3- Muluneh Admass D NUMERICAL MODELING OF FLOW AND

3-  Muluneh Admass D NUMERICAL MODELING OF FLOW AND
3-D NUMERICAL MODELING OF FLOW AND
SEDIMENT TRANSPORT IN RIVERS
Muluneh Admass
May 2005
TRITA-LWR.LIC 2028
ISSN 1650-8629
ISRN KTH/LWR/LIC 2028-SE
Muluneh Admass
TRITA LWR.LIC 2028
ii
3-D Numerical modeling of flow and sediment transport in rivers
TABLE OF CONTENTS
ACKNOWLEDGEMENTS .............................................................................................................................. V
ABSTRACT......................................................................................................................................................... 1
INTRODUCTION............................................................................................................................................. 1
ECOMSED MODEL .........................................................................................................................................2
GOVERNING EQUATIONS ...........................................................................................................................2
Turbulence model............................................................................................................................................... 3
Boundary conditions........................................................................................................................................... 4
NUMERICAL METHODS ............................................................................................................................... 4
MODIFICATIONS AND IMPROVEMENTS.................................................................................................5
The bottom boundary condition......................................................................................................................... 5
Equivalent sand roughness ................................................................................................................................. 6
Shear stress partitioning...................................................................................................................................... 6
Bed load transport and bed evolution ................................................................................................................. 7
MODEL APPLICATION ..................................................................................................................................7
Model description and model boundary .............................................................................................................. 7
Grid generation .................................................................................................................................................. 8
Solution algorithm .............................................................................................................................................. 8
Model calibration................................................................................................................................................ 8
Grid Independence............................................................................................................................................. 8
Sensitivity Analysis ............................................................................................................................................. 9
Validation........................................................................................................................................................... 9
Flow field ........................................................................................................................................................... 9
Sediment transport ........................................................................................................................................... 10
DISCUSSION ................................................................................................................................................... 10
CONCLUSION .................................................................................................................................................11
REFERENCES ................................................................................................................................................ 24
iii
Muluneh Admass
TRITA LWR.LIC 2028
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3-D Numerical modeling of flow and sediment transport in rivers
ACKNOWLEDGEMENTS
First I would like to acknoweledge the Swedish International Development Cooperation Agency
(SIDA) for providing me the financial assistance to do my research work. Then I would like to
extend my deepest gratitude to my main supervisor Bijan Dargahi, who, with never ending patience, has shared with me his great scientific knowledge and research experience. Assistant supervisors Klas Cederwal and Bayou Chane deserve special acknoweledgment. Special thanks to
Britt Chow for all her concern and support. Thanks Kirlna for all your sisterly help. I am also
very greatful to Hans Bergh, Aira Saarelainen, Nandita Singh, and other colleagues in the vattenbyggnad and in the department of Land and Water Resources, KTH. I am indebted to my spiritual father Aba Wolde Meskel, Tin’s family, my family and my friends here in Stockholm and at
home. Finally goes all my available thanks to my wife Tin.
Stockholm, May 2005
Muluneh Admass
v
Muluneh Admass
TRITA LWR.LIC 2028
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3-D Numerical modeling of flow and sediment transport in rivers
ABSTRACT
The fully integrated 3-D, time dependant, hydrodynamic and sediment transport numerical
model ECOMSED was used to simulate flow and sediment transport in rivers. ECOMSED was
originally developed for large water bodies such as lakes and oceans and solves the primitive
equations of RANS along with a second order turbulence model in an orthogonal curvilinear σcoordinate system. The availability of the model as an open FORTRAN source code made modifications and addition of new models possible. A new bed load transport model was implemented
in the code as well as improvements in treatment of river roughness parameterization, bed form
effects, and automatic update of flow depth due to bed evolution. The model was applied to 1km long reach of the River Klarälven, Sweden, where it bifurcates into two west and east channels. The water surface and the flow division in the channels were made in agreement with field
data by spatially varying the roughness. However, the spatial distribution of the bed shear stress
was not realistic. Improvements were made in the bottom boundary condition to represent the
variable effects of bed forms on roughness depending on the flow regime and the flow depth.
The improved model realistically reproduced the flow field as well as the sediment transport
processes in the river Klarälven.
Key words: River model; ECOMSED model; River Klarälven; Bed load transport model
using a Large Eddy Simulation (LES). The
model simulated periodic flow features in
river confluences and agreed well with experimental data. Some recent applications on
use of 3-D sediment transport models are:
Gesseler et al. (1999), Holly and Spasojevic
(1999), Fang (2000), Rodi (2000), Weiming et
al. (2000), Nicholas (2001), and Dargahi
(2004). Gesseler et al. (1999) applied the U.S.
Army Corps Engineering code CH3D-SED
to the Deep Draft Navigation project on the
lower Mississippi River. The code predicted
the sediment deposition in the river with an
accuracy of less 13% in comparison with the
observations. Holly and Spasojevic (1999)
applied and verified the CH3D-SED code to
study water and sediment diversion at the
Old River Control complex on the lower
Mississippi river. Dargahi (2004) predicted
secondary flows and the general flow and
sediment transport patterns in a river that
agreed well with field measurements. A brief
review of the major developments in 3-D
modeling and the limitations is also given by
Lane et al. (2002).
In this paper the fully integrated 3-D, time
dependant, hydrodynamic and sediment
transport numerical model ECOMSED by
Blumberg and Mellor (1987) is used.
ECOMSED was originally developed for
large water bodies such as lakes and oceans
INTRODUCTION
River modelling applications can be grouped
into 2-D depth-averaged hydrodynamic
models and 3-D models. An evaluation of
the extent to which 3-D models improve
predictive ability compared to 2-D models is
well described by Lane et al. (1999) and
Gessler et al. (1999). They stated that a 3-D
model is necessary for predicting sediment
erosion and deposition whenever significant
secondary currents exist, such as in river
bends, crossings, distributaries, or diversions.
Some recent research works on flow modelling are: Weerakoon and Tamai (1989), Olsen
and Stokseth (1995), Weiming et al. (1997),
Sinha et al. (1998), Lane et al. (1999), Bradbrook et al. (2000), and Chau and Jiang
(2001). Olsen and Stokseth (1995) carried out
a 3-D simulation of an 80-m long reach of
the river Sokna in Norway. The model successfully predicted the flow features and the
results were in good agreement with observed data. Sinha et al. (1998) did a comprehensive numerical study of the flow through
a 4-km reach of the Columbia River. They
succeeded in modelling both rapidly varying
bed topography and the presence of multiple
islands. The results agreed well with experiments and field measurements. In a recent
study, Bradbrook et al. (2000) modelled the
flow in a natural river channel confluence
1
Muluneh Admass
TRITA LWR.LIC 2028
∂u ∂v ∂w
+ +
=0
∂x ∂y ∂z
and solves the primitive equations of RANS
along with a second order turbulence model
in an orthogonal curvilinear σ-coordinate
system. The availability of the model as an
open FORTRAN source code made modifications and addition of new models possible.
The first objective of the present study was
to adapt and improve the ECOMSED model
for modeling flow and sediment transport in
rivers. The second objective was to test the
applicability of the improved model to the
river Klarälven in the southwest part of Sweden. A new bed load transport model is implemented in the code as well as improvements in treatment of river roughness
parameterization, bed form effects, and
automatic update of bed evolution. The
improved model realistically reproduced the
flow field as well as the sediment transport
processes in the river Klarälven. The secondary flows at the river bifurcation and around
the bends were well reproduced. The results
also agreed well with the previous simulations
by Dargahi (2004).
(1a-d)
⎛ ∂ 2u ∂ 2u ∂ 2u ⎞
∂u
∂u
∂u
∂u
1 ∂p
+u
+v
+w
=−
+ υ ⎜⎜ 2 + 2 + 2 ⎟⎟
∂t
∂x
∂y
∂z
∂y
∂z ⎠
ρ ∂x
⎝ ∂x
2
2
⎛ ∂ v ∂ v ∂ 2v ⎞
1 ∂p
∂u
∂v
∂v
∂v
+u +v +w = −
+ υ ⎜⎜ 2 + 2 + 2 ⎟⎟
ρ ∂y
∂t
∂x
∂y
∂z
∂y
∂z ⎠
⎝ ∂x
⎛ ∂2w ∂2w ∂ 2w ⎞
∂u
∂w
∂w
1 ∂p
∂w
+u
+v
+w
=−
+ υ ⎜⎜ 2 + 2 + 2 ⎟⎟ + g
∂t
∂x
ρ ∂z
∂y
∂z
∂y
∂z ⎠
⎝ ∂x
In which u , v, w are the velocity components
along x, y, z directions respectively, t is
time, ρ is the fluid density, p is the pressure,
υ is the fluid kinematic viscosity and g is the
gravitional force. The flow is assumed incompressible (constant density) and the
fluid’s coefficient of viscosity is taken constant. It is possible to solve the Navier-Stokes
equations by direct numerical simulation (DNS) if
we can resolve all the relevant length scales
which vary from the smallest eddies to scales
on the orders of the physical dimensions of
the problem domain. For channel flow the
number of grid points needed to resolve all
the relevant scales can be estimated from the
expression (Tannehill et al. 1997)
9/4
(2)
N DNS = (0.088 Re )
ECOMSED MODEL
h
The ECOMSED model is a fully integrated
3-D hydrodynamic, wave and sediment
transport model. It has a free surface and a
bottom following σ-coordinate system (for
better representation of irregular bottom
topography) with an orthogonal curvilinear
grid in the horizontal plane. Here follows a
brief description of the governing equations,
boundary conditions, the turbulence model
and the solution algorithms which are related
to the present study. Details of the model can
be found in Blumberg (2002).
In which Reh is the Reynolds number based
on the mean channel velocity and channel
height. This approach is limited to flows of
simple geometry and very low Reynolds
number. Another promising approach is
known as large-eddy simulation (LES), in which
the large-scale structure of the turbulent flow
is computed directly and only the effects of
the smallest (subgrid-scale) and more nearly
isotropic eddies are modeled. The computational effort required for LES is less than that
of DNS by approximately a factor of 10
using present-day methods. The main thrust
of present-day research in computational
fluid dynamics is through the time averaged
Navier-Stokes equations also known as the
Reynolds averaged Navier-Stokes (RANS) equations. These equations are derived by decomposing the dependent variables in the
conservation equations in to time-mean (obtained over an approximate time interval) and
fluctuating components and then time averaging the entire equation (Tannehill et al.
1997). Time averaging the equations of mo-
GOVERNING EQUATIONS
The fundamental equations of fluid dynamics
are based on the conservation of mass and
momentum. Conservation of mass yields the
continuity equation while conservation of
momentum yields the momentum equation.
Both equations are widely known by NavierStokes equations. Here follows the NavierStokes equations in the Cartesian coordinate
system ( x, y , z axis).
2
3-D Numerical modeling of flow and sediment transport in rivers
Turbulence model
Turbulence models are used to relate the new
correlations that appeared in the RANS equations and in the mean sediment concentration equation with mean values or in other
words to close the system of equations. Here
follows the description of equations used in
ECOMSED to relate the new correlations
with the mean values.
∂
(6a-d)
− (w'u ' , w'v ' ) = K M (U , V )
tion gives rise to new terms, which can be
interpreted as “apparent” stress gradients
associated with the turbulent motion.
ECOMSED model solves the RANS equations with the hydrostatic assumption and the
three-dimensional equation describing the
advection and diffusion of sediment particles
given below,
∂U ∂V ∂W
(3a-e)
+
+
=0
∂x
∂y
∂z
∂U
∂U
∂U
∂U
1 ∂P
+U
+V
+W
=−
+
∂t
∂x
∂y
∂z
ρ ∂x
∂z
(
)
⎛ ∂V ∂U ⎞
⎟⎟
,
− u ' u ' , v ' v ' = 2 AM ⎜⎜
⎝ ∂y ∂x ⎠
⎞ ∂ ⎛ ∂U
∂ ⎛ ∂U
⎞ ∂ ⎛ ∂U
⎞
− u 'u ' ⎟ + ⎜⎜υ
− v 'u ' ⎟⎟ + ⎜υ
− w 'u ' ⎟
⎜υ
∂x ⎝ ∂x
∂
∂
∂
∂
y
y
z
z
⎠
⎝
⎠
⎝
⎠
∂V
∂V
1 ∂P
∂V
∂V
+U
+V
+W
=−
+
ρ ∂y
∂t
∂x
∂y
∂z
⎛ ∂U ∂V ⎞
⎟
− u 'v ' = −v 'u ' = AM ⎜⎜
+
∂x ⎟⎠
⎝ ∂y
⎞ ∂ ⎛ ∂V
∂ ⎛ ∂V
⎞ ∂ ⎛ ∂V
⎞
⎜⎜υ
− u 'v ' ⎟ +
− v ' v ' ⎟⎟ + ⎜υ
− w'v ' ⎟
⎜υ
∂x ⎝ ∂x
⎠ ∂y ⎝ ∂y
⎠
⎠ ∂z ⎝ ∂z
⎛
∂C
∂C
∂C ⎞
⎟
− u 'c ' , v 'c ' , w'c ' = ⎜⎜ AH
, AH
, KH
∂x
∂y
∂z ⎟⎠
⎝
ρg = −
∂P
∂z
In which K M is the eddy viscosity, K H is the
eddy diffusivity, AM is the horizontal viscosity
and AH is the horizontal diffusivity. The
horizontal viscosity AM is calculated according to Smagorinsky (1963)
∂C ∂UC ∂VC ∂ (W − Ws )C
+
+
+
=
∂t
∂x
∂y
∂z
⎞ ∂ ⎛ ∂C
∂ ⎛ ∂C
⎞ ∂ ⎛ ∂C
⎞
− u 'c ' ⎟ + ⎜⎜ Γ
− v 'c ' ⎟⎟ + ⎜ Γ
− w 'c ' ⎟
⎜Γ
∂x ⎝ ∂x
⎠ ∂y ⎝ ∂y
⎠
⎠ ∂z ⎝ ∂z
In which U , V , W are mean velocity components in the x, y , z directions respectively
( x is the main flow direction, y is the stream
wise direction, z is vertical to the bed) while
u ' , v ' , w ' are the corresponding fluctuating
velocity components, C is the mean sediment
concentration and c ' is the flactuating component, Γ is sediment diffusivity and Ws is the
settling velocity of sediment particles. The
new correlations that appeared in the above
equations are related with the mean values
using turbulence models. The settling velocity
of sediment particles is calculated from the
effective diameter of the suspended sediment
using the semi-empirical formulation of
Cheng (1997)
υ
(4)
W =
(25 + 1.2 D 2 )0.5 − 5
s
D50
[
⎡ ( s − 1) g ⎤
D* = ⎢
2
⎥⎦
⎣ υ
∂U j
1 ⎡⎛ ∂U
AM = c∆2 ⎢⎜ i +
2 ⎢⎜⎝ ∂x j
∂xi
⎣
1/ 3
D50
1
⎞
⎟
⎟
⎠
2
(7)
⎤2
⎥
⎥
⎦
In which c = 0.01, ∆2 = ∆x∆y, and Einstein
convention is used. ∆x and ∆y are the grid
spacing in the x and y directions respectively.
The horizontal diffusivity AH is usually set
equal to AM . K M and K H are obtained by appealing to a second order turbulence closure
scheme developed by Mellor and Yamada
(1982) which characterizes the turbulence by
equations for the turbulence kinetic energy,
1 2 , and turbulence macroscale, l , accordq
2
ing to,
∂q 2
∂q 2
∂q 2
∂q 2
∂q 2 ⎞
∂ ⎛
⎟+
+U
+V
+W
= ⎜⎜ K q
∂t
∂x
∂y
∂z
∂z ⎝
∂z ⎟⎠
2
2
3
⎡⎛ ∂U ⎞
∂q 2
∂q 2 ⎞ ∂ ⎛
∂ ⎛
⎛ ∂V ⎞ ⎤ 2q
⎟ + ⎜ AH
2 K M ⎢⎜
+ ⎜⎜ AH
⎟ ⎥−
⎟ +⎜
∂y
∂x ⎟⎠ ∂y ⎜⎝
⎝ ∂z ⎠ ⎦⎥ B1 l ∂x ⎝
⎣⎢⎝ ∂z ⎠
]
1 .5
*
)
(
( )
(5)
( )
( )
( )
(8a,b)
⎞
⎟⎟
⎠
( )
∂ ⎛ ∂ q 2l ⎞
∂ q 2l
∂ q 2l
∂ q 2l
∂ q 2l
⎟+
= ⎜⎜ K q
+W
+V
+U
∂z ⎝
∂z ⎟⎠
∂z
∂y
∂x
∂t
⎡⎛ ∂U ⎞ 2 ⎛ ∂V ⎞ 2 ⎤ q 3 ~ ∂ ⎛
∂q 2l ⎞
∂q 2l ⎞ ∂ ⎛
⎟
⎟⎟ + ⎜⎜ AH
lE1 K M ⎢⎜
⎟ ⎥ − W + ⎜⎜ AH
⎟ +⎜
∂y ⎟⎠
∂x ⎝
∂x ⎠ ∂y ⎝
⎣⎢⎝ ∂z ⎠ ⎝ ∂z ⎠ ⎦⎥ B1
In which D50 is particle diameter for 50%
finer of bed material, D* is the dimensionless
grain size and s is specific density.
In which A1 , A2 , B1 , B2 , E1 , E 2 , S q are empirical
constants. The wall proximity function, W~ , is
defined as
3
Muluneh Admass
~
⎛ l ⎞
W = 1 + E2 ⎜ ⎟
⎝ κL ⎠
2
1
1
1
=
+
L η−z H +z
TRITA LWR.LIC 2028
In which U bt , Vbt , Wbt are the velocity components at the bottom, E − D is the sediment
flux at the water sediment interface which is
calculated using the van Rijn (1984a) procedure and uτb is the bottom friction velocity
associated with the bottom frictional
stress (τ bx ,τ by ) . In ECOMSED the bottom
stress is determined by matching velocities
with the logarithmic law of the wall.
(14)
τ b = ρC D Vb Vb
(9)
(10)
In which κ is the von Karman constant, η is
the water surface elevation and H is the
water depth. While details of the closure
module are rather involved, it is possible to
reduce prescription of the mixing coefficients
to the following expressions,
lq
(11a-c)
K =
M
With the value of the drag coefficient CD
given by
1/ 3
B1
⎛ 6A ⎞
K H = lqA2 ⎜⎜1 − 1 ⎟⎟
B2 ⎠
⎝
⎡ 1 ⎛ H + zb
C D = ⎢ ln⎜⎜
⎣κ ⎝ zo
K q = lqS q
Empirical constants A1 , A2 , B1 , B2 , E1 , E2 , S q are
0.92, 0.74, 16.6, 10.1, 1.8, 1.33, 0.2 respectively (Mellor and Yamada 1982).
⎛ ∂U ∂V ⎞
,
⎟ = (τ ox ,τ oy )
⎝ ∂z ∂z ⎠
q 2 = B1 uτs
2/3
(12a-e)
2
q 2l = 0
W =U
KH
∂η
∂η ∂η
+V
+
∂x
∂y ∂t
∂C
=0
∂z
In which (τ ox ,τ oy ) is the surface wind stress
vector with the surface friction velocity, uτs ,
being the magnitude of the vector.
Bottom boundary: The boundary conditions at
bottom boundary, z = H ( x, y ) , are
⎛ ∂U ∂V ⎞
,
⎟ = (τ bx ,τ by )
⎝ ∂z ∂z ⎠
ρK M ⎜
q 2 = B1 uτb
2/3
KH
(15)
NUMERICAL METHODS
The governing equations and boundary conditions are transformed in to a vertical σ layer and an orthogonal curvilinear horizontal
( ξ1ξ 2 ) coordinate system. The σ - transformation is given by
(13a-e)
2
q 2l = 0
Wbt = −U bt
−2
In which zb and Vb are the grid point and
corresponding resultant horizontal velocity in
the grid point nearest to the bottom. The
parameter z0 depends on the local bottom
roughness.
Open lateral boundary: Two types of open
boundaries exist, inflow and outflow. The
normal component of velocity is specified
while a free slip condition is used for the
tangential component at inflow boundaries.
Turbulence kienetic energy and the macroscale quantity ( q 2l ) are calculated with sufficient accuracy at the boundaries by neglecting
the advection in comparison with other terms
in their respecttive equations. The sediment
concentration data at the inlet is specified,
whereas at outflow boundaries the mixed
boundary condition is used. The clamped
boundary condition in ECOMSED allows
assigning observed water level along the open
boundary grids.
Boundary conditions
The boundary conditions are specified as
surface, bottom and open boundaries.
Free surface: The boundary conditions at the
free surface, z = η ( x, y, t ) , are
ρK M ⎜
⎞⎤
⎟⎟⎥
⎠⎦
∂H
∂H
− Vbt
∂y
∂x
σ=
∂C
=E−D
∂z
z −η
H +η
(16)
In which η is the water surface elevation.
The vertically and horizontally transformed
set of equations is approximated by a finite
4
3-D Numerical modeling of flow and sediment transport in rivers
ing aspects in the model the river Klarälven
was used as a case study. Realistic and comparable results with field measurements and
with previous simulations using other models
were obtained after the following improvements and modifications were made.
difference scheme using a spatially staggered
grid. The leap frog scheme with the Courant-Friedrichs-Levy (CFL) computational
stability condition and a weak filter to remove solution splitting at even and odd time
steps is employed for time differencing.
Three options (upwind difference, central
difference and Multidimensional Positive
Definite Advection Transport Algorithm) are
available for spatial differencing. The hydrodynamic module (ECOM) is 3-D with a split
external-internal mode algorithm; the external mode explicitly solves the depth integrated equations with short time steps to
resolve fast moving waves and to determine
the water surface elevation. The internal
mode uses the computed water surface elevation and implicitly solves the vertical structure of the flow with a shorter time step. The
internal mode then updates some of the
variables of the external mode for the next
time step to begin. At regular intervals specified by the user the sediment transport module (SED) which uses the same numerical
grid, structure and computational framework
as the hydrodynamic model simulates sediment resuspension, transport and deposition
of cohesive and non-cohesive sediments
using the previously computed hydrodynamic
variables.
The bottom boundary condition
The bottom boundary condition used in
ECOMSED as given in equations (14) and
(15) is the generalization of the logarithmic
law which assumes the same roughness in the
whole computational domain. However, in
rivers roughness which is composed of skin
friction due to bed grains and drag form due
to bed forms varies spatially and temporally
depending on the local flow depth and the
local flow regime. The bottom boundary
condition was reformulated as follows to
directly represent the spatial and temporal
variation of the bottom roughness. Starting
from the general equation of the logarithmic
universal velocity distribution (Schlichting
1968)
Vb
u*
=
⎛ ( H + z b )u* ⎞
ln⎜
⎟ + B − ∆B
κ ⎝
υ
⎠
1
(17)
In which B = 5.45 , κ = 0.41 and ∆B , the roughness induced velocity deficit is a function of
the equivalent roughness height k s . Denoting
k s u* / υ by k s+ and using the interpolation formula by Cebeci and Bradshaw (1977)
MODIFICATIONS AND
IMPROVEMENTS
⎧
⎪ 0
⎪
1
⎪ ⎡
⎤
∆B = ⎨ ⎢ B − 8.5 + ln k s+ ⎥ sin 0.4258 ln k s+ − 0.811
κ
⎦
⎪ ⎣
⎪
1
+
−
+
B
8
.
5
ln
k
⎪
s
κ
⎩
The application of ECOMSED to rivers can
be improved by considering some of the
specific features of river hydraulics that differs from large scale motions such as roughness, spatial variations of flow depth, and bed
load transport. In rivers, significant variations
in channel and river bed roughness are
found. Furthermore, the formation of bed
forms directly affects the resistance to flow as
well as the distribution of bed shear stresses.
The other aspect is the spatial variation of the
flow depth that affects the velocity distribution and the boundary layer characteristics.
In ECOMSED model a constant roughness
value is used. The effect of bed form roughness which depends on the local flow regime
is not considered. The bed load transport is
also not modeled. To implement the forgo-
{
[
]}
k s+ < 2.25
2.25 ≤ k s+ ≤ 90
k s+ > 90
(18)
Using equation (17) and (18) the roughness
parameter z 0 in equation (15) becomes
z0 =
υ
u* e
( B − ∆B ) κ
(19)
Equation (19) allows specifying spatially and
temporally variable roughness depending on
the velocity deficit ∆B which is a function of
the local equivalent sand roughness and the
local shear Reynolds number as shown in
equation (18). Available empirical formulas
were used to compute the equivalent sand
roughness of bed grains and bed forms as
5
Muluneh Admass
TRITA LWR.LIC 2028
described in the next section. Further modification was done to solve the problem of
locating the last grid in the narrow range of
the logarithmic region in highly variable river
bed topography. This can be described as
follows. The logarithmic law defined by equation (17) is valid approximately in the
range 40υ ≤ (H + z b ) ≤ 0.2δ where δ is the
z0 =
u*
κ
⎜
⎝
⎟
⎠
υ
κ
⎜
⎝
δ
e
(22)
Equivalent sand roughness
Equivalent sand roughness is composed of
grain, saltation and bed form roughness. The
component of roughness due to saltation and
bed forms depends on the local flow depth
and local flow regime. As a result, the equivalent sand roughness in rivers varies both in
space and time. The problem is how to delineate the areas where the bed forms are
contributing to the resistance and where they
are not. Generally the van Rijn’s (1984b)
procedure can be used to identify the local
bed forms and the corresponding equivalent
sand roughness depending on the dimensionless grain size and transport stage parameter. In this study the average observed
energy gradient and the critical shear velocity
for the average bed grain size of the river
were used to compute the ‘critical’ flow
depth below and above which we have only
grain roughness. A polynomial fit was then
used to smooth out the spatial z 0 distribution of the neighboring cells around the
‘critical’ flow depth. The grain roughness was
taken as 3D90 . Wiberg and Rubin (1985) suggested the saltation roughness in terms of
z 0 value to be approximately 0.1 cm. Field
studies showed that the river bed for the test
case of this study is composed of three dimensional ripples and the equivalent bed
roughness due to ripples was calculated according to Grant and Madsen (1982)
⎟
⎠
In which Π is the profile parameter
and f ( H + z b ) is the wake function. The
δ
boundary layer thickness can be computed
from equation (21) for known values of free
stream velocity, U ∞ and the distance
l (Schlichting 1968).
−1
⎛U l ⎞ 5
δ = 0.37l ⎜ ∞ ⎟
⎝ υ ⎠
u*
⎛
⎛ H + zb ⎞ ⎞
− ( B − ∆B )κ − Π ⎜⎜ 1− cos ⎜⎜ π
⎟⎟ ⎟⎟
⎝ βH ⎠ ⎠
⎝
For a given set of σ - layers equation (22)
needs a minimum depth to be set so that the
first grid off the bed surface lies above z0 in
the whole computational domain.
thickness of the boundary layer. Since
ECOMSED uses the same number of vertical grids, the validity of the logarithmic law
requires the depth, ( H + z b ) , which is half
depth of the last σ - layer to be located within
the logarithmic region throughout the whole
computational domain. In rivers as there are
high spatial and temporal variations in flow
depth and flow regime satisfying the validity
limits everywhere in the computational domain is difficult if not impossible. One possible approach is to use a universal velocity
distribution law which is valid for the whole
flow depth. The validity of the logarithmic
region can be extended to the whole flow
depth by using the velocity defect law (Coles
1956), given by equation (20).
Vb
1 ⎛ (H + z b )u* ⎞
Π ⎛ H + zb ⎞
(20)
f
= ln
+ B − ∆B +
u*
υ
(21)
For rivers equation (21) is difficult to evaluate
as the velocity profiles could be skewed due
to 3-D effects and the flow field may contain
many flow separated regions. For practical
applications the boundary layer thickness can
be assumed to be related to the flow depth,
i.e., δ = β H . A new expression for z 0 can be
obtained from equation (20) for the wake law
with an empirical fit to the wake function
given by Cebeci & Bradshaw (1977)
k s = 27.7
∆2
λ
(23)
In which ∆ is bed form height and λ is bed
form length.
Shear stress partitioning
The other modification made was shear
stress partitioning (in areas where bed form is
contributing to flow resistance) to extract the
6
3-D Numerical modeling of flow and sediment transport in rivers
effective part of the stress that contributes to
sediment transport. The effective shear ve'
locity, u * , for the lower flow regime (ripples
and dunes) was calculated by using the flow
resistance relationship by Engelund and
Hansen (Yang 1996).
4
⎛
⎞
u*
⎟
u* = ⎜⎜ 0.06 g ( s − 1) D50 + 0.4
g ( s − 1) D50 ⎟⎠
⎝
0 .5
'
q b* = 0.053[( s − 1) g ] D50
0.5
' 2
*
(u
)
2
*,crbed
2
*,crbed
)
0.6
(24)
(25)
The improved model was applied to a test
case, the river Klarälven.
Model description and model boundary
The river Klarälven enters Sweden in the
north of the county of Värmland. Its course
in Värmland is south down to the river
mouth on lake Vänern at the city of Karlstad
where it bifurcates into an east and west
channels. Vänern is the largest freshwater
lake in Sweden. The course of the river is a
sequence of regular meanders that are unique
with respect to size and regularity. A one
kilometre long reach of the river was modelled that extends 0.5 km upstream of the
bifurcation (Figure 1). To insure uniform
inlet boundary conditions, the upper limit of
the reach was placed in a straight river section. The river topography was measured in
April 2004 at 11312 points scattered along
the river reach. The mean measurement
resolution ranges from 2 m to 6 m. The flow
discharges at the inlet and the two outlets
defined by east and west channels were
measured. Point velocity measurements were
also taken at the inlet at 10 m intervals across
the stream and at 0.5 m intervals in the vertical direction. To define the inlet boundary
conditions, the velocities along a vertical at
each station were fitted to the wake law velocity distribution to obtain the bed shear
velocity. A continuous distribution of the bed
shear velocity across the stream was obtained
by fitting a curve to the bed shear velocities
(26)
In which q b is the actual bed load transport,
p is the porosity of the bed material, α bξ1 and
α bξ 2 are the direction cosines in the orthogonal curvilinear coordinates ξ1 and
ξ 2 respectively. A formulation for the bed
deformation (Rodi 2000) was used.
(1 − p )
∂z b
1
(qb − qb * )
=
∂t
Ls
(29)
MODEL APPLICATION
Bed load transport and bed evolution
In rivers the contribution of bed load transport to the total load can be significant. Thus
a realistic modeling of sediment transport
would require the inclusion of the bed load
transport. This was done by including the
bed load in the sediment mass-balance equation for the bed evolution.
∂α bξ1 qb ∂α bξ 2 qb
∂z b
+E−D+
+
=0
∂t
∂ξ1
∂ξ 2
(28)
Substituting equation (27) in equation (28)
produces an elliptic equation. First order
upwind scheme discretization of the resulting
equation produces a tri-diagonal matrix
which was solved using the modified strongly
implicit procedure of Stone (1968). The bed
evolution was evaluated using the leap-frog
method using equation (27). The depth of the
water was then adjusted according to the
changes in the bed so that the flow field can
respond to the new situation.
In which u*,crbed is the critical bed shear velocity which depends on the local shear Reynolds number and was calculated using the
Shields curve for initiation of motion. The
critical shear velocity was also corrected for
the bottom and side slopes according to
Chien (1999).
(1 − p )
T 2.1
0.3
D*
Ls = 3D50 D* T 0.9
The transport stage parameter, T which is
used for the computation of suspended and
bed load was calculated using this effective
shear velocity (van Rijn 1984a and van Rijn
1984b)
(u ) − (u
T=
1.5
(27)
The equilibrium bed load transport, qb* and
the non-equilibrium adaptation length for
bed load transport, Ls were calculated according to van Rijn (1984a)
7
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TRITA LWR.LIC 2028
calculated for each station. The velocity at
different σ -layers for each computational
inlet grid was then computed using the wake
law velocity distribution with the shear velocity at the same grid. The observed water
levels at the outlet of the east and west channels were specified. River bottom above the
observed water surface was considered as dry
or inactive cell. Flow depths less than 0.2 m
were considered as 0.2 m to avoid instabilities
due to sediment deposition. At the inlet and
outlet boundaries equilibrium bed load was
assumed while zero suspended sediment
concentration was used at the inlet boundary
as sediment load data was not available.
the water division and the water surface
profile. The normalized error in the computed flow division was with in 0.8 % (West
channel) and 0.6 % (east channel) while the
normalized error in the water surface varied
with in 0.14 % in the deep areas to 3 % in the
shallow areas.
Grid Independence
A grid dependency study was carried out to
minimize numerical errors due to coarse
discretization. In the horizontal direction the
number of grids was dictated by the available
resolution of the bottom topography. Five
vertical layers were taken to start with. By
spatially varying the roughness the normalized error in the water surface profile was
made to lie within 0.2 to 5 % and the flow
division was 0.6 % in the west channel and
0.4 % in the east channel. The number of
grids in the vertical direction was increased
from 5 to 7. By using the same roughness
distribution used for 5 layers the normalized
error in the water surface profile was within
0.17 to 4 % and the flow division was 0.66 %
in the west channel and 0.48 % in the east
channel. There was no significant change in
the water surface profile and flow division.
However, the change in the shear stress was
very significant (the normalized change between the shear stress fields of 5 and 7 layers
was about 10 % and greater). When the
number of grids was increased to 9 the normalized shear stress change between 7 and 9
layers was less than 3%. The results are
summarized in Table 1. Thus, the resolution
126x48x9 is sufficient to obtain reliable numerical results.
Grid generation
The numerical model was built using the
interpolated bottom topography (Figure 1).
The horizontal orthogonal curvilinear grid
was generated by CCHE2D mesh generator
using the Poisson scheme with smoothness
and orthogonality of 91 and 95 %, respectively. The average grid spacing in the computational domain was about 6.0 m. The grid
spacing around the bifurcation and around
the bends was made finer with an average
spacing of 2.0 m.
Solution algorithm
In this study, data for discharge hydrograph
was not available and the flow simulation was
done for an observed steady flow of 285
m3/s which lasted for 2 hours. The model
was run with time steps of 0.2 seconds for
the internal mode and 0.02 seconds for the
external mode with a ramping of 10000 steps.
The model was restarted and run for extra
steps until steady state flow field was obtained. The sediment transport was then
activated and the model was run for 2 hours.
Table 1. Table showing the change in the
normalized errors and shear stress as the
number of vertical layers is increased.
Model calibration
The model was calibrated for a discharge of
285 m3/s using the measured water surface
profiles and discharge division between the
two river channels. The percentage of the
discharge into west and east channels were
42% and 58%, respectively. Model calibration
was done for a grid number of 126x48x9 by
spatially varying the roughness to satisfy both
Layers
8
Normalized
change
in
bottom shear
stress
Normalized
error
in
water
surface [%]
Normalized
error in flow
[%]
West
East
5
0.2 – 5.0
0.6
0.4
7
0.17 - 4.0
0.66
0.48
>= 10 %
9
0.23 – 5.3
0.58
0.42
<= 3 %
3-D Numerical modeling of flow and sediment transport in rivers
files. At the bifurcation point (stagnation)
the local increase in water surface elevation is
shown. The model also predicted the transverse water-surface slope at the river bend in
agreement with the physics of the helicoidal
secondary flows. The water surface superelevation depths between the outside
bank and inside bank ranged between 5 to 8
mm in the east channel bend and from 3 to 9
mm in the west channel bend. They were
comparable to approximate values of 7 mm
(East channel) and 8 mm (west channel)
computed using lppen and Drinker (1962)
method.
The velocity vectors closely followed the bed
topography. A general flow diversion from
the main channel to the east channel was
detected that explains the large portion of the
total flow entering the east channel. The flow
field showed horizontal secondary flows
following the river banks at locations where
the river topography forces flow separations
to occur. The examination of the velocity
vectors and velocity contours in different
horizontal planes showed that the flow separation regions had a complex 3D form that
extended from the water surface to the river
bed. The velocity vectors upstream of the
bifurcation also showed a significant variation with the depth. The surface velocity
vectors (Figure 4) showed no secondary
flows apart from flow line convergence due
to the local increased in the depth. However,
from the fourth sigma layer downwards large
flow recirculation regions appeared (Figures 5
and 6).
The general patterns of the secondary flows
in the vertical directions were predicted by
the model. Few examples are given in Figures 7(a-d) where the velocity vectors, velocity contours and the interpolated secondary
flows are shown. The velocity contours for
different layers are given in figures 8(a-h) to
show the general pattern of velocity variation
in the vertical direction. The bed shear
stresses were computed as total as well as
effective stresses, Figures 9 and 10 showing
the two cases, respectively. The distributions
of shear stresses are similar for both cases
although the magnitudes of total shear
stresses are about four times higher than the
Sensitivity Analysis
Sensitivity analysis was made to check how
far simulation results were sensitive to bed
roughness used for calibration. The water
surface difference between the inlet and the
outlet was found to vary approximately linearly with the roughness while the flow division in the east and west channel was found
not sensitive to changes in roughness. Successively increasing the roughness from a
z 0 value of 1 cm to 2 cm and to 3 cm produced an average water surface gradient of
0.007 %, 0.01% and 0.013%, respectively.
However, the bottom shear stress was found
to be very sensitive to the change in the
roughness. A similar increment of roughness
from a z 0 value of 1 cm to 2 cm produced a
spatial bottom shear stress variation of up to
40 %.
Validation
A comprehensive validation of the model
would require extensive field data on water
surface profiles, velocity profiles, secondary
flow structures and flow circulation regions.
For the purpose of this study only limited
data were available at two different discharge
values of 285 m3/s and 136 m3/s. The model
was validated with the field data for 136 m3/s
by using the roughness distribution obtained
during calibration for 285 m3/s. In comparison with field data, the normalized error in
the observed water surface profile varied
from 0.11 % to 1.33 %. The normalized error
in the flow division was 2.44 % for the west
channel and 1.56 % for the east channel.
Regarding the flow field, stream lines as well
as the local flow circulation region upstream
of the river bifurcation agreed well with the
field observations (Figure 2). The predicted
secondary flow fields also showed a general
agreement with the previous simulations
done by Dargahi (2004).
Flow field
The flow field in the river was successfully
simulated by the model that included the
water surface profiles, velocities, secondary
flows, and bed shear stresses. Figure 3 shows
the contour lines of the water surface pro9
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TRITA LWR.LIC 2028
effective shear stresses. As the bed shear
stress is the primary agent for sediment
transport, the difference in magnitude is
important. The results emphasis the need of
choosing an appropriate distribution for
sediment transport computations. In comparison with west channel, the shear stresses
are higher in the east channel as well as in the
deep main channel that conveys a larger
portion of the flow into the east channel. The
lower shear stress regions correspond to the
area with little sediment transport activities.
computed water surface gradient was lower
than the observed one by a factor of 0.5. This
indicated the presence of significant contribution of resistance from bed forms which
was in agreement with the observed three
dimensional ripples in the field. Moreover,
the bottom shear stress was highly sensitive
to the change in roughness height than the
water surface as discussed in the sensitivity
analysis section. Realistic representation of
roughness is needed as the bottom shear
stress is the main parameter which drives
sediment transport. Two problems appeared
when taking the bed form roughness in to
account. The first problem was related to the
higher shear velocities and the sigma levels.
Locating the last sigma layer in the logarithmic region was not possible due to high bed
shear velocity and high flow depth variation.
The high bed shear velocity required the
depth to the first grid off the surface to be
very small for the shear Reynolds number to
satisfy the validity range. Due to the sigma
transformation the depth of the last sigma
layer is directly proportional to the total
depth of the flow. Locating the last grid in
the logarithmic range in deep areas produced
a situation where the last grid is below the
roughness height in shallow areas. Hence, the
depth in the shallow areas needed to be set to
a certain depth so that the last grid stays
above the roughness height everywhere in
the computational domain. As the number of
vertical grids were increased to obtain a numerically error free grid (checking grid independence) the minimum depth to be set
needed higher values which resulted in excessive modification of the topography. High
roughness values which produced unrealistic
shear stress distribution were required to
compensate for the lowering of the river bed.
The remedy to the first problem was to use
the wake law which is applicable for the
whole depth and to set a certain minimum
depth which will not affect the bed topography so that the last grid will be above the
roughness height in the whole computational
domain. The second problem was related to
the spatial distribution of roughness due to
bed forms. Specification of bed form roughness in the whole computational domain
Sediment transport
The general pattern of bed changes after 2hours is shown in Figure 11. The figure
shows the contour lines expressed in the
units of millimeter (erosion in negative sign).
The main features are the formation of a
sandbank at the entrance to the east channel
and the division of the main channel into two
regions of erosion and deposition. The model
predicted the classical erosion patterns observed in the river bends. The maximum
erosion depth was -46.7 mm, which occurred
in the east channel near the concave curve. A
similar trend was found in the west channel
bend although the maximum erosion was -20
mm. The high values correspond to the regions of high effective bed shear stresses.
Figures 12(a-d) compare the predict bed level
changes at 3 representative river cross sections. For the ease of comparison the vertical scale corresponding to the model values
are exaggerated. The division of river into
distinct regions of deposition and erosion are
well illustrated by these figures.
DISCUSSION
Flow field: The direct application of the
ECOMSED model with no modifications
caused a number of difficulties that produced
unrealistic results both in comparison with
field data and observations and the physics of
the flow. It was not possible to calibrate the
model with a constant roughness value to
produce the measured flow division and the
right water surface profile. The right water
division was produced by modifying the
model to allow specification of spatially variable grain roughness. However, the average
10
3-D Numerical modeling of flow and sediment transport in rivers
resulted in higher shear stresses in shallow
areas. This needed delineation of areas where
the bed form is contributing to the roughness
and where the bed form is not contributing
to the roughness. The observed average
water surface gradient and critical shear velocity for the average bed grain size was used
to determine the ‘critical’ depth above which
the ripples are contributing to the total
roughness. By doing so, a discontinuity in
roughness distribution was introduced. A
polynomial function was fitted to the roughness distribution in the neighboring cells
around the ‘critical’ flow depth to smooth the
discontinuity. This procedure gave a realistic
bed shear stress distribution.
Secondary flows: The secondary flows in the
river cross-sections consisted of multiple
counter-rotating spiral motions (Figures 7ac). The spiral motions (helical cells) are
shown by circles with rotation direction
indicated by arrow. The number of cells
increases as the river bifurcation region is
approached. The increase is partly due to the
anisotropic distribution of wall shear stresses
and the unequal approach velocity. In comparison with the other regions of the flow,
the bifurcation sections had a greater spatial
variation implying an increase in secondary
flow cells. The general patterns agree with
Dargahi’s (2004) finding, although he reports
a higher number of cells especially in the
river bends. The difference between the
present results and his results can be explained partly by a higher grid resolution, and
partly by his use of different turbulence
model.
Sediment transport: ECOMSED considers only
suspended sediment transport as the contribution of bed load movement in large water
bodies such as lakes and oceans is insignificant. However, the bed load contribution can
be considerable in the case of rivers. The bed
load concentration in this study exceeded the
suspended load concentration by a factor of
10. The importance of shear partitioning can
also be seen from the figures 9 and 10. As
shown in figure 9 the total shear stress exceeds the effective shear stress by a factor of
4 times in some areas. As a result the erosion
deposition pattern was exaggerated when the
total shear stress was used. For the bed evolution, using the non-equilibrium bed load
transport in the sediment mass balance equation produced a smooth solution than directly using the bed load transport computed
with empirical formulas which assume local
equilibrium.
CONCLUSION
The study has led to a number of improvements that should increase the applicability of
ECOMSED model to rivers. These are
•
Treatment of river roughness parameterization.
• Bed form effects on the spatial and
temporal roughness distribution
• Bed shear stress partitioning.
• Addition of bed load transport
model.
• Automatic update of bed evolution.
The improved model was successfully applied to simulate flow and sediment transport
in the 1-km long reach of the River
Klarälven. The model reproduced secondary
flows at different locations in the river bank,
at the bifurcation and around the bends. The
study also confirmed the existence of multiply helical motions in the river. The erosion sedimentation patterns simulated were also
realistic. The model predicted the growth of a
large sandbank at the river bifurcation and at
the entrance to the east river channel. With
time, the predicted sandbank can cause a
serious flood problem in combination with
high flow periods.
The improved ECOMSED model is a valuable tool to deal with river engineering problems. The advantages of the model are its
flexibility and the possibilities for modifications or adding new models. The use of
ECOMSED model can eliminate the need of
expensive CFD commercial codes that have
two main disadvantages. Firstly, they are not
primary developed for rivers, and secondly a
“black box” approach does not make their
effective use possible.
11
Muluneh Admass
TRITA LWR.LIC 2028
I=1
Inlet
I = 38
I = 128
I = 82
I = 76
y
East
I = 112
West
x
Figure 1. Model boundary, river bed elevation and horizontal orthogonal curvilinear grids
(x, y and legend in meters).
y
x
Figure 2. Simulated velocity vectors and observed tracer path lines used for model validation (x, y in meters).
12
3-D Numerical modeling of flow and sediment transport in rivers
y
x
Figure 3. Simulated water surface profile showing a local increase at the bifurcation due to
stagnation, and superelevations at the bends (x, y in meters).
0.5 m/s
y
x
Figure 4. Surface velocity vectors at bifurcation showing no recirculation but showing flow
convergence (x, y in meters).
13
Muluneh Admass
TRITA LWR.LIC 2028
0.5 m/s
y
x
Figure 5. Velocity vectors at bifurcation (at the fourth sigma layer from the surface). Secondary flows appeared at this level (x, y in meters).
0.5 m/s
y
x
Figure 6. Velocity vectors at the bifurcation (at the river bed). The intensity of secondary
flows increased from the fourth sigma layer to the bottom (x, y in meters).
14
3-D Numerical modeling of flow and sediment transport in rivers
1 m/s
1 m/s
y
x
a) Section 38
1 m/s
y
x
b) Section 76
15
Muluneh Admass
TRITA LWR.LIC 2028
1 m/s
y
x
c) Section 82
1 m/s
y
x
d) Section 112
Figure 7. Vertical secondary flows showing velocity vectors, velocity contours (m/s) and
interpolated vortices at section 38, 76, 82 and 112 (x, y in meters).
16
3-D Numerical modeling of flow and sediment transport in rivers
y
x
a) Layer 1 (Surface layer)
y
x
b) Layer 2
17
Muluneh Admass
TRITA LWR.LIC 2028
y
x
c) Layer 3
y
x
d) Layer 4
18
3-D Numerical modeling of flow and sediment transport in rivers
y
x
e) Layer 5
y
x
f) Layer 6
19
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TRITA LWR.LIC 2028
y
x
g) Layer 7
y
x
h) Layer 8
Figure 8. Velocity contours in m/s for different layers showing the vertical variation of the
horizontal velocity in the whole computational domain.
20
3-D Numerical modeling of flow and sediment transport in rivers
y
x
Figure 9. Total shear stress distribution in N/m2 (x, y in meters).
y
x
Figure 10. Effective shear stress distribution in N/m2 (x, y in meters).
21
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y
x
Figure 11. Predicted bed level changes after 2 hours in mm (-ve shows erosion while
+ indicates the contour line for which the value is specified. x, y in meters).
y
x
a) Section 38
y
x
b) Section 76
22
3-D Numerical modeling of flow and sediment transport in rivers
y
x
c) Section 82
y
x
d) Section 112
Figure 12. Predicted bed level changes at section 38, 76, 82 and 112 after 2 hours simulation (x, y in meters, the predicted values are exaggerated,
Original bed level,
Predicted bed level).
23
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