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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.

5

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


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.

7


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

8


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

9


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

10


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;

11


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. )

12


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


(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


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


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.

16


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


(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


(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


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


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


, and

, which

22


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


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


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


Second term in RHS

, (2. )

, (2. )

Third term in RHS

, (2. )

Fourth term in RHS

(2. )

(2. )

(2. )

(2. )

27


Fifth term in RHS

(2. )

(2. )

(2. )

(2. )

28


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


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


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................................................................................................................................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


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

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flow, sediment, and nutrient transport in a riparian … · web viewnumerical solution the sds-2dh...

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