Determining the best mathematical model for stable hydraulics slop of the
Sepidroud river basin
A.R.mardookhpour*
*-Ph.D.department of civil and water engineering.Islamic Azad University,branch of
Lahijan-Iran-Lahijan-pobox:1616
Tel : 00982144211051
Fax ; 00982144247905
Alireza.mardookhpour@yahoo.com
Abstract
This paper deals with an analysis of selected equations used for the determination of a
stable longwise slope calculation of torrential rivers of Sepidroud in guilan province in
north of Iran .Irregularity of the gradient, accompanied by heavy bed-load experiencing
abrupt changes of the flow as a result of heavy rainfalls of short duration and high
intensity, these are typical features impacting the behavior and characteristics of
torrential rivers. Three methods are used to determine the stable slope: the first is based on
tangent tension (shear stress theory), the second observes a (critical) non-scouring crosssectional velocity (critical mean channel velocities), and the third applies the bottom layer
velocity, (the critical bed velocities).The mathematical hydraulic model HEC-RAS v. 3.1.3 has
been used for the verification of the methods in this research study.
Keywords: stable bed slope, shear stress theory, critical means channel velocities, HEC-RAS
software
1
1-Introduction
Typically, the hydraulics of torrential rivers is quite different if compared with those of
lowlands. Irregularities of the lengthwise river-bed gradient and a significantly varying
grain-size distribution of the bed-load are specific features of such rivers(Akan.2006). The
bed-load is usually blended and it consists of sandy, gravelly, and cobble-formed grain
particles. Sudden changes in the flow rate triggered by flash rainfall of short duration and
high intensity usually hit solely small drainage areas(;Chaudry.2008) this is also a typical
feature of such channels behaviors High flow rate results in losses of the bed-load from the
channel bottom and from the river banks(.Chadwick.2004) Thus, the sediment deposition
during the decrease of the driving force becomes an unavoidable consequence. One of the
basic objectives of the respective studies is the determination of the stable bed slope of the
channel that would resist the driving force during the design floods(Chanson2004). The
creation of a sustainable bottom slope depends not only on the sediment grain-size distribution,
but also on the saturation with water of the bed-load(Sturm 2001). The theoretical scope of the
study aims mainly at three methods of the stable bed slope analysis(Krovak2007). The methods
are based on the shear stress theory, on the critical mean channel velocities distribution, and on
the critical bed velocity that is based on the bottom velocities.
2- Materials and methods
The methods are based on the shear stress theory, on the critical mean channel velocities
distribution, and on the critical bed velocity that is based on the bottom velocities(Chapra, 2006).
The hydraulic model HEC-RAS v. 3.1.3 has been used for the method verification in the
Sepidroud river.( HEC-RAS,2006)
2
3-Theory
On the basis of shear stress theory, Shields theorem dominates the situation by the following
formula (1):
Is = [0.06 × (ρm – ρw) ×de] / (ρw × R)
(1)
On the basis of critical mean channel velocities distribution, Manning and Strickler theorem
dominates the situation b by he following formula (2):
Is = Vv 2 / [Ks 2 ×R4/3]
(2)
On the basis of the critical bed velocity that is based on the bottom velocities, Novak theorem
dominates the situation by the following formula (3):
Is = 0.0035× C2 × de / R
(3)
Also the following equation has been used for the conversion of k and n coefficients,
n = R1/6 / [18 × log (a × 12.2 × R / K)]
(4)
where:
Is – stable bed slope (m/m)
ρm – bed-load material density (kg/m3)
ρ – water density (kg/m3)
de – effective grain diameter (m)
R – hydraulic radius (m)
vv – critical mean channel velocities (m/s)
k – coefficient, bottom roughness (m)
n – Manning’s roughness coefficient
ks – mean velocity coefficient of wetted perimeter, unpaved channel bed
a – the constant in Manning- Stickler’s equation related to the value of de
3
C – characteristic of the sediment load
The hydraulic model HEC-RAS has been used to quantitatively analyze the above equations.
4- Description of the hydrological and hydraulics data of the basin and river
The Sepidroud river is a sinistral tributary river at its fluvial kilometer 2.0 and the mean
slope of the channel is 4%.Some of the other data’s of river are as follows:
Total catchments area = 8.964 km2
Forest coverage= 67%
Length of watershed =6.73 Km
Watershed shape factor= 0.653
Torrential coefficient = 0.118
Figure1 shows the sketched diagram of the river. Also Table 1 lists the N-year discharges.
Table 2 shows the hydraulics characteristics of sediments in each reach or investigated
span of river for grain size distribution.
5-Results and discussion
The evaluation of the channel capacity, the flow velocity, and the proposed stable slope of
the river-bed covers 3 choices of the river reaches, all characterized by the effective grain
size of de =0.06 m , de= 0.10 m and de= 0.16 m .
The above-mentioned data set-ups were computed for 2 flow rates of Q100 = 9.9 m3/s and
Q5 = 5.2 m3/s.figure 2 shows the velocity distribution in cross section of the river. It
should be noted that the span of the river for computations has altitude between 1864 till
1863 and for briefing it is shown by deducting 1000m.
4
Steady-state calculations under non-uniform flow conditions were performed for three
selected river-reaches, all characterized by the effective grain size de. The discharge, mean
flow velocity, and geometric characteristics of the cross sectional profiles were identified by
virtue of the model HEC-RAS. On the basis of these data the individual equations have been verified
and the results summarized into the following pictures and graphs (Figures 3 and 8).
The basis for the calculation of the stable bottom slope became the classical Shields
equation (1) based on the shear stress philosophy. A good agreement was also shown
with the calculation after Manning-Strickler (2). The Eq. (3) of Novak indicates
somewhat higher values of the slope stability. The difference between the extreme results
amounts to 19% (Figures 7 and 8). In general, it can be stated that the model accuracy
increases with the reliability of granulometric analysis of the bed-load and discharges. For the
Sepidroud river both Shields an Manning-strickler theories have good agreements with analyzing
results by HEC-RAS software. It is proposed that the mathematical formulas of Shields and
Manning-strickler that have both good convergence with HEC-RAS analysis would be applied
by researchers , although the Novak formula has good agreement with analyzing results by
HEC-RAS software within Q<5.2m3/s and de<0.1m.
6-conclusion
The theoretical scope of the study aims mainly at three methods of the stable bed slope
analysis. The methods are based on the shear stress theory, on the critical mean channel
velocities distribution, and on the critical bed velocity that is based on the bottom
velocities. The hydraulic model HEC-RAS v. 3.1.3 has been used for the method
verification in the Sepidroud river. The results obtained from utilizing Shields, ManningStrickler, and Novak formulas shows the Novak theory indicates somewhat higher values
of the slope stability. The difference between the extreme results amounts to 19% .It is
5
recommended that for submitting a mathematical method for Sepidroud river , the
researches uses Shields or Manning-strickler formulas because of good agreement and
convergence with HEC-RAS analysis.
References
1-Akan.O.(2006).Open-channel hydraulics.Elsevier.Amsterdam.Netherland.
2-Chadwick.A. (2004).Hydraulics in civil and environmental engineering.4 ed. Taylor and
Francis. London
3-Chanson.H.(2004).The hydraulics of open channel flow.2 ed. Elsevier. USA
4-Chapra.S.C.(2006).Numerical methods for hydraulics engineering.5 ed. Mc-Graw Hill.
New York.USA
5-Chaudry.M.(2008).Open –channel flow.2 ed. Springer.USA
6-HEC-RAS.(2006).Hydraulics Engineering Centers River Analysis System.US Army
Corps of Engineers.USA
7-Krovak.F.(2007).On the determination of the stable bed slop of a channel using
mathematics models. Czech university. Czech Republic.
8-Sturm.T.W.(2001).Open channel hydraulics.Mc-Graw Hill . New York.USA
6
Figure1 .the sketched diagram of the river
Figure 2. Velocity distribution in cross section, flooded by Q5 and Q100
7
0.07
0.06
Is(m/m)
0.05
Shields
0.04
Manning-Strickler
0.03
Novak
0.02
0.01
0
de=0.06m
Figure 3. Analysis of Shields,Manning strickler and Novak for Q5=5.2m3/s and de = 0.06m
0.03
0.025
Is(m/m)
0.02
0.015
Shields
Manning-Strickler
Novak
0.01
0.005
0
de=0.1m
Figure 4. Analysis of Shields,Manning strickler and Novak for Q5=5.2m3/s and de = 0.1m
8
0.06
0.05
Is(m/m)
0.04
0.03
Shields
Manning-Strickler
Novak
0.02
0.01
0
de=0.16m
Figure 5. Analysis of Shields,Manning strickler and Novak for Q5=5.2m3/s and de = 0.16m
0.09
0.08
Is(m/m)
0.07
0.06
Shields
0.05
Manning-Strickler
0.04
Novak
0.03
0.02
0.01
0
de=0.06m
Figure 6. Analysis of Shields,Manning strickler and Novak for Q100=9.9m3/s and de = 0.06m
0.02
0.018
Is(m/m)
0.016
0.014
Shields
0.012
Manning-Strickler
0.01
Novak
0.008
0.006
0.004
0.002
0
de=0.1m
Figure 7. Analysis of Shields,Manning strickler and Novak for Q100=9.9m3/s and de = 0.1m
9
0.035
0.03
Is(m/m)
0.025
0.02
Shields
Manning-Strickler
Novak
0.015
0.01
0.005
0
de=0.16m
Figure 8. Analysis of Shields,Manning strickler and Novak for Q100=9.9m3/s and de = 0.16m
10
N(years) 1
(m3/s)
0.9
Table 1. Design discharges
2
5
10
20
1.2
5.2
6.9
7.7
50
8.4
Table 2. Hydraulic characteristics of sediments
reach de
a
ρm
ρw
C
3
0.06 18.29 2650 1000 5.58
2
0.1 16.78 2650 1000 5.58
1
0.16 15.31 2650 1000 5.58
11
100
9.9