Application of Finite Volume Method in Modeling The Flood Propagation Generated by Dam-Break On The Non-Uniformly

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Application of Finite Volume Method in Modeling The Flood Propagation Generated by Dam-Break On The Non-Uniformly

Building Layout

M. Syahril B.Kusuma¹, Dantje K. Natakusumah¹, Dhemi Harlan¹, Bobby Minola Ginting¹

1Civil Engineering Study Program Faculty of Civil and Environmental Engineering, Institut Teknologi Bandung, Indonesia

(email: minola_06@yahoo.co.id)

ABSTRACT

Numerical scheme developed by Jameson (1981) was used first to solve Unsteady Compressible Euler equations on various aero plane wings sections (aerofoil). The governing equations used in this scheme cover the mass conservation, momentum and energy balance. This scheme is modified into different manner to solve the shallow water equations in conservation forms which cover mass and momentum conservation in both directions. Once water elevation as boundary condition is specified, the characteristic method is used to determine the velocity both in x and y direction meanwhile wall boundary condition is treated with normal velocity at the wall is zero condition. Artificial viscosity as numerical dissipation is used to handle the numerical instabilities. The hybrid Runge Kutta fourth order as time stepping scheme is used to minimize the computational cost, since the artificial terms was only computed once. To validate the model, comparisons are made between its numerical and laboratory results. A good agreement is shown here, where the numerical dissipation can muffle the oscillations well. The advantages of using this scheme are the strong shock-capturing ability, simple computation and well applied for complex domain.

Keywords: dam-break, finite volume method, artificial viscosity, hybrid Runge Kutta 1. BACKGROUND

Indonesia is a country with its big potential in water resources. Dam construction is a kind of river potency utilization which has been widely applied in the world. One of the most important issues that develops in construction world is climate change problem. This issue closely relates to the natural disaster phenomenon. The effects of climate change can affect the hydrology characteristic, especially in precipitation pattern which tends to become larger. In the construction point of view, this negative impact must be handled correctly to prevent disaster.

In order to preventing the negative impacts of climate change, particularly in dam construction, some basic activities may be done. The structure capacity of the dam may be rechecked again to analyze if it has the sufficient capacity to flow the flood discharge that has increased due to climate change. Another activity that can be done is to make the disaster mitigation if a dam collapses. Numerical modeling in flood propagation of dam-break phenomenon is one of the parts and useful tool in disaster mitigation.

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Generally, the fluid flow characteristic can be derived mathematically from Navier Stokes equation.

Particularly, the flow characteristic in two dimensional on shallow water such on river and estuary can be explained from equation (1) – (3). These equations are well known as St. Venant Equation (SVE).

Equation (1) describes the mass continuity equation and equation (2) and (3) describe the momentum conservation equation in x and y direction respectively.

These equations are hard to solve analytically, therefore need the other methods. The numerical method is one of the methods to solve SVE. Previously, many numerical methods are developed by some researchers, include characteristic method, finite difference method (FDM), finite element method (FEM) and finite volume method (FVM). Most of schemes in FDM has the easy computational procedure but not in domain geometrical flexibility. For complex domain, it seems hard to apply FDM since the meshing is only in rectangular shape. FEM has the powerful flexibility in modeling complex domain but generally needs the high computational memory. FVM has the powerful both of flexibility in modeling complex domain and simple computational procedure.

Jameson (1981) developed finite volume cell center to solve the Navier Stokes equation for Unsteady Compressible Euler equations on various aero plane wings sections (aerofoil). This scheme is changed into different form to solve SVE.

2. GOVERNING EQUATIONS

The integration of three dimensional Navier Stokes equation in vertical direction with the assumption of vertical velocity is distributed uniformly yields SVE with mathematically can be written as:

0 (1)

gH S S (2)

gH S S (3)

with H is water depth , u and v are velocities in x and y directions, g is gravity acceleration, Sx and Sy

are channel slopes in x and y directions, Sfx and Sfy are roughness of the channel which can be estimated as:

S ^√ S ^√ (4)

with nM is Manning coefficient. Equation (2) and (3) hasve the other terms such wind force, Coriolis force, etc but they are neglected in this case since their effects are small.

3. NUMERICAL MODEL

The numerical model developed in this research has two components, space discretization and time discretization. Equation (1) – (3) are changed into matrix form written as equation (5).

W H

uH F

uH

u H gH G

vH

uvH S

0

gH S S (5)

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3

Equation (1) – (3) are written in other forms then integrated to closed domain Ω yield equation (6).

∬ W dΩ ∬ div H dΩ ∬ S dΩ (6)

Gauss divergence theorem is applied into equation (6), then equation (5) changes as follow:

∬ W dΩ ∮ H . dГ_Г ∬ S dΩ (7)

The symbol n defines the normal vector perpendicular with domain boundary and mathematically is written as:

ı ȷ (8)

Equation (7) can be written into other form as equation (9). Equation (9) is well known as “Time Dependent Euler Equation”. This equation is discritized in space and time.

∬ W dΩ ∮ F dy_Г G dx ∬ S dΩ (9)

3.1 Space Discretization

If the domain (Ω) is divided into several sub-domains as shown in Figure 1 that are not overlapping each other then Ω Ω1 Ω2 Ω3, equation (9) can be re-written as follows:

∬ W dΩ ∮ H . dГ ∬ S dΩ

∬ W dΩ ∮ H . dГ ∬ S dΩ (10)

∬ W dΩ ∮ H . dГ ∬ S dΩ

Figure 1: Domain Discretization (Ginting, 2011)

The variables in vector W are not directly defined on the mesh point. As simplification, these values are defined in center of the mesh. For the area of a mesh is defined as Ak, then these values can be determined as follows:

W ∬ W dΩ (11)

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The value of Ak can be determined in the form of “trapezoidal rule” or “Breithscneider” formula. The convective term in equation (9) can be solved as follow:

∮ F dy_Г G dx

, ∑ F ∆y G ∆x (12)

with N is number of sides in a mesh. Substituting equation (12) into equation (9) yields:

A W ∑ F ∆y G ∆x A S (13)

The velocity flux (Qi) is defined as:

Q u ∆y v ∆x (14)

If equation (12) is defined as C(Wk) and equation (14) is substituted into equation (12), then the convective term can be written as:

C W ∑

Q H Q u H gH ∆y Q v H gH ∆x

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Now, equation (13) is re-written as:

A W C W A S (16)

One of the most important terms in equation (16) which neglected is viscous term with the capability to muffle the oscillations when hydraulic jump occurs (Natakusumah, 2004). Therefore, the numerical dissipation term D(Wk) is added into equation (16) and yields:

A W C W D W A S (17)

The numerical dissipation term used in this research is artificial viscosity developed by Jameson (1981), which consists of two operators, Laplacian (D²(Wk)) and Biharmonic (D⁴(Wk)) which can be written as:

D W D W D W (18)

D W ∑ ∈ W W (19)

D W ∑ ∈ W W (20)

with ∈ is Biharmonic coefficient determined empirically. Both of these operators are function of mesh area, time stepping and adaptive coefficient. These coefficients are determined as follow:

∈ ∈ ^∑_∑ ^|_| ^|_| (21)

with Hk is average water depth on k^th cell and Hi is the water depth on each sides of k^th cell. The value of Biharmonic coefficient must be zero when hydraulic jump occurs, then value of ∈ is corrected as:

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∈ max 0, ∈ ∈ (22)

3.2 Time Discretization

Equation (17) can be classified as Ordinary Differential Equation (ODE) which only consists of first derivation in time. Equation (17) is changed into other form as:

W C W D W S (23)

Runge Kutta fourth order is used to solve equation (23) which mathematically is wriitten as:

W W

W W α C W D W W W α C W D W W W α C W D W W W α C W D W

W W

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3.3 Initial Condition and Boundary Condition

Initial condition is determined as cold start, where water depth is specified and velocity in both directions are defined as zero. There are two kinds of boundary condition, open boundary and wall boundary. For open boundary, if discharge or water level is defined, then the velocity is determined from characteristic method. In addition, the water level and velocity can be defined directly on open boundary.

Wall boundary is defined as a slip condition and there is no flow through in or out from channel wall.

This condition is written mathematically as:

Q . 0 (25)

with Qw is flux velocity. When water flow from wet area to dry area, there should be a treatment to handle the zero depth condition since the instability numerical computation will occur. Therefore, the wet and dry treatment will be given as (Casulli, 2008):

H x, y, η p x, y, z dz max D , h x, y η (26) with h is water depth (n‐1)^th time step and η is water level increase on n^th time step. Dmin is a limiter value for dry area depth. If the value of water depth is less than Dmin, then H becomes zero and velocity is also zero.

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4. LABORATORY MODEL 4.1 Model Set-up

The flood propagation is modeled as flash flood due to the water in large amount suddenly flow in short time period. The laboratory model is built and researched by Setiawati (2011).

The aim of this work is to know the impact of the building on the non-uniformly layout to flood wave.

The sketch of this physical model is depicted in Figure 2.

Figure 2: Laboratory Model Set-up (Setiawati, 2011)

The channel is 10 m long and 1m wide. The reservoir length and width are 2 m and 4 m respectively and filled with 30 cm of water. There is no water in channel downstream from gate. To simulate the dam-break, the gate is opened suddenly. The building size is 10 x 10 cm².

Figure 3: Reservoir and Gate (Left); Channel (Right) (Setiawati, 2011) 4.2 Measurement Device

The water depth is measured with wave probe for reservoir. This device is used together with data logger to record data. Piezometer is used to measure water depth on channel. The change of water level in piezometer is recorded by handycam.

Reservoir Channel

Downstream Pond Pump

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Figure 4: Piezometer on Laboratory Model (Setiawati, 2011) 4.3 Measurement Point

The measurement points on channel are divided into 8 grids, each grid is 1 m, measured from gate. All grids on the channel are connected with piezometer. The measurement point naming is based on Figure 5.

Figure 5: Measurement Points On Channel (Setiawati, 2011)

Figure 6: Building Layout for Configuration with 1 Building, 3 and 5 Buildings

(Setiawati, 2011) 5. RESULT AND ANALYSIS

The numerical model is built to simulate the dam-break phenomenon and the results will be calibrated with the laboratory measurement results. The simulated cases are the configuration without building, 1 building, 3 buildings and 5 buildings.

5.1 Configuration without Building

In this section the comparison between numerical model and laboratory results are 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 Dmin is set as 0.001 mm. The average Manning coefficient 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).

1 m

GATE DOWNSTREAM

1 m

0.10m

0.10m 0.05m

1 m

0.10m

0.10m 0.10m

0.10m 0.05m 0.05m

0.10m 0.10m

0.05m 0.05m

0.10m 0.10m

0.05m 0.05m

0.10m 0.10m

0.05m 0.05m

0.10m 0.10m

0.05m 0.05m

0.10m 0.10m

0.05m 0.05m 0.10m 0.10m

0.05m 0.05m 0.10m

0.10m

0.10m 0.10m

0.05m

A E B

A CD FGHIJ B CDEFGH IJA BCDEFGHIJABCDEFGH I J

ABCDEFGH I J ABCDEFGH I JABCDEFGHIJAB CDEFGH IJ

GRID 3 GRID 4 GRID 5 GRID 6

GRID 2

GRID 1 GRID 7 GRID 8

J I G

F H

D C

A B E

3 1 2 5 4 7 6

GRID 3 9

8

J 2

D A 1

B C E FGH I 6

4 3 5 8 7 9

= building Note:

8

J D

A B C E FGH I 4

2 1 3 6 5 7 9

GRID 3 GRID 3

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Figure 7-a shows the emptying curve of reservoir and the numerical model agrees with the experimental results, it is clearly shown that there is no significant difference between them. Figure 7-b shows 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 7-c - Figure 7-f shows 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-significant differences.

Figure 7: Time Evolution of Water Level for Configuration without Building

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Water Elevation (cm)

Time (s)

Station -2.00 m (Resevoir)

Numerical Observation

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Time (s)

Station +0.45 m (Channel 1E5)

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Time (s)

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Time (s)

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Time (s)

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Water Elevation (cm)

Time (s)

a b

c d

e f

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9 5.2 Configuration with 1 Building

In this section, the initial condition is set same with section 5.1, but the building is placed in channel 2.55 m from gate. Actually, when simulating the physical model for initial water depth at reservoir is 30 cm, the water propagates along channel and overtops the building which has the dimension 10 x 10 x 15 cm. Therefore, the experimental results are not compared with numerical results since the building in this numerical model is set as wall boundary which there is no water allowed to overtop the building.

The time evolution of water level for this configuration is shown in Figure 9.

Figure 8: Numerical Results of Flood Propagation for Configuration without Building When time is 1 s, water from reservoir propagates along channel but has not reached the building. The water depth upstream of building reaches approximately 0.5 - 3 cm. After 2 s, water crashes the building.

The water depth upstream of building reaches approximately 22 cm, while 5 cm approximately at downstream of the building. The results shown here indicate that the artificial viscosity as numerical dissipation term is able to muffle the shock wave phenomenon when water crashes the building and when hydraulic jump occurs. Figure 9 shows the comparison of several values of Laplacian and Biharmonic coefficient in time evolution of water level. It can be shown that the different values of these coefficients will give the different values when shock wave occurs but not significant. Actually many values are tried on this model, but only significant results are presented here.

1s

2s

3s

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5.3 Configuration 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 gate. Numerical model shows that after 2 s, water crashes 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 due to building is more significant. It is clearly shown from Figure 11 that during 3 - 5 s, the backwater affects the flow further towards upstream of the channel. Also, the artificial viscosity shows the good performance in handling the numerical instabilities and shock wave phenomenon.

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Time (s)

Station -2.00 m (Reservoir)

Numerical Eps1=0.25, Eps2=0.015 Numerical Eps1=0.95, Eps2=0.030 Numerical Eps1=0.55, Eps2=0.030

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Water Elevation (cm)

Time (s)

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Time (s)

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Time (s)

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Water Elevation (cm)

Time (s)

Numerical Eps1=0.25, Eps=0.015 Numerical Eps1=0.95, Eps=0.030 Numerical Eps1=0.55, Eps2=0.030

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

Time (s)

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Figure 9: The Effect of Different Values of Laplacian (Eps1) and Biharmonic (Eps2) Coefficient for Time Evolution of Water Level for Configuration with 1 Building

5.4 Configuration with 5 Buildings

The 5 buildings are taken on the channel with the non-uniformly layout, while the initial condition is set same with the previous case. Numerical models show that after 2 s, water crash the 3 first 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 (configuration with 3 buildings). The significant difference is shown from the water depth at the downstream of 3 first building.

Figure 10: Numerical Results of Flood Propagation Near Building for Configuration with 1 Building

In this case, the water depth at 2 s is about 19 cm at downstream of the building 1 (3F7) and building 3 (3F3) and about 22 cm at downstream of building 2 (3F5). This is caused by the flow against building 4 (3H6) and building 5 (3H4). The water depth at the upstream of building 4 and 5 become approximately 25 cm, while 7 cm approximately at downstream of the building 4 and 5. The backwater effect in this case is almost same with the previous case that during 3 - 5 s the backwater affects the flow further towards upstream of the channel.

1s

2s

3s

4s

5s

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Figure 11: Numerical Results of Flood Propagation Near Building for Configuration with 3 Buildings

1s

2s

3s

4s

5s

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Figure 12: Numerical Results of Flood Propagation Near Building for Configuration with 5 Buildings

6. CONCLUSIONS

Finite volume method for space discretization and Runge Kutta fourth order time stepping with artificial viscosity as numerical dissipation can simulate the dam-break flow phenomenon and gives the good results. It is clearly shown from the numerical results that show good agreement with laboratory results.

In this case, the instability numerical computation especially when shock wave occurs can be handled well.

1s

2s

3s

4s

5s

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The shape of front wave is affected by building. In addition, it is also affected by the building configuration. Numerical model shows that although the building has the significant effect, but the effect is only locally, it is clearly shown that the water depth (with or without building and with different building configurations) is almost same at the downstream of the channel. This phenomenon had been researched also by Soarez (2002). The physical model built there shows that the effect of building to dam-break wave is only locally and its propagation velocity remains almost unchanged.

The good numerical model for dam-break case can be used as a tool for a disaster mitigation. With this model, some real dam-break cases can be simulated, especially for the areas with have the big negative effects if the dam is broken. The water depth and velocity of flow due to dam-break phenomenon can be predicted along the channel or river where the dam is built, then the negative effect can be estimated and so the optimum solution to anticipate this disaster can be reached.

REFERENCES

Casulli, V. (2008) A High Resolution Wetting and Drying Algorithm for Free Surface Hydrodynamic.

International Journal of Numerical Method in Fluids.

Ginting, B.M., Natakusumah, D.K., Kusuma, M.S., Harlan, Dhemi. (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, Schmidt, Friedrichshafen (1981) Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes. Springer-Verlag, 1981.

Mingham, C.G., Causon, D.M. 1(998). High-Resolution Finite Volume Method for Shallow Water Flows. Journal of Hydraulic Engineering, 124(6), 605-613.

Natakusumah D.K., Nuradil, C. (2004) Simulasi Aliran di Perairan Dangkal dengan Menggunakan Metoda Volume Hingga pada Sistem Grid tak Beraturan. Jurnal Teknik Sipil, Volume 11 April 2004, No. 2.

Setiawati, T. (2011) Kajian Model Fisik Rambatan Banjir Akibat Keruntuhan Tanggul Pada Tata Letak Bangunan Tidak Seragam. Magister Thesis, Insitut Teknologi Bandung, 2011, Bandung, Indonesia.

Soares Frazão, S., and Zech, Y. (2004) Discussion of Numerical Prediction of Dam-Break Flows in General Geometries with Complex Bed Topography by Jian G. Zhou, Derek M. Causon, Clive G.

Mingham, and David M. Ingram. ASCE, April 2004, Vol. 130, No. 4, pp. 332–340.

Soares Frazão, S., and Zech, Y. (1999) Effects of a Sharp Bend on Dam-Break Flow. Proc., 28th IAHR Congress CD-ROM, Graz, Austria, August 1999.

Tahershamsi, A., and Namin, M. (2010) Two Dimensional Modeling of Dam-Break Flows. River Flow 2010 - Dittrich, Koll, Aberle & Geisenhainer (eds) - 2010 Bundesanstalt für Wasserbau, ISBN 978-3- 939230-00-7.

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