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Study on the Morphological Behavior in Meandering River with

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					          Study on the Morphological Behavior in Meandering River with
              Erodible Bank at Sorachigawa River, Hokkaido, Japan
                                            Januar Fery IRAWAN
                                        Candidate for Master Degree
                                     Supervisor: Prof. Yasuyuki Shimizu
                      Division of Engineering and Policy for Cold Regional Environment

INTRODUCTION
Morphological behavior with erodible bank in                   zone. Jagers [3] has reviewed on the angle of
meandering river has been a major attention in the             repose and mechanism of bank erosion in
river   engineering     since   a   last   decade.   The       meandering channel. As for planform model, the
morphological behavior with erodible bank is an                kinematic model has been applied in modeling of
evolution of the river either lateral or vertical in the       meandering river by some researchers such as
stream channel which is influenced by secondary                Ferguson, Howard, Howard and Knutson. The
flow. The secondary flow due to effect of centrifugal          difference between model proposed by Struiksma
force over the streamline and longitudinal direction           and Johansen is the use of a cosine shaped
causes the upper fluid moves from inner to outer               perturbation in the transverse direction and a
bank. Surajate [1] as well mentioned that the effect           linear transverse profile. Johannesson and Parker
of cross circulation can be calculated by using the            developed model based on the Ikeda’s model
estimation of Engelund’s Formula. The other way                (Jagers [3]). The Dynamic model using non-linear
has been done to simulate flow and sand bar                    equation has been developed by Mosselman
movement by Chang Lae Jang and Shimizu [5] in                  (Jaggers [3]). The model is based on the 2D water
the experiment flume in which the secondary flow is            shallow water equations with a rigid lid
estimated by using strength coefficient proposed by            approximation.
Engelund.
                                                               BASIC EQUATION
Bank erosion and planform model has been special               In order to obtain the accurate computation for
attention toward river engineers and often included            depth and velocity as well as sediment transport,
in the simulation either in a natural river or an              the channel coordinate system which has been
experiment channel. On the basis of early work of              used to study is Moving Boundary Fitted
Bank erosion model proposed by Erik Mosellman                  Coordinate System (MBFC). This coordinate has
[6] has applied the two-dimensional morphological              been used by many researchers [1], [2], [5], [6]
model with bank erosion model and provision to                 and [7]    to express the flow dynamic in the
account for the resulting planform change and input            meandering river with the realistic shape.
of bank erosion products Yalin et al. [4] investigated         Flow Model
that bank erosion happens along the outer bank                 The depth-averaged flow equation in the MBFC
subsequently by converging flow, while deposition is           system read as follows
                                                                ∂ ⎛h⎞ ∂ ⎡              ∂ ⎡
on the inner bank along the diverging flow. The site                       (
                                                               ∂τ ⎝ J ⎠ ∂ξ ⎣
                                                                                 )
                                                                                ξ h⎤
                                                                                         (
                                                                                  J ⎦ ∂η ⎣
                                                                                               )
                                                                                             η h⎤
                                                                  ⎜ ⎟ + ⎢ ξt + u ⎥ + ⎢ ηt + u ⎥ = 0
                                                                                               J⎦
                                                                                                            (1)
of the erosion and deposition zone in the flow field
                                                               h is water depth, J is coordinate transformation
is located in the location of convergence-divergence           Jacobin, u ξ dan u η are contra variant of


                                                           1
velocity components in η direction, respectively.                              is critical non-dimensional shear stress
Then, for momentum equation in ξ direction                                     derived by Iwasaki’s formula; d is grain
                                                                               diameter.
become:

∂uξ              ∂uξ            ∂uξ
                                                                               Planform Model and Bank Erosion Model
∂τ
    + (ξt + uξ )
                 ∂ξ
                     (       )
                     + ηt + uη
                                ∂η
                                    +α1uξ uξ +α2uξ uη +α3uηuη − Dξ

       ⎡                ∂H                          ∂H ⎤                       Bank erosion is calculated on the basis of Parker
= − g ⎢ (ξ x2 + ξ y2 )         + (ξ xη x + ξ yη y )
       ⎣                ∂ξ                          ∂η ⎥⎦                      Model which is erosion happened in the outer
 c uξ
− d   (ηyuξ −ξyuη )2 + (−ηxuξ −ξxuη )2 + Dξ                                    bends and deposition at the inner bends. The inner
  hJ
                                                                               and         outer             banks         are       allowed            to             migrate
For momentum equation in η direction is:                                       independently of each other. The transverse
                                                                               positions of the inner bank and outer bank are
∂uη           ∂uη       ∂uη
∂τ            ∂ξ
                     (
    +(ξt +uξ ) + ηt +uη
                        ∂η
                             )
                            +α4uξuξ +α5uξuη +α6uηuη −Dη                        thus functions of stream wise position and time.

    cduη                                                                       The local change in channel bed region for width
−        (ηyuξ −ξyuη )2 +(−ηxuξ −ξxuη)2 + Dn
     hJ                                                                        B is given as:
      ⎡                      ∂H               ∂H ⎤                   (3)
= − g ⎢ (η x ξ x + η y ξ y )
                             ∂ξ
                                 (        )
                                + ξ x2 + ξ y2
                                              ∂η ⎥
      ⎣                                          ⎦                                  ⎡ ( B t +1 − B t ) Y t + 1 − Y t ⎤
                                                                                    ⎢                  +             ⎥
                                                                               ∂η             ∂t              ∂t                                                               (6)
                                                                                  =−⎣                                ⎦
Dξ and Dη                are the momentum equation for                         ∂t                  B ave
                                                                               Where,
diffusion in the ξ and η direction, respectively.
Cd is the friction coefficient in bed load sediment                            Bt +1               = New width of channel due to the
and g is the gravity coefficient and α is the
coefficient of the equation. The ξ axis is drawn                                                      bank erosion mechanism
                                                                               Bt                  = Old width of channel in the previous
along the channel for a given initial channel with a
plane shape and the η axis is set to intersect the                                                    calculation over time
ξ axis. Then, the plane ( ξ , η ) is divided into parts                        Yt +1               = New position of Y coordinates
to form the initial grid for computations.                                                            because of bank migration
                                                                               Yt                  = old position of Y coordinates in the
Sediment Transport Model                                                                              previous calculation over time
                                                                               Bave                = Width average of Bt +1 and Bt
The continuity equation of bed load transport in a
two-dimensional general coordinate system is :
                                                                               The scheme assumes that the centerline                                                                is
 ∂ ⎛ zb ⎞ 1 ⎡ ∂ ⎛ qξ ⎞   ∂ ⎛ qη ⎞ ⎤            (4)
   ⎜ ⎟+          ⎜ ⎢ ⎟+    ⎜    ⎟ =0          ⎥                                always located halfway between the bases                                                              of
∂t ⎝ J ⎠     1 − λ ⎣ ∂ξ ⎜ J ⎟
                        ⎝   ⎠        ∂η ⎜ J ⎟ ⎦
                                        ⎝   ⎠
                                                                               the inner and outer banks, so that no = ni                                                            at
                                                                               any given time. Besides, the erosion                                                                  is
Where zb is bed elevation; λ is porosity of the
                                                                               estimated on the basis of formulas                                                                    as
bed material; q ξ and qη are contra-variant                                    follows :
components of bed load transport rate per unit width
in ξ and η       directions respectively. Bed load                                                       ⎡
                                                                                                         ⎢
                                                                                                                                                                           ⎤
                                                                                                                                                                           ⎥
                                                                                               1                1          (1 + Cn o ) q + ∂ η b
transport rate in s and n component directions is                              no =
                                                                                       (S os
                                                                                                         ⎢
                                                                                               + S tbo ) ⎢ ( − λ p )B os ⎛
                                                                                                           1                   1      ⎞
                                                                                                                                        os , n
                                                                                                                                               ∂t
                                                                                                                                                                           ⎥
                                                                                                                                                                           ⎥
                                                                                                                                                                                 (7)
                                                                                                         ⎢               ⎜1 + C n o ⎟                          n = no
                                                                                                                                                                           ⎥
calculated by formula of Ashida and Michiue                                                              ⎣               ⎝     2      ⎠                                    ⎦
                                                                                                           ⎡                                                           ⎤
                                                                                               1           ⎢       1           (1 + Cn i ) q         ∂η b              ⎥
                                                                               no =
                                                                                       (S is
                                                                                                           ⎢
                                                                                                         ) ⎢ (1 − λ p )B is ⎛ 1 + 1 C n i ⎞ is , n
                                                                                                                                                   +                   ⎥
                                                                                                                                                                                 (8)
                  ⎛ τ *c ⎞⎛              ⎞                                                     + S tbi                                                ∂t               ⎥
                          ⎜ τ *c         ⎟ s gd 3            (5)                                           ⎢                ⎜             ⎟                 n = − ni
                                                                                                                                                                       ⎥
q = 17τ           ⎜1 −
                  ⎜ τ ⎟⎜1 − τ
    b      3/ 2                                                                                            ⎣                ⎝      2      ⎠                            ⎦
           *             ⎟               ⎟ g
                  ⎝    * ⎠⎝   *          ⎠                                     In which λ p is bed porosity; Cn is curvature in
                                                                               the centerline; Bo is width of outer side bank;
where τ * is non-dimensional shear stress; τ *c                                 Bi is width of inner side bank; S os is slope of


                                                                           2
outer side bank; S is is slope of inner side bank;
∂ηb                                                                To solve the flow equation, the specified
         is bed deformation;       qos ,n     is sediment
 ∂t                                                                discharge is given. The initial bed roughness,
transport outside bank in the stream wise direction.               initial velocity of the computational domain and
 qis ,n is sediment transport inside bank in the stream            initial water depth are derived by using the
wise direction; no is transverse position of base of               Manning’s Formula as it describes in the previous
inner bank and no is transverse position of base the               chapter. Friction coefficient is calculated by using
outer side bank.
                                                                   ratio between initial velocity and velocity friction.
METHODOLOGY
                                                                   The sediment transport formula of Ashida and
The model was applied to Sorachigawa River from
                                                                   Michue was coupled by secondary flow effect
Kilo-Post 70 up to 78.8 to calculate bed morphology
                                                                   estimated by Engelund’s Formula. The sediment
and     bank   erosion.   The   calculation       for   bed
                                                                   transport formula is bed load sediment transport.
morphology and bank erosion will be compared with
                                                                   Because the grain size is in gravel size, it assumes
field data and aerial photo data to validate the model.
                                                                   that there is no suspended sediment in the water.
The calculation procedure is ordered as follows :
                                                                   Bed elevation change is calculated from the
1. compute the depth averaged flow field in the
                                                                   continuity equation in which the quantity of
      given plane shape of the water channel
                                                                   deposition balance with the quantity of erosion.
2. compute the secondary flow perpendicular to the
      stream line of the depth-averaged flow
                                                                   The simulation of the bed morphology calculation
3. compute the sediment transport rate and river
                                                                   was calculated by using steady discharge that
      bed evolution
                                                                   represent an overall-averaged discharge of a
4. determine how bank erosion and sediment
                                                                   maximum-annual discharge which is recorded
      deposition alter the shape of channel
                                                                   from 1972 to 2005. The model is intended to
5. set a coordinate system using the new boundary
                                                                   forecast bed morphology for long-term average
      and update the computational data
                                                                   condition by assuming one-day calculation equal
6. Updating time
                                                                   to one-year bed evolution in the real river. A
                                                                   ten-years averaged discharge is defined at 388
RESULTS AND DISCUSIONS
                                                                   m 3 / s for bed morphology simulation and 500
The computation results consist of two simulation i.e.
                                                                   m 3 / s for bank erosion calculation
bed morphology simulation and bank erosion
simulation. The bed morphology simulate only in the
                                                                   CONCLUSION AND RECOMMENDATIONS
area of main channel, while the bank erosion include
floodplain into calculation domain. The bank erosion
                                                                   The simulation results have a good agreement
model      takes   vegetation    and        structure   into
                                                                   with field data and aerial photo data. The gravel
consideration. The computational grid consists of 24
                                                                   bar on the Sorachigawa River has characteristic
x 176 points either for the calculation under low
                                                                   1-3 meter, formed from alternate bar with
water channel and for the calculation high water
                                                                   wavelength of bar is 150 meter – 500 meter.
channel with bank embankment.


                                                               3
   FIELD AND AERIAL PHOTO DATA




                     BED MORPHOLOGY SIMULATION




                     BANK EROSION SIMULATION




   Figure 1. Comparison of Simulation Results and Field data mapped on aerial photo, flow is from right to left


As for Bank Erosion Simulation, the model can                 [3] H.R.A. Jagers.         Modelling Planform
simulate the sites of erosion in a good agreement                 Changes of Braided Rivers. PhD thesis,
                                                                  University of Twente, January 2000.
after comparing the simulation result with sites              [4] M.S. Yalin and A.M. Ferreira da Silva.
erosion mapped in year 2006 and plotted on the                    Fluvial Processes. IAHR International
                                                                  Association of Hydraulic engineering and
aerial photo of year 2005. In the point of view of
                                                                  Research Monograph, Delft, The Netherlands,
river engineering, the determination of site to locate            2001.
bank protection structure should be integrated with           [5] Chang-Lae Jang and Y. Shimizu. Numerical
                                                                  Simulation of Relatively Wide, Shallow
the simulation model. It could be necessary to reduce             Channels with Erodible Banks. Journal of
the cost of bank protection by applying the                       Hydraulic Engineering, ASCE, July 2005.
                                                              [6] Erik Mosselman. Morphological Modelling
vegetation as bank protection in the future. In the
                                                                  of Rivers with Erodible Banks. Journal of
point of the hydraulic research, this is important to             Hydrologycal Processes, 12, 1357-1370,
improve the model applying the different material                 1998.
                                                              [7] Stephen E Darby, Andrei M. Alabyan and
bank and under unsteady discharge.                                Marco J. Van de Wiel. Numerical Simulation
                                                                  of Bank Erosion and Channel Migration in
                                                                  Meandering Rivers. Water Resources
REFFERENCES
                                                                  Research, Vol. 38, No. 9, 1163,2002.
[1] Surajate B.A. Computation of Turbulence and
    Bed Morphology in Meandering River. PhD
    thesis, Hokkaido University, September 2005.
[2] Chang-Lae Jang. Study on the Morphological
    Behavior of the Channel with Erodible Banks.
    PhD Thesis, Hokkaido University, September
    2003.


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