Chapter 2
Methodology2.1 General
The summary of the methodology used in this research is presented in Figure 2. 1.
Application Large Eddy Simulation SDS- 2DH model to Flow in Compound Channel
Numerical Simulation Experiment
· Governing EquationLESSDS-2DH Depth Integrated
· Numerical technique
· Wave Height Gauge (WHG)Measuring water level
· Electro Magnetic Velocimeter (EMV)Measuring velocity
Simulation Result Experiment Result
Comparison(Result and Discussion)
Conclusion
Figure 2. 1 Schematic diagram of the methodology.
Detailed explanation of each phase is presented in the preceding discussion.
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CHAPTER 2 METHODOLOGY
2.2 Experimental Set Up
2.2.1 Flume Description
In order to support the experimental part of this work, the tilting laboratory flume is used.
This flume is 0.40 m wide, and its overall length equals 14 m. The longitudinal slope of the
bed is 0.001. The major hydraulic variables are summarized in Table 2. 1. Sketch and
picture of the laboratory flume cross section are shown in Figure 2. 2 and Figure 2. 3.
Table 2. 1 Major hydraulic variables of experiments.
Channel Width (B)Main channel width (Bm)Flood channel width (Bf)Main channel depth (Hm)Flood channel depth (Hf)Longitudinal Bed Slope (I)
40 cm24 cm16 cm6.2 cm1.2 cm1.0 x 10-3
Figure 2. 2 Laboratory flume cross section.
A
A
16 cm
24 cm
Bf
Bm
position of WHG
1 cm
EMV, 2 cm interval
16 cm24 cm
Hf = 1.2 cm
WHG
Hm = 6.2 cm
BfBm
1 cm
1 cm
EMV
1 cm
2 cm interval
Figure 2. 3 Channel sketch, its cross section and location of measurement.
6
Bm Bf16 cm24 cm
Flow direction
CHAPTER 2 METHODOLOGY
2.2.2 Measuring Devices
The classical measuring devices used during this research are summarized in Table 2. 2.
The performed measurements include: (1) water level and (2) velocity. The water levels
were measured using an automatic wave height gauge mounted on the measurement trolley
(Figure 2. 5). The equipments are from Kenek Company. The electromagnetic velocimetry
is used for measuring velocity in the edge boundary of flood channel and main channel
(Figure 2. 4).
Table 2. 2 Measuring devices used in the flume channel.
Device Producer(serial number)
Capacity type wave height gauge (WHG)
electromagnetic velocimetry (EMV)
CHT4 30
VM-201HVMI2-200-08PS
Figure 2. 4 EMV/electromagnetic velocimetry series VMI2-200-08PS.
Figure 2. 5 WH/capacity type wave height meter series CHT4 30.
The frequency for obtaining the data is put to 100 Hz with 30 second time recording.
Velocity measurements are conducted in transverse direction of the channel for 2 cm
interval while height measurement is done for one point near the boundary of main and
flood channel. The location of both measurements can be seen in Figure 2. 3, the broken
line is for velocity measurement and the rectangular point is for wave height measurement.
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CHAPTER 2 METHODOLOGY
2.3 SDS-2DH Simulation
2.3.1 Governing Equation
2.3.1.1 Large Eddy Simulation (LES) Method
LES is a compromise between DNS and RANS. The main idea behind LES is to filter out
the fine or high frequency scales of motion and leave the large scales to be solved directly,
while the effects of the small eddies on the large eddies are modeled. This approach is
motivated by one of the most important features of turbulent flows, irregularity. Indeed,
homogenous, isotropic turbulence (when sufficiently far away from the walls) is believed
to have a random nature. The fact that it is random suggests that it has a universal character
and the effects of the smaller scales should be capable of being represented by a model and
thus predictable. On the other hand, the larger eddies in a turbulent flow are widely
believed to be deterministic, hence predictable once the effects of the smaller eddies on
them is known. Furthermore, these larger eddies are often the most important flow
structures and carry the most energy.
The LES method consist the following steps:
i) decompose flow variables into large and small scale parts, with the large scale part
purportedly defined by a filtering process;
ii) filter the governing equations, and substitute the decomposition from part i) into the
nonlinear terms to construct the unclosed terms to be modeled;
iii) model these unresolved stresses;
iv) solve for the large-scale contribution (while essentially ignoring the small-scale part).
LES decomposition
The LES decomposition was introduced by Deardorf [2] and was first analyzed in detail or
the incompressible Navier Stokes equations by Leonard [7]. It is constructed by applying a
local spatial filter (or in the simplest case, spatial average) to all appropriate variables. The
LES is written decomposition as
(2. )
In this decomposition is usually termed the large or resolved scale part of the solution,
and is called the small-scale, or subgrid-scale (SGS), or unresolved part. It is important
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CHAPTER 2 METHODOLOGY
to note that both resolved and unresolved scales depend on both space and time, and this is
a major distinction and advantage compared with the Reynolds decomposition.
Filter
In LES a low-pass, local, spatial filter is applied to the Navier-Stokes equations, instead of
an ensemble or temporal average. The main idea is similar to that of Reynolds-averaging in
which the equations governing the mean components of the flow are derived. The mean
components can be thought of as the largest of the scales in the turbulence. With spatial
filtering, the equations governing the larger components of the turbulent scales are
derived. A Filtered variable results from the convolution of a resolved variable with a
filter kernel as shown in (2. ):
(2. )
The filter kernel, , is a weighting function whose support varies depending on the
filter type. The most commonly used filters in LES are the Tophat, Gaussian, and Sharp
Spectral filters (Ikeda, [4]).
The effect of filtering can be seen in the sketch shown in Figure 2. 6, which the filtered
component of a function and the original function are depicted. The filtering operation
serves to damp scales on the order of the filter width denoted as . The width is a certain
characteristic length of the filter. The filter kernel is scaled such that if the
function to be filtered is a constant, the resulting filtered function is that same constant.
Figure 2. 6 Sketch of function and its filtered component .
In equation (2. ) is formally the filtered solution corresponding to equation (2. ). From
this it is easily shown that, in general, and
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CHAPTER 2 METHODOLOGY
GS element
The filtering method described above is applied to the Navier-Stokes equations, which now
describe only the motion of the large scales.
Continuity Equation
The continuity equation for incompressible fluid is written as:
(2. )
Because the continuity equation is linear, filtering does not change it significantly:
or (2. )
Momentum Equation
The momentum equation is filtered in the same manner. The obtained equations may be
written as:
(2. )
Where :
(2. )
(2. )
The last term in equation (2. ) appears additional term, which called subgrid scale (SGS).
This additional terms need to be modeled.
Subgrid Scale (SGS)
As described above, Lij , Cij , Rij are the subgrid scale (SGS) Leonard, Cross and
Reynolds stresses, respectively. The Leonard stresses represent the interaction among the
resolved scales and can be computed directly. The Cross terms represent the interaction
among the resolved and unresolved scales while the Reynolds stresses describe the
interaction among the unresolved ones. In RANS modeling, the Leonard and Cross terms
go to zero. This is in general the case for LES, although using the cutoff filter in spectral
space results in only the Reynolds term. The decomposition affects the derivation of the
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CHAPTER 2 METHODOLOGY
turbulent kinetic energy equations. Many modeling approaches guided by RANS modeling
are based on only the Reynolds terms.
The Leonard term and Cross term are approximately equal. They were typically dropped
from consideration because their order of magnitude was the same as the order of
magnitude of the discretisation error. The last, Reynolds- stresses need to be modeled.
Smagorinsky model is used to solve the remaining term. This model is based on Eddy
viscosity concept as written as:
(2. )
(2. )
(2. )
By applying the Smagorinsky model and only Reynolds stress affected, equation (2. )
becomes:
(2. )
2.3.1.2 SDS-2DH Equation
As the phenomenon to be investigated is mainly two-dimensional, a depth-averaged model
will be preferred to a complete three-dimensional model solving the Navier- Stokes
equations, in order to limit the programming complexity and the computational cost. The
model that will be used is the so-called SDS-2DH model, originally proposed by Nadaoka
and Yagi [12]. This model, whose principle will be described below, produces indeed
satisfactory results when modeling horizontal vortices due to transverse shearing in partly-
vegetation-covered channels.
According to Nadaoka and Yagi [12], the turbulence structure of a shallow-water flow is
characterized by the coexistence of 3D turbulence, having length scales less than the water
depth, and horizontal two-dimensional eddies with much larger length scales. As a result,
the spectral structure of such a flow can be depicted as on Figure 2. 7: a first peak
corresponds to the horizontal 2D vortices generated by the transverse shearing. In this area,
an inverse cascade of spectral energy can be observed, due to processes like vortex pairing;
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CHAPTER 2 METHODOLOGY
while a direct attenuation also exists, due to dissipation by bottom friction. A part of this
dissipated energy may be supplied to 3D turbulence, at higher wave-number ; while
bottom friction may also directly provide 3D turbulent energy.
Figure 2. 7 Turbulent energy spectrum in a depth-averaged flow with a shear layer, according to Nadaoka
and Yagi [9].
This proposed SDS-2DH model, in principle, is similar to Large Eddy Simulation (LES),
according to the length scales to be modeled. Indeed, similarly to the SDS-2DH model,
LES models solve explicitly the large turbulence scales, while the smaller scales are
modeled implicitly, using a so-called subgrid model. However, when the grid size reduces,
LES results tend towards the results obtained from a Direct Navier-Stokes (DNS)
simulation, in which all turbulence scales are modeled, from the larger one to the smaller
one, which corresponds to molecular dissipation. This means that, when decreasing the
grid size, an LES subgrid model will converge towards molecular viscosity.
Based on equation (2. ) and (2. ) the SDS-2DH equations will be derived. Rewrite these
equations as written as:
(2. )
(2. )
(2. )
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CHAPTER 2 METHODOLOGY
Definitions Sketch
Figure 2. 8 Definition sketch of the axis directions and velocity components.
Where x, y and z are respectively the longitudinal, transverse and vertical directions; u, v
and w are the local velocity components, respectively in the x-, y- and z-directions (see
Figure 2. 8); p is the pressure; is the density of water; g is the gravity constant; and is
the molecular viscosity.
A spatial filtering (LES method) has been used where , , and are the resolved
(filtered or larger) components and , , and are the residual (subgrid or smaller)
components. These forms are similar to that of Reynolds-averaging where , and are
the Reynolds averaged velocities; and u', v' and w' are their turbulent fluctuations, whose
products define Reynolds turbulent stresses. In the present work, the shear stresses due to
molecular viscosity will be neglected compared to the Reynolds stresses, as they are
usually several orders of magnitude smaller.
The depth-integrated will be performed along the z-direction, between the bed level –h and the free-surface
water level . The depth-integrated longitudinal U and transverse V velocity components are thus defined as
:
13
H
uv
w
xy
z
h
CHAPTER 2 METHODOLOGY
(2. )
The total water column height is defined as:
(2. )
Free Surface Boundary
The free-surface boundary condition is defined by assuming that a particle present on the surface at a given
time will remain on it. The free-surface is thus defined by
S(x,y,z,t)= η (x,y,t)-z = 0 (2. )
simply expressing that the variable z gets the value defining the free-surface. The
substantial derivative D/Dt of this equation equals zero, which means that a particle on the
free-surface remains on the surface, giving thus
: , , hence
(2. )
Bottom Surface Boundary
The bottom boundary condition is obtained similarly :
S(x,y,z,t)= z0 (x,y,t)-z = 0, with :
at z = z0
The velocity in the bottom is: U(-h) = V(-h) = W(-h) = 0, then
(2. )
14
CHAPTER 2 METHODOLOGY
a. Depth Averaged Continuity Equations
Integrating the continuity equation (2. ) along the depth gives
where the integration and differentiation operators have to be inverted using the Leibnitz rule :
(2. )
(2. )
The three terms in the left-hand side of (2. ) are thus written as
Grouping again those three terms, and using the definitions of depth-integrated velocities
(2. )
U and V given by (2. ), boundary conditions given by (2. ) and (2. ) the continuity equation becomes
(2. )
(2. )
b. Depth Integrated Momentum Equations
When the momentum equation in the x-direction (2. ) is integrated along the depth z,
15
CHAPTER 2 METHODOLOGY
one obtains(2. )
Term 1
As for the continuity equation, the Leibnitz rule is used to invert the integration and
derivation operators. Using the fixed bed hypothesis and the definition of depth averaged
longitudinal velocity U (2. ), the acceleration term, the first term in the left hand side of
(2. ), gives
(2. )
Term 2
The first convection term , the second term in the left-hand side of (2. ), gives
(2. )
The integration of the velocity product in the first term of the right-hand side of (2. )
will generate the first dispersion term. Indeed, one expects to express this term as a
function of the depth-averaged longitudinal velocity U. The local velocity varies along
the depth z (Figure 2. 9). The depth-integration of its squared value is thus different from
the square of the depth-averaged velocity U. Several authors suggest to use the so-called
Boussinesq coefficient in order to take into account this difference (Liggett [11]) :
However, most of these authors then assume that this Boussinesq coefficient equals = 1,
neglecting thus the dispersion effect.
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CHAPTER 2 METHODOLOGY
Figure 2. 9 Typical vertical velocity profiles.
The integration of the square of in (2. ) can be written as :
(2. )
where the second term in the right-hand side equals zero, as the integration of along the
depth equals U; and the third term is the so-called dispersion term. Equation finally gives
(2. )
Term 3
In the same way, the second convection term in (2. ) term 3) becomes
(2. )
Term 4
The third convection term of (2. ) term 4) simplifies to
(2. )
Term 5
The Leibnitz rule applied to the pressure term in (2. ) term 5), first term in the right-hand
side, gives :
where the pressure pa at the free-surface is set equal to zero.
17
CHAPTER 2 METHODOLOGY
(2. )
where the x-direction (longitudinal) channel bed slope can be defined as
(2. )
Term 6
Lastly, using again the Leibnitz rule, the shear-stress terms become:
(2. )
It is then assumed that the shear stress at the free-surface is negligible. The second, fifth
and seventh term in the right-hand side of (2. ) equal thus zero. On the other hand,
regarding the shear stresses at the bed, the third and sixth terms (stresses along vertical
planes) will be assumed negligible compared to the eightieth term (stress along the
horizontal plane). The shear stress terms (2. ) reduce thus to
(2. )
The depth-averaged x-wise momentum equation (2. ) is obtained by the addition of (2. ),
(2. ), (2. ), (2. ), (2. ) and (2. ) :
18
CHAPTER 2 METHODOLOGY
(2. )
where the two last terms of the left-hand side equal zero, due to the boundary conditions at
the free surface (2. ) and the bed (2. ). Using the definitions (2. ) of the bed slope S0 and
grouping the x-derivatives, one obtains
(2. )
The so-called "non-conservative" form of (2. ) is obtained by subtracting the continuity
equation (2. ) multiplied by U, and by dividing the resulting equation by H :
(2. )
(2. )
Reynolds stresses is defined as :
(2. )
where t is the eddy viscosity; ij is the Kronecker symbol (ij = 1 for i = j; and ij = 0 for
i j); and k is the kinetic turbulent energy.
, (2. )
(2. )
(2. )
19
CHAPTER 2 METHODOLOGY
Where is defined as bottom stresses due to bottom friction and vegetation drag.
Writing back equation (2.39), one gets at last:
(2. )
(2. )
c. SDS Turbulence
The depth-averaged kinetic energy of SDS turbulence, k is evaluated with the following
energy-transport equations:
(2. )
The eddy viscosity and the energy dissipation rate are evaluated by k and l according
to the usual k-equation model.
(2. )
(2. )
For the model parameters, , and , the standard values , and
are adopted here.
The turbulence length-scale l is expressed as
, which
20
CHAPTER 2 METHODOLOGY
Pkh and Pkv are calculated with the following relations from Rastogi and Rodi [13] with
additional term in Pkv due to vegetation drag by Ikeda [13]:
(2. )
(2. )
The Pkh term corresponds to the turbulent kinetic energy production, due to the interaction
between the turbulent shear stress and the depth-averaged velocity gradient.
The terms Pkv is source term, who absorb all the secondary terms originating from non-
uniformity of vertical profiles. The main contribution to this term arises from significant
vertical velocity gradients near the bed. It expresses therefore the turbulent kinetic energy
production due to bed friction and vegetation drag.
SynthesisAs a result, the SDS-2DH equations can be summarised as:
21
CHAPTER 2 METHODOLOGY
, and
, which
22
CHAPTER 2 METHODOLOGY
2.3.2 Numerical Solution
The SDS-2DH equation is solved with finite difference method which successive over
relaxation (SOR) is applied to numerical computation. There exist a number of approaches
for the discretization of those equations. A stable finite difference method is based on
using a so called staggered grid (type Arakawa C, McKibben , J. F. [10]), when the
unknown variables u, v and lie at different grids shifted with respect to each other.
Figure 2. 10 shows the staggered grid scheme. That simple model of staggered grid gives
possibility to use simple discretization and prevent numerical instabilities forming within the
model.
The first spatial discretisation makes use of a staggered "marker-and-cell" (MAC) mesh
(Bousmar, D. [1]), slightly adapted for shallow-water flow modeling. In such a mesh, the
velocities u and v are defined for positions situated at a middle distance between the points
where the water level are defined (Figure 2. 10). This location enables an easy
estimation of the water level value at any point of interest ( , U, V) using a linear
interpolation. Such a staggered mesh provides a good coupling between the velocities and
the water depth, insuring a very good mass and momentum conservation during the
resolution, this condition is indeed required for the uniform-flow modeling with cyclic
boundary condition.
Additionally, the values of the viscosity t, and of the turbulent kinetic energy k are
defined at the same locations as the water level . Each equation from 2.24, 2.44, 2.45 and
2.49 are then discretised with a computational molecule centred on the location where the
value varying with the time is defined : on the water-level definition point for the
continuity equation (2.24) and for the turbulent kinetic energy transport equation (2.49); on
the longitudinal-velocity U definition point for the x momentum equation (2.44); and on
the transverse-velocity V definition point for the y momentum equation (2.45).
Momentum equations are written using upwind sheme while the first order derivative in
the continuity equation is written using centred difference operator. When the value of a
variable is needed on a point different of its definition point, this value is interpolated from
adjacent values.
23
CHAPTER 2 METHODOLOGY
Figure 2. 10 Staggered Grid MAC (Marker And Cell).
a. Continuity equation discretization
The continuity equation discretized on staggered grid can be written as follows:
(2. )
24
dy
i-2
y
x
u
vi-1 i i+1 i+2
j+1
j
j-1
dx
CHAPTER 2 METHODOLOGYi-2 i-1 i i+1 i+2
dy
y
x
dx
i-2 i-1 i i+1 i+2
j+2
j+1
j
j-1
j+1u
v
ui,j ui+1,j ui+2,jui-1,j
Hi-1,j Hi+1,jHi,j
vi,j
vi,j+1
vi,j+2
vi,j-1
Figure 2. 11 Points on a grid used for continuity equation solving.
b. Momentum equation discretization
The momentum equations discretized on staggered grid can be written as follows:
X- direction
(2. )
(2. )
Y- direction
(2. )
(2. )
25
CHAPTER 2 METHODOLOGY
Term in Left Hand Side (LHS)
(2. )
(2. )
First term in Right Hand Side (RHS)
(2. )
(2. )
Similar for y direction:
(2. )
(2. )
In this computation is been used. The scheme is called K-K (Kawamura-Kuwahara)
scheme.
26
CHAPTER 2 METHODOLOGY
Second term in RHS
, (2. )
, (2. )
Third term in RHS
, (2. )
Fourth term in RHS
(2. )
(2. )
(2. )
(2. )
27
CHAPTER 2 METHODOLOGY
Fifth term in RHS
(2. )
(2. )
(2. )
(2. )
28
CHAPTER 2 METHODOLOGY
c. Turbulent kinetic-energy transport (k) equation discretization
The turbulent kinetic-energy transport equations discretized on staggered grid is written as
follows:
(2. )
(2. )
Term in Left Hand Side (LHS)
(2. )
First term in Right Hand Side (RHS)
(2. )
(2. )
29
CHAPTER 2 METHODOLOGY
Second term in RHS
(2. )
(2. )
Third term in RHS
(2. )
(2. )
Fourth term in RHS
(2. )
(2. )
Fifth term in RHS
(2. )
(2. )
With stability criteria:
(2. )
30
CHAPTER 2 METHODOLOGY
2.3.3 Computational Condition
Simulation is done with the following condition:
Table 2. 3 Computational domain and grid size and time step
Channel Width (B)
Slope (I)
Main channel depth (Hm)
Roughness (n)
Longitudinal domain size
Longitudinal grid size ( )
Transverse domain size
Transverse grid size ( )
Time step ( )
40 cm
1.0 x 10-3
6.0 cm
0.0103
15 m
1.0 cm
40 cm
0.5 cm
0.01
A white noise, the magnitude of which is 1% of the velocity of flow in the main channel, is
imposed at the boundary to stimulate the development of horizontal vortices.
31
CHAPTER 2 METHODOLOGY
Chapter 2................................................................................................................................5
Methodology..........................................................................................................................5
2.1 General...................................................................................................................5
2.2 Experimental Set Up..............................................................................................6
2.2.1 Flume Description..........................................................................................6
2.2.2 Measuring Devices.........................................................................................7
2.3 SDS-2DH Simulation.............................................................................................8
2.3.1 Governing Equation.......................................................................................8
2.3.1.1 Large Eddy Simulation (LES) Method......................................................8
2.3.1.2 SDS-2DH Equation..................................................................................11
2.3.2 Numerical Solution......................................................................................23
2.3.3 Computational Condition.............................................................................31
32
CHAPTER 2 METHODOLOGY
Figure 2. 1 Schematic diagram of the methodology.........................................5
Figure 2. 2 Laboratory flume cross section...........................................................6
Figure 2. 3 Channel sketch, its cross section and location of measurement....................................................................................................................6
Figure 2. 4 EMV/electromagnetic velocimetry series VMI2-200-08PS.....7
Figure 2. 5 WH/capacity type wave height meter series CHT4 30............7
Figure 2. 6 Sketch of function and its filtered component ................................9
Figure 2. 7 Turbulent energy spectrum in a depth-averaged flow with a shear layer,
according to Nadaoka and Yagi [9].....................................................................................12
Figure 2. 8 Definition sketch of the axis directions and velocity components....................13
Figure 2. 9 Typical vertical velocity profiles.......................................................................17
Figure 2. 10 Staggered Grid MAC (Marker And Cell)........................................................24
Figure 2. 11 Points on a grid used for continuity equation solving.....................................25
Table 2. 1 Major hydraulic variables of experiments........................................6
Table 2. 2 Measuring devices used in the flume channel................................7
Table 2. 3 Computational domain and grid size and time step...................31
33
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