study to compute the current by applying nested grid technique

7/31/2019 Study to compute the current by applying nested grid technique

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Study to compute the current by applyingnested grid technique

Duong Thi Thuy Nga, Nguyen Ky Phung

Abstract Current model is an important model in modelling. It helps us to predict the current and to solve some other

problems in environment such as: computing and predicting the sediment transport, the oil spill, the accretion, erosion at the

estuary. This model is based on a 2D finite-difference grid. It takes a lot of time to compute the current in a large area in several

days. We apply nested grid technique to the model to improve the speed of computing. The computational results in study area

are agreement to our experimental results and the real data in Ca Mau coastal zone. It shows that the model is faster than the

old one.

Index Terms current model, velocity, nested grid.

1 INTRODUCTION

To compute the current in a coastal zone is a verycomplex problem. There are many natural factors affect-ing the current such as tide, wind. So we must build agood model to compute the current correctly and fast.

In the world, there are many research groups re-searched this problem. Most of them used Defant andHansen methods [1, 7] at Atlantic, Pacific. Some of themused ADI method to solve the problem [2, 3, 4, 6, 9, 10].Their models are still run slowly.

Therefore, the main objectives for the development ofthis model is:

- Building the nested grid method to improve thespeed of the computing, and

- Testing the model in Ca Mau coastal zone to decide ifthe model can be applied in reality.

The remainder of this paper is following, section 2 pre-sents detailed expositions for the model. The experimentresults are undertaken in section 3. Section 4 focuses onfuture developments of the model. Finally, section 5 givesconclusions of the study.

2 METHODOLOGY

2.1 Current model

The model solves the depth averaged 2D shallow wa-ter equations. They are equations of floods, ocean tidesand storm surges. They are derived by using the hypoth-eses of vertically uniform horizontal velocity and negligi-ble vertical acceleration (i.e. hydrostatic pressure distribu-tion). The assumptions are valid where the wave length ismuch greater than the depth of water. In the case of oceantide the equations are applicable everywhere.The 2D equations in the horizontal plane are describableby the following partial differential equations of masscontinuity and momentum conservation in the X and Ydirections for an in-plan Cartesian coordinate frame of

reference.where:u, v: depth averaged velocity components in X and Y di-rections [m/s] : water surface elevation [m]

h+ : depth of water [m]

t: time [s]x, y: distance in X and Y directions [m]C: Chezy coefficient

f: Coriolis force coefficient, : horizontal diffusion of momentum coefficient inX and Y directions

fx(t), fy(t): sum of components of external forces in X andY directions

2.2 Algorithm

The model usees an alternating direction implicit (ADI)finite different method to solve the problems. The methodinvolves two stages. In each stage, a tri-diagonal matrixfor the computational domain is built to solve the values:The form of the equation:

There are millions of nodes to compute the values. There

are also millions of equations to be solved. The tri-diagonal matrix method will increase the speed of com-puting because values out of the diagonals are 0.

x y

JOURNAL OF COMPUTING, VOLUME 4, ISSUE 5, MAY 2012, ISSN 2151-9617

https://sites.google.com/site/journalofcomputing

WWW.JOURNALOFCOMPUTING.ORG 184


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We have the equations:b1* + c1* =

a2* + b2* + c2* = d2. . .

ai* + bi* + ci* = di. . .

aI-1* + bI-1* + cI-1* = dI-1aI* + bI* =where = d1 a1*

= dI cI*

, : known values of the two boundaries Method to solve the equations:Equation (1): devide b1:

+ W1* = G1with W1 = c1/b1; G1 = /b1Equation (2): take the result of (1) to remove a2:

+ W2* = G2with W2 = c2/ (b2 a2*W1); G2 = (d2 a2*G1)/ (b2 a2*W1)Continue to remove ai, get Wi and Gi:

Wi = ci/ (bi ai*Wi-1) , i = 2, 3, , I-1Gi = (di ai*Gi-1)/ (bi ai*Wi-1) , i = 2, 3, , I-1andWI = cI/(bI aI*WI-1)GI = ( aI*GI-1)/(bI aI*WI-1)The last equation: = GIGet , , , by reverse alternating:

= Gi Wi* with i = I-1, I-2, , 1 Initial conditionsAt time t = 0:u = 0, v = 0,

C(x,y,0) = C0(x,y) or C(x,y,0)= constant Boundary conditions- Compute tidal components:

, or

- Compute Q = U * WThen, compute the velocities at the boundaries.

Nested gridThe model often run slowly when being applied in a

large domain. Hence, we use nested grid technique to

help them run faster.

Figure 1. The finite grid

A computational grid is overlapped into the domain.The model will compute values such as u,v, ,h. Loca-tions of those values are described as follow:

Figure 2. Location of a closed grid on a sparse grid

Linking gridsValues in close grids (lower case characters) are in-

terpolated by values in the sparse grid (upper case char-acters). Then, the values in closed grids are used to editthe ones in the sparse grid.

1 1( 12, 6) ( 6, 3) ( 6, 4)

2 2u j k U J K U J K

1 1( 12, 6) ( 6, 3) ( 7, 3)

2 2v j k V J K V J K

1 1 2 1( 12, 6) ( 6, 3) ( 6, 4) ( 12, 6)

2 2 1 2j k Z J K Z J K j k

1

nx 2

nx

1d

nx

nx 1nx

1

1

n

ix1n

ix1

1

n

ix

2

n

Ix

1

n

I

xn

I

x

1

nIx

nIx Id

1d

1nxd

1nx1

0

nx1n

Ix

1

1

nx

1

2

nx

1d

2

nx 1

3

nx

Id

nIx

1

n

Ix 2

n

Ix 1

nx

1nx 1nx

0

N

i

iii tA1

)sin(





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Diagram The complexity of the algorithmfor (i = 0; i < Time to make the current model run stably;i++){

for (j = 0; j < Total rows; j++){

for (k = 0; k < Total columns; k++)Solve the tri-diagonal matrix to com-

pute u in the study area}

Compute

,v in the study areafor (j = 0; j < Total columns; j++){

for (k = 0; k < Total rows; k++)Solve the tri-diagonal matrix to com-

pute v in the study area}

Compute

,u in the study area}for (i = 0; i < Time to predict the current; i++){

for (j = 0; j < Total rows; j++){

for (k = 0; k < Total columns; k++)Solve the tri-diagonal matrix to com-

pute u in the study area}

Compute

,v in the study areafor (j = 0; j < Total columns; j++)

{for (k = 0; k < Total rows; k++)

Solve the tri-diagonal matrix to com-pute v in the study area

}

Compute

,u in the study area}The complexity is O(n3). It is affected by time to predictthe current, the width and the height of the study area.

3. Experimental results

Applying the model in Ca Mau coastal zone under the

following conditions:- The study area: 46200 m * 66000 m- A sparsed grid: myx 300 , ss 30 - A closed grid: (23000 m * 24000 m at Bay Hap estuary)

myx 100 , ss 10





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Figure 3. Map of the study area

Figure 4. Location of the closed grid

Figure 5. 3D Topography of Ca Mau coastal zone

Boundary condition the for current modelOn the open boundary, the water levels are given bycomputing tidal components as shown in table I and II.The parameters of these tidal components are from [11,12, 13].

Table 1. TIDAL CHARACTERISTICS AT EASTERN SEA

No. Name oftidal com-ponents

Amplitude(m)

Phase(rad)

1 M2 0.72 0.59

2 N2 0.15 0.08

3 S2 0.3 1.3

4 K2 0.08 1.3

5 K1 0.59 5.4

6 O1 0.42 4.6

7P1 0.19 5.48 Q1 0.01 4.2

9 M4 0.01 4.8

10 MS4 0.01 5.8

11 M6 0.004 2.6

Table 2. TIDAL CHARACTERISTICS AT THAI LAN BAY

No. Name oftidal com-ponents

Amplitude(m)

Phase(rad)

1 M2 0.15 1.35

2 N2 0.15 0.08

3 S2 0.12 1.35

4 K2 0.08 1.3

5 K1 0.38 0.18

6 O1 0.25 1.8

7 P1 0.49 5.4

8 Q1 0.07 4.2

9 M1 0.08 3.5

Wind: Southern West direction, 4.2m/s.





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Results Do not use the closed grid

Figure 6. The current after 1 month, 15 days

Figure 7. The current after 1 month, 15 days and 6 hours


Figure 9. The current after 1 month, 15 days and 18

hours

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20 40 60 80 100 120 140

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Use the closed grid

Figure 10. The current after 1 month, 15 days


Figure 12. The current after 1 month, 15 days and 12hours

Figure 13. The current after 1 month, 15 days and 18hours

Evaluate:Using the closed grid will help the model compute the

interested area in detail. The speed of the model

The model runs on the computer Core Duo 3GB. Theprogramming language used to compute the values is C#.- Using a sparsed grid in the study area (

myx 300 ): It takes 102 hours to compute thecurrent for 3 months in reality.- Using a closed grid in the study area (

myx 100 ) : It takes 805 hours to compute thecurrent for 3 months in reality.- Using a sparsed grid and a closed grid in the study area:It takes 257 hours to compute the current for 3 months inreality.





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4. Testing the modelThe model was tested in two new ways: experimentalanalysis and U-channel.

The first testThe model was used to compute water flow in a rectangle

channel: L=100m, W= 6m, h=1m, t = 0.1s, x = y =

1m. Velocity and water surface elevation of the modelwere compared with those of experimental analysis.

Figure 14. The rectangle channel used for testing

Figure 15. Graph of velocity

Figure 16. Graph of water surface elevation

Values of the model are nearly similar to those of the ex-perimental analysis. So the results are reliable.

The second testThe model run on U-channel with some parameters:h=10, the width of the channel is always 10m, fit on a rec-

tangle (L = 45m, W = 30m), t = 0.03s, x = 0.5m, y =0.5m. The velocity and water surface elevation were com-pared with those of Shukry experiment as follow:

Figure 17. Velocity of the model

Figure 18. Velocity of Shukry experiment

Figure 19. Water surface elevation of the model

Figure 20. Water surface elevation of Shukry experiment





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The shape and direction of the water flow, as well as thewater surface elevation of the two experiments are simi-lar.

FUTUREDEVELOPMENTS

The model has been tested in Ca Mau coastal zone.To be more reliable, the model needs to be tested with alot of real data.

CONCLUSIONS

This paper presents the current model in Ca Maucoastal zone. To improve the speed of the computing, wehave built the new algorithm to use the nested grid andapplied it to the model. The results were tested with sim-ulation experiments in Ca Mau coastal zone. With thereliability and good speed of computation, we can applyit to reality.

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Scientific Background.




study to compute the current by applying nested grid technique

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