difference in the shear stress distribution given by the two
models for b/h = 7.37 except in bed corners (see Fig. 9c).
For b/h = 1.34 (Fig. 9d) very good agreement is given by
the proposed model. The average RE between the proposed
equation and Sterling and Knight relationship and the
experiments of Knight et al. (1984) at channel wall is 4.7
and 21.8, respectively, and the RMSE is 5.5 and 28.6,
respectively (Table 6).
0.6
y/P
1
0.8
0.6
0.4
0.2
0
0
0.2
(b) h/H=0.46
0.4
0.6
y/P
5 Shear stress in a compound channel
Fig. 11 Shear stress distribution in compound channel
The shear stress distribution at the compound channel
boundaries, shown in Fig. 10, is calculated for two different heights of h/H = 0.324 and 0.46 (where h = floodplain
depth and H = flow depth) with the aid of the Sterling and
Knight (2002) method and the proposed model (see
Fig. 11a, b). The results were compared by using the laboratory outcomes of Rajaratnam and Ahmadi (1981). In
general, the two models did not match the experimental
data and recorded high errors. It should be mentioned that
the proposed equation had higher errors in predicting the
shear stress distribution in comparison with Sterling and
Knight method, as shown in Table 7.
123
In compound channels, the flow behavior is much more
complex than in simple geometry due to the presence of
secondary currents and vortices occurring in the cross
section which develops along the main channel–floodplain
interface. The two methods give very similar results, with
no good predicted results relative to the measured values
on the left side of the main channel wall. From the Fig. 7 a
reverse prediction is observed, a lesser prediction at low
range of the y/P values and higher estimation at the parts
near the bed. On the bed of the main channel, the Sterling
and Knight method showed better agreement with the
Stoch Environ Res Risk Assess (2015) 29:1–11
Table 7 Comparison of the
experimental results in a
compound channel with the
results obtained by the proposed
model and the Sterling and
Knight equation
Border
9
Present model
Eq. Sterling and Knight (2002)
h/H = 0.324
h/H = 0.46
h/H = 0.324
h/H = 0.46
RE
RMSE
RE
RMSE
RE
RMSE
RE
RMSE
Left side wall
14.83
21
26.88
43.6
14.83
21
26.88
43.6
Bed
11.95
15.04
15.32
19.48
5.54
8.8
7.13
9.8
1.27
2.9
Main channel
Right side wall
3.72
6.3
1.27
2.9
10.63
12.87
Floodplain
Bed
10.63
14.49
8.36
11
16.17
22.19
1.12
10.15
Wall
10.76
9
4.97
6.9
10.76
9
4.97
6.9
experimental results as compared to the proposed model.
On the right side of main channel wall, the results derived
from the proposed method were closer to the experimental
results in comparison with the Sterling and Knight method.
Although, with an increment of flow height in the main
channel, the difference between the two methods with
laboratory outcomes decreases. There is a lateral transfer of
momentum from the fast flowing main channel to the
floodplain at their interface. This phenomenon causes the
high velocity gradient between the floodplain and main
channel. Therefore, the two methods predict the shear
stress less than the measured data at the main channel
interface with the floodplain. While, with increment of flow
height in main channel interface with the floodplain, more
fitness between the Sterling and Knight equation and
experimental data is obtained. On channel wall of the
floodplain, the two methods give very similar results, and
with the increment of flow height, the difference between
the methods and experimental results is reduced.
corner region. In rectangular channels, the bed shear stress
calculated by the Shannon relation is nearly uniform for
b/h = 2 and 1.34, while for compound channels the Tsallis
model ought to be extended further. This study illustrates
how the differences between the two models are distinguishable case by case in open channels.
Acknowledgments The authors would like to express their appreciation to the anonymous reviewers for their helpful comments and to
Ellen Vuosalo Tavakoli for the painstaking editing of the English text.
Appendix 1
This appendix presents the way Eq. (15) is derived. The
first constraint applied concerns the general probability of
the variable s, i.e.,
Z smax
f ðsÞds ¼ 1;
ð21Þ
0
6 Conclusion
The Shannon and Tsallis entropy concepts based on the
probability theory were applied to predict the distribution
of the boundary shear stress. These computational methods
are described and the results compared with experimental
data. The comparison between the two models for computing the boundary shear stress distribution in circular,
circular with flat bed and rectangular channels showed that
these two methods produce similar results for the circular
cross section, with no significant difference between the
two models. The bed shear stress calculated by both models
in circular with flat bed and rectangular channels for all
aspect ratios reaches its maximum value in the center;
however, the Tsallis model provides more reliable results
in comparison with those got through the Shannon relation.
The model based on Shannon entropy is not effective in
predicting the distribution of boundary shear stress in the
by substitution of the probability density function
[Eq. (12)] into Eq. (21), one obtains:
1
Z smax iq1
q1h 0
k þ k2 s
ds ¼ 1;
ð22Þ
q
0
integrating between 0 and smax gives:
#
k h
ik smax
1 1 0
k þ k2 s
¼ 1;
k k2
ð23Þ
0
the definite integral over that interval 0 and smax is given
by:
k h
ik 1k 1 0
1 1 0
k þ k2 smax ½k k ¼ 1;
ð24Þ
k k2
k k2
thus:
h 0
ik
0
k þ k2 smax ½k k ¼ k2 kk :
ð25Þ
123
10
Stoch Environ Res Risk Assess (2015) 29:1–11
thus:
Appendix 2
This appendix presents the way Eq. (16) is derived. The
second constraint concerns the continuity of the variable s,
i.e.:
1
Z smax iq1
q1h 0
k þ k2 s
s
ds ¼ smean ;
ð26Þ
q
0
this can be written in the form:
1Z
1
iq1
q 1 q1 smax h 0
s k þ k2 s
ds ¼ smean ;
q
0
by applying integration by parts one obtains:
1
q
q1
q 1 q1
q1 1 0
smax
k þ k2 smax
q
q
k2
3
q
Z smax
h
i
q1
7
0
q1 1
k þ k2 s
ds7
5 ¼ smean ;
q k2
0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
after integrating between 0 and smax we will have:
smax
2q1
q1 1 q1 0
q1
k þ k2 s
;
A¼
q k22 2q 1
0
ð27Þ
ð28Þ
ð29Þ
ð30Þ
the definite integral over that interval 0 and smax is given
by:
2q1
q1 1 q1 0
q 1 1 q 1 0 2q1
q1
A¼
k
ðk Þ q1 ;
þ
k
s
2
max
2
q k2 2q 1
q k22 2q 1
ð31Þ
by inserting Eq. (31) into Eq. (28) one obtains:
1
q
q1
q 1 q1
q1 1 0
smax
k þ k2 smax
q
q
k2
2q1
0
q1 1 q1
q1
k
þ
k
s
2
max
q k22 2q 1
#
q 1 1 q 1 0 2q1
ðk Þ q1 ¼ smean ;
þ
q k22 2q 1
ð32Þ
this can be written in the form:
k
ik 1k 1 1 h 0
ikþ1
1 smax h 0
k
k þ k2 smax þ
k
s
2
max
k
k k22 k þ 1
k2
k
0
1 1 1
½k kþ1 ¼ smean ;
þ
ð33Þ
k k22 k þ 1
123
ð34Þ
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A
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Z smax
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ikþ1
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k þ k2 smax 2
k2
k2 k þ 1
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½k þ 2
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