2013-Numerical Solution of River Flood and Dam Break Problems by Cell Centred Finite Volume Scheme

8/18/2019 2013-Numerical Solution of River Flood and Dam Break Problems by Cell Centred Finite Volume Scheme

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NUMERICAL SOLUTION OF RIVER FLOOD AND DAM

BREAK PROBLEMS BY CELL CENTRED FINITE VOLUME

SCHEME

Dantje K. Natakusumah1*, M. Syahril Badri Kusuma1, Dhemi Harlan1,

M. Rizky Ramadhan1, and Bobby Minola Ginting2

1Civil Engineering Department, Institute of Technology Bandung

5Civil Engineering Department, Parahyangan Catholic University, Bandung

*Email: [email protected]

Abstract

This paper presents the development of a numerical solution for simulating ood phenomena caused by dam break problem. This dam break phenomena should

be examined because of the potential damages and on the greater dam can cause

casualties. This numerical solution presented in this paper is developed using cell-

centered nite volume scheme rst introduced by Jameson (1981) for solving the

two dimensional compressible Euler equations (1981). This numerical solution has

been implemented for several cases of dam break and the ow on a channel and

the results has been compared with the results obtained by other researchers. The

conclusion from this comparison is the simulation results shows good agreement.

This numerical solution is then applied in a hypothetical case at the Lawe-Lawe

Dam and being a part of mitigation effort just in case the disaster phenomena’shappen.

Keywords: Finite Volume Method (FVM), Jameson Scheme, Unstructured Grid,

St. Vennant Equation, nearly dry bed ow problem and Dam Break

Problems.

INTRODUCTION

Background

One form of the utilization of water resources is the construction of dams in order

to use water as an energy source for power generator, irrigation and water supply

during dry season. The dam construction has been carefully planned in advance but

the physical changes around the dam either naturally or due to human intervention

will gradually affect the condition of the dam itself. The most dangerous impacts

caused by changes condition of the dam is a dam failure or dam break, which could

cause ooding and even in large dam can cause the loss of life. Dam break was

caused by overtopping due to the capacity of the dam no longer accommodate the

increased discharge into dams, collapse, earthquakes and liquefaction or landslides

on the dam. To reduce the impact of dam break problems a concept of numerical

methods as a tool for dam break mitigation was developed.


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By applying the Gauss divergence theory, Equation (4) converted to surface integral

that can be written as:

∫∫ Ω=∫ Γ⋅+∫∫ Ω∂∂

ΩΓΩ

dSdnHdWt

................................................................... (5)

Where n is a normal vector of surface Γ and it value is:

............................................................................................ (6)

So that equation (6) can be written as:

.................................................................. (7)

Space Discretization

FVM space discretization process begins by dividing a large domain into smaller

sub domains. Figure 1 shows a domain Ω is divided into a small cell volume Ω1,

Ω2 and Ω

3. In practice, the space discretization mostly performed using triangular

or square shape cell or a combination of the two (hybrid grids).

Figure 1 : Division of domain Ω into a number of sub-domain or cell.

By referring to Figure 1, Equation (7) can be written as:

........................................................... (8)

Second part on right hand of Equation (8) is convective term which obtained by

count the ux that passing domain or can be written as:

........................................................(9)

Variable W on Equation (9) is placed at the center of the cell volume and assumed

to represent the average value of W on the cell volume Ωi,j and is dened as:


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.....................................................................................(10)

Where Ai,j is an extensive cell volume Ω

i,j and calculated by the trapezoidal rule

formula:

( )( )∑=

++ ++≈

D

Ak

k k k k ji y y x x A 11,2

1.............................................................. (11)

The Convective terms on Equation (9) was approximated by the trapezoidal rule

formula as

............................................................ (12)

Form Equation (10), ux is calculated depends on the number of elements surrounding

domain Ω. By assuming cell volume Ai,j to be constant and by ssubstituting Equation

(11) and (13) into equation (11) so we can obtain discretization space for FVM as

follows:

( ) ji ji

DA

ABk

k k k k ji ji S A xG y F W t

A,,,,

⋅=∆−∆+∂

∂∑=

........................................... (13)

Calculation of the convective on FVM highly dependent on the ux that passes

through the cell volume or expressed by:

( ) ( )∑=

∆−∆= Np

i

iiiik xG y F W C 1

......................................................................(14)

Where N p is a number of edges that form the cell volume k and C(W

k ) is a convective

operator which is discrete approached equilibrium ux through all sides of the

cell volume k . By following the work of Nuradil (2004) and Natakusumah et al.

(2004), a velocity ux Qi for the ux that passing through the cell volume side- I is

introduced as:

iiiii xv yuQ ∆−∆= ...................................................................................(15)

The convective operator for cell volume k can be written as follows:

..................................................................(16)


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So that equation (16) can be written as:

( ) ( ) k k k k k S AW C W A =+∂∂

......................................................................... (17)

Or in expanded the above equation can be written as

.................... (18)

Equation (16) and (17) does not have natural viscous term that can reduce oscillations

naturally when hydraulic jump occurs. The addition of articial numerical dissipation

performed to reduce oscillations that arise and nd solutions that are not defective.

The addition of numerical dissipation can be written as follows:

( ) ( ) ( ) k k k k k k S AW DW C W t

A =−+∂

∂ ......................................................... (19)

where C(W k ) is convective operator, while D(W

k ) is an articial numerical

dissipation operator. The concept of articial numerical dissipation developed by

Jameson for the Euler equations has been successfully implemented by Nuradil

(2004) and Natakusumah et.al (2004) for shallow water problems governed by St.

Vennant equations.

Following these earlier works, articial numerical dissipation operator in Equation

(18) consists of the combined of second order and fourth order dissipation terms

dened as follows:

( ) ( ) ( )k k k W DW DW D 42

−= .......................................................................(20)

Where operator D2(W k ) and D4(W

k ) are Laplacian and Biharmonik dissipation

operator.

................................................................(21)

The Biharmonic operator was added so that the calculation is smoother but turned

off in the vicinity of hydraulic jump and the Laplacian operator turned on to reduce


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oscillations. Substitution is performed by using a hydraulic jump sensor and

expressed by:

................................................................................ (22)

................................................................................ (23)

Time Discretization

The time step calculation uses the explicit three-stage Runge-Kutta method. Theidea of this method is to calculate the value of n

k W on the right hand of Equation

(24) in the interval n∆t and (n+1)∆t to obtain the value of 1nk

W + . In this research is

used the three stage Runge-Kutta methods in three stages as follows:

( )

( ) ( ) ( )( ) ( )( )[ ]

( ) ( ) ( )( ) ( )( )[ ]

( ) ( ) ( )( ) ( )( )[ ]

( )31

02

3

03

01

2

02

00

1

01

0

k

n

k

k k

k

k k k

k k

k

k k k

k k

k

k k k

n

k k

W W

W DW C A

t W W

W DW C

A

t W W

W DW C A

t W W

W W

=

−∆

−=

−∆

−=

−∆

−=

=

+

α

α

α

.................................................. (24)

With coefcients α1, α2 and α3 as follows:

6.01 =α 6.02 =α 0.13 =α

Initial and Boundary ConditionThe initial condition used is cold start with prescribed water depth at various points.

There are two kinds of boundary conditions, wall boundary conditions and ow

boundary conditions. In wall boundary condition, there is no ow that penetrates

the surface of the wall. This condition expressed by normal velocity ux ow across

the surface of the wall is equal to zero. Inow and outow boundary conditions are

determined by using the characteristics method.


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Wet and Dry Treatment

The technique used in this paper to solve the problem of nearly dry bed is the

porosity function approach proposed by Casulli (2008) which stated as follows:

( ) ( )

otherwise

z y xh z y x p

0

0,1,,

>+=

( ) Ω∈ y x, ....................................................(25)

Where the integral evaluation horizontally every cell volume on ni

z η = expressed

as follows:

( ) ( )∫Ω

Ω= d x y x p p i ,,η ...............................................................................(26)

Equation (23) show as p(η) = 0 the cells is in the dry state otherwise as p(η) = P 1 the

cells volume is in the wet state. Thus, the total depth H(x, y, z) at each cell volume

expressed as:

........................................... (27)

RESULTS AND DISCUSSION

The results presented in the following paragraphs were taken from master thesis

and earlier papers written by Bobby Minola Ginting (use Quadrilateral mesh

only) and master thesis of Mochamad Rizky Ramadhan (use both Quadrilateral

and Triangular mesh). They work using completely different computer code but

both are based on the same Jameson Scheme.

Case 1: Flow over a Triangular Obstacle (Small Scale) Case

Ramadhan (2013) has successfully implemented his nite volume code to simulate

the experimental set up for testing the dam break ow over a bump proposed by

Hiver (2000) which involving dry bed. This aim to improve the results obtained

earlier by Ginting (2011) and there are some improvement although they are not

always in a perfect agreement with experiment. In this simulation, the domain is set

as rectangular channel with 5.60 m length and 0.50 m width and discritized using

quadrilateral mesh (1120 cells and 1243 points). The reservoir is lled by 0.111 m

water depth, meanwhile the water depth at the channel before the triangular bed is

set as zero. After the triangular bed, the channel contains 0.025 m water depth and

downstream part is closed by wall. In other words, the domain is a closed system

with a wall at upstream and downstream. The symmetrical bump is 0.065 m highand has bed slopes of ± 0.14.


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Figure 2 : Numerical simulation results of ow depth along the channel at (a) t =

1.8 s, (b) t = 3 s, (c) t = 8.4 s and (e) t = 15.5 s.

From the simulation results, after the opening of the gate, water ows suddenly

with high velocity on dry channel, and while reaches the upstream bump, the wave

is reected back in upstream direction. After 1.8 s, the wave propagation is formed

on an upward dry slope, and after it reaches the top of bump, the water starts to

ow downward on dry slope, until reaching the wet bed on the downstream. At

3.0 s – 3.7 s, the water reaches the downstream wall, and there is a reected wave

in upstream direction. At 8.4 s, the water depth on upstream and downstream of

bump is relatively the same, but the water from the top of bump starts to ow

downward, and it clearly depicted by the results at 15.5 s. From this result, there is

a little difference between numerical and experimental results on downstream at 3.0

s and 3.7 s. It can be caused by the complexity in determining the wall boundary

conditions.

Case 2: Reservoir and L-Shaped Channel Case

Ramadhan (2013) used quadrilateral mesh (906 cells and 1021 points) to improve

the results obtained by Ginting (2011) in modeling model experimental test proposed

by CADAM team which aimed at verifying the capability of numerical schemes

to simulate the dam break ow problem. There some improvement in the results

although they are not always in a perfect agreement with experiment. The bed ofreservoir is 0.33 m lower than the channel bed. The reservoir and the channel are


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separated by a gate, which is opened suddenly to produce dam break ow. The initial

water depth in the reservoir is 0.53 m, and 0.0 m. This laboratory result has been

compared before by Soarez (1998) and Tahershamsi (2010), etc. In this research,

Manning coefcient is used 0.0095. The results shown there is a little difference in

the results from the lab experiments but obtained the same pattern in this case. The

biggest difference was found in G2 point where the difference of 0.05 m between

the simulation results and lab experiments was observed. At another point there is a

little difference but still have the similar pattern.

Figure 3 Mesh used in the numerical simulation and comparison results of ow

depth along the channel at points G1, G and G4.

Case 3: Circular Dam Break Problem

In order to test the capability of the presented scheme, we consider a circular dam- break test case involving a water cylinder dam with the radius at the center of

a square computational domain of 40 m by 40 m. The computational domain is

divided into 2085 points connected to form 4008 triangular elements. The initial

water depth inside the cylinder dam is 2.5 m, and the water level outside the dam is

0.5 m. At time t = 0 s, the cylindrical wall dam in middle collapses instantaneously

and then a circular wave propagates in all radial direction.

In this case the numerical simulations will be compared with the results of

numerical simulations using much better nite volume scheme based on WeightedAverage Flux (WAF) approach used by Loukili and Soulaimani (2007). Based on


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the graph shown in Figure 4, at t = 0.4 s shock wave propagates outward dam and

rarefaction wave propagates inward and reach the middle of the dam. At t = 0.7 s

rarefaction wave reection causes the water level falls in the middle dam. At t =

1.4 s rarefaction wave reections resulting water level is lower than the elevation

around the dam and also began forming a second shock wave. After that, at t = 3.5

s, the second shock wave propagates out of each dam and at t = 4.7 s shock wave

reection occurs in the middle and continue to propagate out of the dam.

Please note that the differences between the two results lie in the speed of change

in water level and the shock wave propagation. Differences in changes of water can

be observed from at t = 0.4 s which shows that the results of Jameson Scheme was

lower than that of Superbee scheme simulations. Although there are differences, both shows similar pattern. This show that the numerical model used in this study is

good enough to simulate the process of formation of a complex wave.

Figure 4. Mesh used for the simulation and comparison of present numerical results

with that of WAF Superbee Scheme at t=0 s, t=0.4 s, t=0.7 s, t=3.5 s and t=4.7 s.

Case 4: Application on Lawe-Lawe Dam Break Problem

Ramadhan (2013) made an attempt to simulate hypothetical case of Lawe-Lawe

dam break problem as part mitigation effort. After Situ Gitung disaster, dam break

simulation is now become important part of dam construction plan especially when

there is a settlements downstream of the location of the dam. In performing this

application topographic data is availability. Lawe-Lawe dam will be built with high

+14 m so the simulation domain used is the contour elevation at 13 m height and

assume that sections of the dam that collapse along the 30 m. The computational

domain is divided into 1656 points connected to form 1550 quadrilateral elements.


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a) Conguration without Building

In this section the comparison between numerical model and laboratory results is

presented. The computational mesh is taken as rectangular grid. The initial condition

is set as 30 cm at reservoir. The time step is set as 0.01 s. The value of D min is set

as 0.001 mm. The average Manning coefcient is set as 0.010 since the material of

channel bed is steel. Measurement points are taken at reservoir (-2 m), 1E5 (+0.45

m), 3E5 (+2.45 m), 5E5 (+4.45 m), 7E5 (+6.45 m) and 8E5 (+7.45 m).

Figure 6 : Time Evolution of Water Level for Conguration without Building

Figure 6-a show the emptying curve of reservoir and the numerical model agrees

with the experimental results, it is clearly shown that there is no signicant difference

between them. Figure 6-b show that numerical model gives the bigger value than

experimental results, especially for time 1 – 5 s but for time 10 – 40 s, the numerical

model gives the good results. Figure 6-c - Figure 6-f show the numerical model

gives the slightly difference results with the laboratory model especially for 1 – 5

s. After 10 s, the numerical model shows good agreement results with laboratory

model that indicated by no-signicant differences.

b) Conguration with 3 Buildings

The initial condition is same with previous case, except the 3 buildings are taken

at the channel on 2.55 m in front of the gate. Numerical model shows that after 2

s, water crash the buildings and the water depth at the upstream of them becomes

about 25 cm, while 3 cm approximately at downstream of the building 1 (3F7) and

building 3 (3F3) and about 4 cm at downstream of building 2 (3F5). On grid 3F4

and 3F6, the water depth is approximately 16 cm. In this case, the backwater effect


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due to building is more signicant. It is clearly shown from Figure 8 that during

3 - 5 s, the backwater affects the ow further towards upstream of the channel.

Also, the articial viscosity shows the good performance in handling the numerical

instabilities and shock wave phenomenon.

Figure 7 : Time Evolution of Water Level for Conguration with 3 Building

c) Conguration with 5 Buildings

The 5 buildings are taken on the channel with the non-uniformly layout, while theinitial condition is set same with the previous case. Numerical models shows that

after 2 s, water crash the rst 3 buildings and the water depth at the upstream of

them becomes about 25 cm and this result does not differ much with the previous

case (conguration with 3 buildings). The signicant difference is shown from the

water depth at the downstream of 3 rst buildings.

Figure 8 : Time Evolution of Water Level for Conguration with 5 Buildings


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CONCLUSION AND RECOMMENDATION

The simulation results described earlier shows that Finite volume method with

three Runge-Kutta method for time dicretization and articial numerical dissipation

give the good results in solving the dam break problem with nearly dry bed. Due to

explicit nature of the computation, this method is much affected by time stepping

regarding its stability computation and nishing time. There is no strict rule in

determining the value of Dmin regarding wet and dry treatment. The oscillation

still appears especially for the case with high shock wave therefore the other kind

of numerical dissipation may be adopted. The grids should be built smaller to give

the better result, but it may need more computational time and greater computer

memory.

ACKNOWLEDGEMENTS

The authors are grateful to LPPM-ITB for funding the research on “The application

of nite volume method for mitigation of disaster due to ood and dam break

problems”, provided by the Institute of Technology Bandung, through the ITB

Innovation Research Program 2012.

REFERENCESCasulli, V., (2008), “A High Resolution Wetting and Drying Algorithm for Free

Surface Hydrodynamics”, International Journal for Numerical Methods in

Fluids, 2008

Ginting, B.M., (2011), “Two Dimensional Flood Propagation Modeling Generated

by Dam Break Using Finite Volume Method”. Master Theses, Institute of

Technology Bandung, Desember 2011, Bandung, Indonesia.

Ginting, B.M., Natakusumah, D.K., Kusuma, M.S.B., Harlan, D. (2011), “Model

2 Dimensi Propagasi Aliran Banjir Akibat Keruntuhan Bendungan Dengan

Metode Volume Hingga”. Konferensi Nasional Pasca Sarjana Teknik Sipil,Desember 2011, Bandung, Indonesia.

Jameson A., Schmidt W., and Turkel E. (1981), “Numerical Solution of the Euler

Equations by Finite Volume Methods Using Runge-Kutta Time Stepping

Schemes”, AIAA paper, 1981.

Natakusumah D. K., 1992, “Finite Volume Solutions of The Two-dimensional

Compressible Flow Equations on Structured, Unstructured and Hydrid

Grids”, PhD Dissertation, Departement of Civil Engineering, University of

Wales, Sweansea, United Kingdom.


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Natakusumah D.K., Nuradil C, (2004), “Simulasi Aliran di Perairan

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2013-Numerical Solution of River Flood and Dam Break Problems by Cell Centred Finite Volume Scheme

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