Download pdf - Research Article Discretization of Multidimensional ...downloads.hindawi.com/journals/jam/2013/642485.pdfResearch Article Discretization of Multidimensional Mathematical Equations

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 642485 12 pageshttpdxdoiorg1011552013642485

Research ArticleDiscretization of Multidimensional MathematicalEquations of Dam Break Phenomena Using a Novel Approach ofFinite Volume Method

Hamid Reza Vosoughifar1 Azam Dolatshah2 and Seyed Kazem Sadat Shokouhi1

1 Department of Civil Engineering Islamic Azad University South Tehran Branch 17776 Tehran Iran2Department of Civil Engineering Islamic Azad University Dezful Branch Khuzestan 313 Dezful Iran

Correspondence should be addressed to Hamid Reza Vosoughifar vosoughifarazadacir

Received 11 July 2012 Accepted 2 February 2013

Academic Editor Chein-Shan Liu

Copyright copy 2013 Hamid Reza Vosoughifar et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper was concerned to simulate both wet and dry bed dam break problems A high-resolution finite volume method (FVM)was employed to solve the one-dimensional (1D) and two-dimensional (2D) shallowwater equations (SWEs) using an unstructuredVoronoi mesh grid In this attempt the robust local Lax-Friedrichs (LLxF) scheme was used for the calculating of the numericalflux at cells interfaces The model named V-Break was run under the asymmetry partial and circular dam break conditions andthen verified by comparing the model outputs with the documented results Due to a precise agreement between those output anddocumented results the V-Break could be considered as a reliablemethod for dealing with shallowwater (SW) and shock problemsespecially those having discontinuities In addition statistical observations indicated a good conformity between the V-Break andanalytical results clearly

1 Introduction

Floods induced by dam failures can cause significant loss ofhuman life and property damages especially when locatedin highly populated regions These entail numerical andlaboratory investigations of dam break flows and theirpotential damage The shallow water equations (SWEs) areconventionally used to describe the unsteady open channelflow such as dam break These equations are named as SaintVenant equations for one-dimensional (1D) problem and alsoinclude the continuity and momentum equations for two-dimensional (2D) studies

Many researchers studied the dam break problem suchas Toro [1] Wu et al [2] Wang and Shen [3] Mohapatra etal [4] Zoppou and Roberts [5] Wang et al [6] Wang andLiu [7] Venutelli [8] Ponce et al [9] Zhou et al [10] Lianget al [11] Quecedo et al [12] Begnudelli and Sanders [13]Loukili and Soulaımani [14] Dıaz et al [15] and Aliparast[16] especially using computational fluid dynamic (CFD)methods

Recently Shamsai and Mousavi evaluated the dam breakparameters such as breachwidth side slop time of failure andpeak outflow using 142 case studies of previous researchesThen using those evaluations and also the BREACH andFLDWAV application software the studies of Aidoghmoshearth dam breach were examined [17] Xia et al developeda 2D morphodynamic model for predicting dam break flowsover a mobile bed [18] Shamsi et al developed a 2D flowmodel based on SWEs They used finite volume method(FVM) total variation diminishing (TVD) and weightedaverage flux (WAF) schemes as well as Harten-Lax-van Leer-Contact (HLLC) Riemann solver [19] Erpicum et al pre-sented a 2D finite volume (FV) multiblock flow solver whichwas able to deal with the natural topography variation [20]Baghlani utilized a combination of the robust and effectiveflux-difference splitting (FDS) and flux-vector splitting (FVS)methods to simulate dam break problems based on FVM ona Cartesian gridThemethod combined the effectiveness androbustness of the FDS and FVSmethods to precisely estimatethe numerical flux at each cell interface [21] Zhang and Wu

2 Journal of Applied Mathematics

developed a hydrodynamic and sediment transportmodel fordam break flows The 2D SWEs were solved based on theFVMwith an unstructured quadtree mesh grid [22] Singh etal developed a 2D numerical model to solve the SWEs for thesimulation of dambreak problems [23] Chang et al proposeda meshless numerical model to investigate the shallow water(SW) dam break in 1D open channel A numerical modelwas used to solve the SWEs based on smoothed particlehydrodynamics (SPH) The concept of slice water particleswas adapted in the SPH-SWE formulation [24] Shakibaeiniaand Jin developed a new mesh-free particle model based onthe weakly compressible MPS (WC-MPS) formulation formodeling the dam break problem over a mobile bed [25]Sarveram and Shamsai investigated the dam break problemin converge and diverge rectangular channels in the unsteadystance using Saint Venant equations and a quasi-Lagrangianmethod [26]

This paper attempts to present a novel development for1D and 2D dam break problems in both wet and dry bedsA high-resolution FVM is employed to solve the SWEs onunstructured Voronoi meshThe local Lax-Friedrichs (LLxF)scheme is used for the estimation of fluxes at cells and thenumerical approximation of hyperbolic conservation laws

2 Research Methodology

21 Governing Equations The continuity and momentumequations of the SW can be written in different formsdepending upon the requirements of the numerical solutionof governing equations The 2D SWEs with source terms aregiven in the vector form considering a rigid bed channel [1]as follows

120597119880

120597119905+

120597119865

120597119909+

120597119866

120597119910+ 119878 = 0 (1)

119880 = [

[

ℎ

119906ℎ

Vℎ

]

]

(2)

119865 =

[[[[[

[

119906ℎ

1199062ℎ +

1

2119892ℎ2

119906Vℎ

]]]]]

]

(3)

119866 =

[[[[[[

[

Vℎ

119906ℎ

V2ℎ +1

2119892ℎ2

]]]]]]

]

(4)

119878 =

[[[[[[

[

0

minus119892ℎ (1198780119909

minus1198992

ℎ43119906radic1199062 + V2)

minus119892ℎ (1198780119910

minus1198992

ℎ43119906radic1199062 + V2)

]]]]]]

]

(5)

In this study the friction slopes were estimated using theManningrsquos formulas In the case of the dam break flow theinfluence of bottom roughness prevailed over the turbulentshear stress between cells Therefore the effective stress termswere neglected in the computation

22 Numerical Modeling Algorithm The main advantage ofthe FVM is that volume integrals in a partial differentialEquation (PDE) containing a divergence term are convertedto surface integrals using the divergence theorem Theseterms are then evaluated as fluxes at the surfaces of each FVBecause the flux entering a given volume is identical to thatleaving the adjacent volume these methods are conservativeAnother advantage of the FVM is that it is easily formu-lated to allow for unstructured meshes Unstructured gridmethods utilize an arbitrary collection of elements to fill thedomain These types of grids typically utilize triangles in 2Dand tetrahedrals in 3D although quadrilateral hexahedralVoronoi and Delaunay meshes can also be unstructuredIn the Voronoi mesh the chosen point has lower distancein the devoted domain rather than other points If onepoint has the same distance from several domains it willbe divided between domains Indeed these points createVoronoi cell boundaries Consequently internal sections ofthe Voronoi mesh consist of nodes belonging to one domainand boundaries include nodes that belong to several domains[27]

In this paper the studied domain was discretized usingunstructured Voronoi meshes Delaunay triangulation wascreated and then the Voronoi mesh was established usingthe Qhull program in MATLAB software The governingequation was discretized applying the FVM In this approachthe studied domainwas divided into several separated controlvolumes without any overlapping By integrating the gov-erning differential equation over every control volume thesystem of algebraic equations was created so that each ofits formulations belonged to one control volume and eachequation linked a parameter in the control volume node todifferent numbers of the parameter in adjacent nodes Thisconsequently led to the computation of the parameter in eachnode [28]

In order to solve discrete equations the parameter ineach node was computed considering its discrete equationand newest adjacent nodesrsquo values Solution procedure can beexpressed as follows

(1) Assuming an initial value in each node as an initialcondition

(2) Calculating the value in a node considering its dis-crete equation

(3) Performing pervious step for all nodes over thestudied domain one cycle is performed by repetitionthis step

(4) Verifying the convergence clause If this clause issatisfied the computing will end otherwise the com-putations will be repeated from the second step

As exact values of boundary conditions were not distinctthe Riemann boundary condition was utilized for computing

Journal of Applied Mathematics 3

the investigating parameter Therefore by assuming a layerwhich is close to the boundary layer 120597120597119909 = 0 and 120597120597119910 =

0 were defined for the investigating parameter and thencalculated values for boundary adjacent nodes transform torelated boundary nodes This procedure will continue untilthe results difference is converged to zero

23 Discretization of Governing Equations

231 FV Discretization Various methods can be used todiscretize the governing equations among which the FVMdue to its ability to satisfy mass andmomentum conservationis frequently adopted In this research the discretization of(1) was performed using the FVMwith unstructured Voronoimesh as shown in Figure 1 It should bementioned that119891 facevertexes nomination direction is counterclockwise from 119887 to119886 centering p (see Figure 1) [1] as follows

∬119860

120597119880

120597119905119889119860 + ∬

119860

(nabla sdot ) 119889119860 = ∬119860

119878119889119860 (6)

By implementing divergence theorem (6) is yielded to (7) asfollows

∬119860

120597119880

120597119905119889119860 + ∮

119897

( sdot ) 119889119897 = ∬119860

119878119889119860 (7)

where = 119894 + 119895 Equation (7) can be written as (8) byapproximating the line integral for all control volumes andnodes generally

119889119880119894

119889119905= minus

1

119860119894

sum

119895

( sdot 119894119895

Δ119897119894119895

) + 119878119894 (8)

232 Voronoi Mesh By applying the Voronoi mesh (8) isyielded to (9) for an investigated control volume howeverFigure 1 illustrates the 2D Voronoi mesh grid used fordescribing these equations as follows

120597119880

120597119905= minus

1

119860sum

119891

( sdot 119860 119890120576)119891

+ 119878 (9)

= 119867120576

sdot 119890120576

+ 119867120578

sdot 119890120578 (10)

Then

120597119880

120597119905= minus

1

119860sum

119891

119867120576119860119891

+ 119878

int

119905+Δ119905

119905

120597119880

120597119905119889119905 = int

119905+Δ119905

119905

(minus1

119860sum

119891

119867120576119860119891

) 119889119905 + int

119905+Δ119905

119905

119878119889119905

119880119899+1

119901minus 119880119899

119901= minus

1

119860sum

119891

119867120576119860119891

Δ119905 + 119878Δ119905

119891

119875

119890120585

119860119891

119890120578

Δ120585

Δ119909119891

Δ119910119891

119909

119886

119887

119910

nb

119860119910119891

119860119909119891

Figure 1 The 2D schematic Voronoi mesh cell used for describingthe discretization of the governing equations

119867120576

= 1198991119865120576

+ 1198992119866120576

1198991

=(119909nb minus 119909

119901)

radic(119909nb minus 119909119901

)2

+ (119910nb minus 119910119901

)2

1198992

=(119910nb minus 119910

119901)

radic(119909nb minus 119909119901

)2

+ (119910nb minus 119910119901

)2

(11)

The discrete equation can be written as followS

119880119899+1

119901= 119880119899

119901minus

Δ119905

119860sum

119891

[1198991119865120576119860119891

+ 1198992119866120576119860119891

]119899

+ 119878119899Δ119905 (12)

24 The Local Lax-Friedrichs (LLxF) High-Order Scheme Inshock capturing schemes the location of discontinuity iscaptured automatically by the scheme as a part of the solutionprocedure These slope-limiter or flux-limiter methods canbe extended to systems of equations In this paper thealgorithm is based upon central differences with comparableperformance to Riemann type solvers when used to obtain asolution for PDErsquos describing systems FV and Finite Differ-ence (FD) methods are closely related to central schemes likethe most shock capturing schemes [29] Rusanov scheme isoften called the LLxFmethod because it has the same form asthe Lax-Friedrichs (LxF) method but the viscosity coefficientis chosen locally It can be shown that this is a sufficientviscosity to make the method converge to the vanishing-viscosity solution Itmeans that it is less diffusive than normalLxF since it locally limits the numerical viscosity insteadof having a uniform viscosity on the entire domain LLxFformulates the FVM in space and the explicit Euler methodin time

Many researchers (eg Lin et al [30] van Dam andZegeling [31] and Lu et al [32]) utilized LLxF splitting


Yes

No

Start

Read the hydraulic and geometric properties of the channel

Read the CFL

Read the total run time (Tstop)

Create the initial matrixes according to hydraulic and geometric properties of the channel

Create the Voronoi computational mesh grid based on Delaunay triangulationusing Qhull program in MATLAB

velocity and so forth

Transfer results as initial and boundary conditions

Plot results

End

Read the start time (119905 = 0)

119905 lt Tstop

Calculate Δ119905 according to CFL

Calculate coefficients of all matrixes flux interface LLxF scheme water depth

Figure 2 The V-Break flowchart

scheme in different problems such as 2D SWEs 1D adaptivemoving mesh method and its application to hyperbolic con-servation laws from magnetohydrodynamics (MHD) andthe performance of the weighted essential nonoscillatory(WENO) method

In this research LLxF is used as a flux calculator Byexpanding (12) (13) can be written as follows

119880119899+1

119901= 119880119899

119901minus

Δ119905

119860

[

[

sum

119891

((119865119891

)nbsdotout

119860119891

1198991119891

)


Table 1 General summarization of the equipment used by some other researchers and present study

Reference Wang and Liu [7] Liang et al [11] Loukili and Soulaımani [14] Baghlani [21] V-BreakNumerical method FVM FVM FVM FVM FVM

Riemann solverRoe-MUSCL Roe-upwindHLL-MUSCL composite

methods (CFLF8)HLLC Lax-Fredrichs HLL HLLC

WAF FDS-FVS Local Lax-Fredrichs

Mesh grid Unstructured triangular Rectangular(quadtree)

Unstructured triangularunstructured quadrilateral Rectangular Unstructured

VoronoiDimensionalapproach 2D 1D 2D 1D 2D 1D 2D 1D 2D

ℎ(m

)

119909 (m)

ℎ119906

DAM

0 05 1

ℎ119889

Figure 3 1D Studied domain for verification

+ sum

119891

((119865119891

)nbsdotin

119860119891

1198991119891

)

+ sum

119891

((119866119891

)nbsdotout

sdot 119860119891

sdot 1198992119891

)

minus sum

119891

((119866119891

)nbsdotin

119860119891

1198992119891

)]

]

+ 119878Δ119905

(13)

Intercell fluxes can be estimated by implementing the follow-ing equations

(119865119891

)nbsdotout

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

nb minus 119906119899

119901)

(119865119891

)nbsdotin

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

119901minus 119906119899

nb)

(119866119891

)nbsdotout

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

nb minus V119899

119901)

(119866119891

)nbsdotin

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

119901minus V119899

nb)

(14)

After computing intercell fluxes by utilizing the LLxF schemein Voronoi mesh equations can be solved and the final resultcan be calculated for each time stepThe Δ119905 can be computedusing the Courant Friedrichs Lewy (CFL) for each time stepas follows

Δ119905 = CFL times min(radic(119909nb minus 119909119901

)2

+ (119910nb minus 119910119901

)2

)

times (1

max1003816100381610038161003816100381610038161003816119906 + radic119892ℎ

1003816100381610038161003816100381610038161003816

+1

max1003816100381610038161003816100381610038161003816V + radic119892ℎ

1003816100381610038161003816100381610038161003816

)

(15)

The CFL should range over [0 1] for achieving to the stability(0 lt CFL lt 1)

25 Preparation and Validation of the Numerical AlgorithmThe CFD code named V-Break was prepared on the unstruc-tured Voronoi mesh grid using MATLAB programmingFigure 2 illustrates the running process of the V-Break as aflowchart

V-Break was then validated using Stokerrsquos analyticalsolution in 1D [33] and previous results were obtained byother researchers in 2D (eg Wang and Liu [7] Loukili andSoulaımani [14] Liang et al [11] and Baghlani [21]) Table 1presents key solvers algorithms and methods used in theseprevious studies

3 Results

31 1D Dam Break Test At the instant of the dam breakwater is released through the breach forming a positivewave that propagates downstream and a negative wave thatmoves upstream Here Stokerrsquos analytical solution of dambreak problem can be used to illustrate the accuracy of thenumerical schemes Stoker derived this theory just for thewet bed dam break problem but it is possible to develop


0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)

V-Break (119905 = 002 s)Stokerrsquos analytical

solution (119905 = 002 s)

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)

V-Break (119905 = 01 s)

solution (119905 = 01 s)Stokerrsquos analytical

(b)

Figure 4 1DWater depth values obtained using V-Break and Stokerrsquos analytical solution (a) 1D water depth at 119905 = 002 s (b) 1D water depthat 119905 = 01 s

0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)


solution (119905 = 002 s)

(a)

0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)

V-Break (119905 = 01 s)


(b)

Figure 5 1D Velocity values obtained using V-Break and stokerrsquos analytical solution (a) Velocities at 119905 = 002 s (b) velocities at 119905 = 01 s

it for dry bed considering a downstream water depth veryclose to zero [33] A 1D horizontal rectangular channel having1m in length with wall at either ends and no roughness wasconsidered however the source term equaled to zero Theinitial velocity is zero and a barrier is present at 119909 = 05mwhich is removed at 119905 = 0 sThe water depths of the upstreamand downstream are 1 and 05m respectively Also the CFLparameter is equal to 09 Figure 3 illustrates the 1Ddambreakdomain [34]

Figures 4 and 5 show the water depth and velocity of theflow obtained using V-Break and Stokerrsquos analytical solutionat 119905 = 002 and 01 s Furthermore Table 2 presents theresults of the Mann-Whitney test for a statistical comparisonbetween the obtained results using V-Break and Stokerrsquosanalytical solution

In terms of comparing two groups of data statisticallythe null hypothesis (119867

0) is usually a hypothesis of ldquonondif-

ferencerdquo It means that there is no difference between the

ranks of the two comparing groups These two groups canbe defined as V-Break and Stokerrsquos analytical results Whencomputing the value of Mann-Whitney 119880 test the number ofcomparisons equals to the product of the number of data inthe first group (119873

119860) times the number of data in the second

group (119873119861) which is equal to 119873

119860times 119873119861 If the null hypothesis

is true then the value of Mann-Whitney 119880 test should beabout half thit value The smallest possible value of Mann-Whitney 119880 test is zero The largest possible value is equalto (119873119860

times 119873119861)2 If this value is much smaller than that the

119875-value will be small The 119875 value or calculated probabilityis the estimated probability of rejecting the null hypothesisof a study question when that hypothesis is true while thesignificance level is usually considered as 120572 = 005 So if the119875 value is more than 005 the null hypothesis will be trueespecially when119875 value is closer to 1 Also the null hypothesisis true when RS1 le Wilcoxon W le RS2 Here RS1 is the sumof ranks for the first group and RS2 is the sum of ranks for


Table 2 Mann-Whitney test parameters for comparison between V-Break and stokerrsquos analytical solution results

Output result type ParameterMann-Whitney 119880 Wilcoxon 119882 119885 119875 value

Water depth at 119905 = 002 s 4314500 8685500 minus0030 0976Water depth at 119905 = 01 s 5200500 10453500 minus0004 0997Velocity at 119905 = 002 s 4269500 8734500 minus0643 0520Velocity at 119905 = 01 s 5904500 11899500 minus0082 0934

200

175

150

125

100

75

50

25

0

2001751501251007550250

95m

75m

119909 (m)

119910(m

)

Figure 6 2D studied domain for verification

the second group Similarly the null hypothesis will be trueif |119885| le 119885

0 while 119885

0is the critical value extracted from the

119885-distribution graph statistically This value for the presentstudy equal to 196

In this research all the above conditions were satisfiedin Table 2 So the null hypothesis is true and there is nodifference between V-Break and Stokerrsquos analytical solutionresults Consequently V-Break results are acceptable havinga good agreement with Stokerrsquos analytical solutions

32 2D Partial Dam Break Test

321 V-Break For this case study a 2D partial dam breakproblem with asymmetrical breach was considered Thecomputational domain was defined in a channel with 200min length and 200m inwidthThebreach is 75m in length andthe dam is 15m in height The initial upstream water depth is10m The downstream water depth is 5m in a wet bed and01m in a dry bed The roughness coefficient was assumedzero implying a frictionless surface In this example a domainwith 40 times 40 node points it was proposed and a time of 72 swas considered as the total time for the calculation procedureAlso theCFL parameter is equal to 09 Figure 6 illustrates the2D dam break domain Figure 7 shows the results of previousstudies

Different conditions of the domain were simulated usingV-Break Figure 8 shows the output results of V-Break

322 Mesh Configuration Comparison The output resultswere then compared with the previous studies A section wasconsidered at 119910 = 130m and the leading points of waterdepth contours were exported as a function of distance (119909)Figure 9 shows the comparison among results obtained usingV-Break on Voronoi results obtained by Liang et al [11] onrectangular and results obtained by Loukili and Soulaımani[14] on triangular mesh grids in both wet and dry beds

323 Riemann Solver Comparison AspreviouslymentionedWang and Liu implemented four typical FVMs includingthe Reo-MUSCL Reo-upwind HLL-MUSCL and CFLF8composite methods on unstructured triangular meshes tosimulate a 2D dam break problem [7] In this research theRiemann solver used in the current study was compared withsolvers used byWang andLiu considering the particularmeshgrids used at each study [7] For a better visual comparisona section was considered at 119910 = 130m and the leading pointsof water depth contours were exported and then results areshown in Figure 10

33 2D Circular Dam Break Test In this test 200 compu-tational cells were used where initial conditions consist oftwo states separated by a circular discontinuity The radiusof the circle is 50m and it is centered at 119909 = 100mBoth components of the velocity u and v were set to zeroeverywhere and ℎ was set to 10m within the circle and1m outside the circle The computational grid consisted of40 times 40 cells and the solution was sought after 2 s Thedomain was defined for the V-Break and then the outputresults were compared with the previous studies Figures 11and 12 illustrate output results of V-Break including the watersurface formation and water depth contours respectivelyAlso Figure 13 shows results performed by Baghlani [21] inorder to use them in comparison

Figures 11 12 and 13 show that the V-Break can simulatethe 2D circular dam break well as compared with previousstudy by Baghlani [21]

4 Conclusions

In the current research a novel and friendly user code namedV-Break was written showing that the LLxF scheme alongwith the FVM on the unstructured Voronoi grid is a suitablecombination in order to simulate 1D2Ddambreak problems


200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200100500 200100500 150

200100500 150200100500 150

150200

200

200

150

150

10 10

1010

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

2150

150

150 5050

0

0100100

10050119909

2150

150

150 5050

0

0100100

10050

150

150

150 5050

0

0 100100

10050119909

2150

150

150 5050

0

0

100

100100

150100

50119909

(a) Roe-MUSCL (b) Roe-upwind

(c) HLL-MUSCL (d) CFLF8



119909 (m)

119909 (m)

119910(m

)200

1208040

0

160119910

(m)

200

1208040

0

160

20012080400 160

20012080400 160

200

200

120120 8080

40

40

0

002

6

10

160

160

8

4

6

108

4

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

200

200

120120 8080

40

40

0

0160

160

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

(A1)

(A2)

200

100

50

0

150

200100500 150

200

100

50

0

150

200100500 150

200150

500

100

200150

10050

864200

200150

500

100

200150

10050

8

6

04

(B1)

(B2)

150

50100

150100

50

15100100

50

(C)

Figure 7 2D asymmetrical partial dam break test in a frictionless horizontal and wetdry bed domain at 119905 = 72 s by some other researchers(A1) Liang et al Wet bed [11] (A2) Liang et al dry bed at 119905 = 5 s [11] (B1) Loukili and Soulaımani-Wet Bed [14] (B2) Loukili and Soulaımani-Dry Bed [14] and (C) Wang and Liu [7]

The advantages of this method are very promising especiallyin reconstructing the conducted tests For 1D dam breakStokerrsquos analytical solution was considered for validationIllustrations and computed 119875 values demonstrated that thereare no significant differences between Stokerrsquos analyticalsolution results andV-Break outputs In addition to 1Dpartialdam break the 2D partial dam break tests were done Resultswere compared with some other previous studies for valida-tion at 119905 = 72 s Each test can be done in the wet or dry bedconditions In all scenarios considered here no significantnumerical dispersion problemor nonphysical alternationwasobserved in the results The comparison showed a goodagreement between V-Break results with previous studies In

terms of mesh grid comparison it was seen that the Voronoimesh grid results are closer to triangular mesh grid resultscompared with rectangular mesh grid results In additionthe 2D circular dam break test was done and its results werecompared with previous studies at 119905 = 2 s which seemsaccepted Generally the unstructured Voronoi mesh grid isable to model inlet and outlet fluxes in every direction ofcontrol volume facesThe results indicated a higher efficiencyand precision of the discrete equations resulted from theVoronoi mesh Thus it could be recommended to utilizethe Voronoi mesh in the numerical discrete equations TheVoronoi mesh grid is able to model complicated geometriesand also it could produce the final discrete equations leading


10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)

Contour of partial dam break by LLxF at time 119905 = 72 s

(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)

by LLxF at time 119905 = 72 sContour of partial dam break

(b)

Figure 8 2D asymmetrical partial dam break test and in a frictionless horizontal domain at 119905 = 72 s obtained by V-Break (a) Wet bed and(b) dry bed

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

VoronoiRectangularTriangular

119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)

Rectangular Liang et al 2004Triangular Loukili and Soulaiumlmani 2007

(b)

Figure 9 The comparison among Voronoi rectangular [11] and triangular [14] mesh grids (a) Wet bed and (b) dry bed


456789

1011

Wat

er d

epth

(m)

Roe-MUSCLRoe-upwindHLL-MUSCL

CFLF8LLxF

0 50 100 150 200119909 (m)

Figure 10 The comparison among various Riemann solvers usedin Wang and Liursquos code (Reo-MUSCL Reo-Upwind HLL-MUSCLand CLFL8) and V-Break (LLxF)

119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)

Figure 11 2D Circular dam break problem at 119905 = 2 s using V-Break

0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)

Figure 12 2D Circular dam break water depth contours obtainedusing V-Break

119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)

Figure 13 2D Circular dam break problem at 119905 = 2 s by Baghlani[21]

to accurate results within a lower computational demandcompared to other unstructured meshes Furthermore ahigh capability of LLxF scheme was demonstrated in the 2Ddam break pneumonia evaluation compared with the otherpresented Riemann solutions

Nomenclatures

Adjacent surface vector of the investigated Voronoicell

119860 Area of the adjacent surface vector of theinvestigated Voronoi cell ()

119860119909 The component of 119860 in 119909 direction


119865 119866 Flux vector functions119865120576 119866120576 The Voronoi cell normal flux vectors

119880 The vector of conserved variables119867 Input and output fluxes to a Voronoi cell119878 The vector of source terms1198780119909 Bed slope in the 119909 direction

1198780119910 Bed slope in the 119910 direction

119886 119887 Nodes of the both sides of the investigated Voronoicell 119891

119890120585 Unit outward normal vector in each Voronoi cell 119891

119890120578 The unit tangent vector in each Voronoi cell 119891

119891 Joint surface element between investigated cell andother adjacent cells

Gravity accelerationℎ The water depthℎ Themean water depthℎ119906 The upstream water depth at 119905 = 0

ℎ119889 The downstream water depth at 119905 = 0

119897 The boundary of the 119894th control volume119899 The Manningrsquos roughness coefficient The outward unit vector normal to the boundarynb The central node of adjacent cells119905 Time119906 Velocity vector component in 119909 direction119906 The mean velocity vector component in 119909 directionV Velocity vector component in 119910 directionV The mean velocity vector component in 119910 direction119909 Horizontal coordinate component


119910 Vertical coordinate componentsum119891

The sum over the all Voronoi cells119901 Central node of investigated Voronoi cellΔ119905 Time intervalΔ120585 The distance between the central node of

investigated Voronoi cell and the adjacent cells1198670 The null hypothesis

119885 The value of 119885-Test1198850 The critical value of 119885 extracted from the statistical

119885-distribution graphRS1 Sum of ranks for the first comparing groupRS2 Sum of ranks for the second comparing group120572 The significance level119873119860 The number of data in the first comparing group

119873119861 The number of data in the second comparinggroup

Subscripts

119891 Denotes the parameter at the Voronoi cellssidersquos area 119891

119894 Counts all central control volumes119895 Counts all nodes of the central control volumesnb Denotes parameters at the central node of

adjacent Voronoi cells119901 Denotes parameters at central node of

investigated Voronoi cellin Denotes the parameter outside the Voronoi

cells sidersquos area 119891

out Denotes the parameter outside the Voronoicells sidersquos area 119891

120585 The parameter component in 119890120585direction


Superscripts

119899 Denotes parameters belonging to time of 119905

119899 + 1 Denotes parameters belonging to time of 119905 + Δ119905

References

[1] E F Toro Shock-Capturing Methods for Free Surface ShallowFlows John Wiley amp Sons New York NY USA 2001

[2] C Wu G Huang and Y Zheng ldquoTheoretical solution of dam-break shockwaverdquo Journal ofHydraulic Engineering vol 125 no11 pp 1210ndash1215 1999

[3] Z Wang and H T Shen ldquoLagrangian simulation of one-dimensional dam-break flowrdquo Journal of Hydraulic Engineeringvol 125 no 11 pp 1217ndash1220 1999

[4] P K Mohapatra V Eswaran and S M Bhallamudi ldquoTwo-dimensional analysis of dam-break flow in vertical planerdquoJournal of Hydraulic Engineering vol 125 no 2 pp 183ndash1921999

[5] C Zoppou and S Roberts ldquoNumerical solution of the two-dimensional unsteady dam breakrdquo Applied Mathematical Mod-elling vol 24 no 7 pp 457ndash475 2000

[6] J S Wang H G Ni and Y S He ldquoFinite-difference TVDscheme for computation of dam-break problemsrdquo Journal ofHydraulic Engineering vol 126 no 4 pp 253ndash262 2000

[7] J W Wang and R X Liu ldquoA comparative study of finitevolume methods on unstructured meshes for simulation of 2Dshallow water wave problemsrdquo Mathematics and Computers inSimulation vol 53 no 3 pp 171ndash184 2000

[8] M Venutelli ldquoA fractional-step Pade-Galerkin model for dam-break flow simulationrdquo Applied Mathematics and Computationvol 134 no 1 pp 93ndash107 2003

[9] V M Ponce A Taher-Shamsi and A V Shetty ldquoDam-breachflood wave propagation using dimensionless parametersrdquo Jour-nal of Hydraulic Engineering vol 129 no 10 pp 777ndash782 2003

[10] J G Zhou D M Causon C G Mingham and D MIngram ldquoNumerical prediction of dam-break flows in generalgeometries with complex bed topographyrdquo Journal of HydraulicEngineering vol 130 no 4 pp 332ndash340 2004

[11] Q Liang A G L Borthwick and G Stelling ldquoSimulation ofdam- and dyke-break hydrodynamics on dynamically adaptivequadtree gridsrdquo International Journal for Numerical Methods inFluids vol 46 no 2 pp 127ndash162 2004

[12] M Quecedo M Pastor M I Herreros J A F Merodo and QZhang ldquoComparison of two mathematical models for solvingthe dam break problem using the FEM methodrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 36-38 pp 3984ndash4005 2005

[13] L Begnudelli and B F Sanders ldquoUnstructured grid finite-volume algorithm for shallow-water flow and scalar transportwith wetting and dryingrdquo Journal of Hydraulic Engineering vol132 no 4 pp 371ndash384 2006

[14] Y Loukili and A Soulaımani ldquoNumerical tracking of shallowwater waves by the unstructured finite volume WAF approx-imationrdquo International Journal of Computational Methods inEngineering Science andMechanics vol 8 no 2 pp 75ndash88 2007

[15] M J CDıaz T C Rebollo ED Fernandez-Nieto JMGVidaandC Pares ldquoWell-balanced finite volume schemes for 2Dnon-homogeneous hyperbolic systemsApplication to the dambreakof Aznalcollarrdquo Computer Methods in Applied Mechanics andEngineering vol 197 no 45ndash48 pp 3932ndash3950 2008

[16] M Aliparast ldquoTwo-dimensional finite volume method fordam-break flow simulationrdquo International Journal of SedimentResearch vol 24 no 1 pp 99ndash107 2009

[17] A Shamsai and S H Mousavi ldquoEvaluation and assessmentof dam-break parameters of embankment dams case studiesaidaghmosh damrdquo Journal of Civil Engineering Islamic AzadUniversity vol 2 no 1 pp 65ndash73 2010

[18] J Xia B Lin R A Falconer and G Wang ldquoModelling dam-break flows over mobile beds using a 2D coupled approachrdquoAdvances in Water Resources vol 33 no 2 pp 171ndash183 2010

[19] A T Shamsi M G Hessaroyeh and M M Nasim ldquoTwodimensionalmodeling of dam-break Flowrdquo in Proceedings of theInternational Conference on Fluvial Hydraulics BraunschweigGermany September 2010

[20] S Erpicum B J Dewals P Archambeau and M PirottonldquoDam break flow computation based on an efficient flux vectorsplittingrdquo Journal of Computational and Applied Mathematicsvol 234 no 7 pp 2143ndash2151 2010

[21] A Baghlani ldquoSimulation of dam-break problem by a robustflux-vector splitting approach in Cartesian gridrdquo Scientia Iran-ica vol 18 no 5 pp 1061ndash1068 2011

[22] M Zhang and W M Wu ldquoA two dimensional hydrodynamicand sediment transport model for dam break based on finitevolume method with quadtree gridrdquo Applied Ocean Researchvol 33 no 4 pp 297ndash308 2011


[23] J Singh M S Altinakar and Y Ding ldquoTwo-dimensionalnumerical modeling of dam-break flows over natural terrainusing a central explicit schemerdquo Advances in Water Resourcesvol 34 no 10 pp 1366ndash1375 2011

[24] T J ChangHMKaoKHChang andMHHsu ldquoNumericalsimulation of shallow-water dam break flows in open channelusing smooth particle hydrodynamicsrdquo Journal of Hydrologyvol 408 no 1-2 pp 78ndash90 2011

[25] A Shakibaeinia and Y C Jin ldquoA mesh-free particle modelfor simulation of mobile-bed dam breakrdquo Advances in WaterResources vol 34 no 6 pp 794ndash807 2011

[26] H Sarveram and A Shamsai ldquoModeling the 1D dam-breakusing Quasi-Lagrangianmethodrdquo in Proceedings of the 1st Inter-national and 3rdNational Conference onDams andHydropowerTehran Iran February 2012

[27] T A Prickett ldquoModeling techniques for groundwater evalua-tionrdquo Advances in Hydroscience vol 10 pp 1ndash143 1975

[28] C A J Fletcher Computational Techniques for Fluid Dynamicsvol 2 Springer Berlin Germany 2nd edition 1991

[29] V V Rusanov ldquoThe calculation of the interaction of non-stationary shock waves with barriersrdquo Journal of Computationaland Mathematical Physics USSR vol 1 pp 267ndash279 1961

[30] G F Lin J S Lai and W D Guo ldquoFinite-volume component-wise TVD schemes for 2D shallow water equationsrdquo Advancesin Water Resources vol 26 no 8 pp 861ndash873 2003

[31] A van Dam and P A Zegeling ldquoA robust moving meshfinite volume method applied to 1D hyperbolic conservationlaws from magnetohydrodynamicsrdquo Journal of ComputationalPhysics vol 216 no 2 pp 526ndash546 2006

[32] C Lu J Qiu and R Wang ldquoA numerical study for the per-formance of the WENO schemes based on different numericalfluxes for the shallow water equationsrdquo Journal of Computa-tional Mathematics vol 28 no 6 pp 807ndash825 2010

[33] J J Stoker Water Waves The Mathematical Theory with Appli-cations Wiley-Interscience New York NY USA 1957

[34] J Hudson Numerical techniques for morphodynamic modeling[PhD thesis] The University of Reading West Berkshire UK2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


developed a hydrodynamic and sediment transportmodel fordam break flows The 2D SWEs were solved based on theFVMwith an unstructured quadtree mesh grid [22] Singh etal developed a 2D numerical model to solve the SWEs for thesimulation of dambreak problems [23] Chang et al proposeda meshless numerical model to investigate the shallow water(SW) dam break in 1D open channel A numerical modelwas used to solve the SWEs based on smoothed particlehydrodynamics (SPH) The concept of slice water particleswas adapted in the SPH-SWE formulation [24] Shakibaeiniaand Jin developed a new mesh-free particle model based onthe weakly compressible MPS (WC-MPS) formulation formodeling the dam break problem over a mobile bed [25]Sarveram and Shamsai investigated the dam break problemin converge and diverge rectangular channels in the unsteadystance using Saint Venant equations and a quasi-Lagrangianmethod [26]

This paper attempts to present a novel development for1D and 2D dam break problems in both wet and dry bedsA high-resolution FVM is employed to solve the SWEs onunstructured Voronoi meshThe local Lax-Friedrichs (LLxF)scheme is used for the estimation of fluxes at cells and thenumerical approximation of hyperbolic conservation laws

2 Research Methodology

21 Governing Equations The continuity and momentumequations of the SW can be written in different formsdepending upon the requirements of the numerical solutionof governing equations The 2D SWEs with source terms aregiven in the vector form considering a rigid bed channel [1]as follows

120597119880

120597119905+

120597119865

120597119909+

120597119866

120597119910+ 119878 = 0 (1)

119880 = [

[

ℎ

119906ℎ

Vℎ

]

]

(2)

119865 =

[[[[[

[

119906ℎ

1199062ℎ +

1

2119892ℎ2

119906Vℎ

]]]]]

]

(3)

119866 =

[[[[[[

[

Vℎ

119906ℎ

V2ℎ +1

2119892ℎ2

]]]]]]

]

(4)

119878 =

[[[[[[

[

0

minus119892ℎ (1198780119909

minus1198992

ℎ43119906radic1199062 + V2)

minus119892ℎ (1198780119910

minus1198992

ℎ43119906radic1199062 + V2)

]]]]]]

]

(5)

In this study the friction slopes were estimated using theManningrsquos formulas In the case of the dam break flow theinfluence of bottom roughness prevailed over the turbulentshear stress between cells Therefore the effective stress termswere neglected in the computation

22 Numerical Modeling Algorithm The main advantage ofthe FVM is that volume integrals in a partial differentialEquation (PDE) containing a divergence term are convertedto surface integrals using the divergence theorem Theseterms are then evaluated as fluxes at the surfaces of each FVBecause the flux entering a given volume is identical to thatleaving the adjacent volume these methods are conservativeAnother advantage of the FVM is that it is easily formu-lated to allow for unstructured meshes Unstructured gridmethods utilize an arbitrary collection of elements to fill thedomain These types of grids typically utilize triangles in 2Dand tetrahedrals in 3D although quadrilateral hexahedralVoronoi and Delaunay meshes can also be unstructuredIn the Voronoi mesh the chosen point has lower distancein the devoted domain rather than other points If onepoint has the same distance from several domains it willbe divided between domains Indeed these points createVoronoi cell boundaries Consequently internal sections ofthe Voronoi mesh consist of nodes belonging to one domainand boundaries include nodes that belong to several domains[27]

In this paper the studied domain was discretized usingunstructured Voronoi meshes Delaunay triangulation wascreated and then the Voronoi mesh was established usingthe Qhull program in MATLAB software The governingequation was discretized applying the FVM In this approachthe studied domainwas divided into several separated controlvolumes without any overlapping By integrating the gov-erning differential equation over every control volume thesystem of algebraic equations was created so that each ofits formulations belonged to one control volume and eachequation linked a parameter in the control volume node todifferent numbers of the parameter in adjacent nodes Thisconsequently led to the computation of the parameter in eachnode [28]

In order to solve discrete equations the parameter ineach node was computed considering its discrete equationand newest adjacent nodesrsquo values Solution procedure can beexpressed as follows

(1) Assuming an initial value in each node as an initialcondition

(2) Calculating the value in a node considering its dis-crete equation

(3) Performing pervious step for all nodes over thestudied domain one cycle is performed by repetitionthis step

(4) Verifying the convergence clause If this clause issatisfied the computing will end otherwise the com-putations will be repeated from the second step

As exact values of boundary conditions were not distinctthe Riemann boundary condition was utilized for computing






∬119860

120597119880

120597119905119889119860 + ∬

119860

(nabla sdot ) 119889119860 = ∬119860

119878119889119860 (6)


∬119860

120597119880

120597119905119889119860 + ∮

119897

( sdot ) 119889119897 = ∬119860

119878119889119860 (7)


119889119880119894

119889119905= minus

1

119860119894

sum

119895

( sdot 119894119895

Δ119897119894119895

) + 119878119894 (8)


120597119880

120597119905= minus

1

119860sum

119891

( sdot 119860 119890120576)119891

+ 119878 (9)

= 119867120576

sdot 119890120576

+ 119867120578

sdot 119890120578 (10)

Then

120597119880

120597119905= minus

1

119860sum

119891

119867120576119860119891

+ 119878

int

119905+Δ119905

119905

120597119880

120597119905119889119905 = int

119905+Δ119905

119905

(minus1

119860sum

119891

119867120576119860119891

) 119889119905 + int

119905+Δ119905

119905

119878119889119905

119880119899+1

119901minus 119880119899

119901= minus

1

119860sum

119891

119867120576119860119891

Δ119905 + 119878Δ119905

119891

119875

119890120585

119860119891

119890120578

Δ120585

Δ119909119891

Δ119910119891

119909

119886

119887

119910

nb

119860119910119891

119860119909119891


119867120576

= 1198991119865120576

+ 1198992119866120576

1198991

=(119909nb minus 119909

119901)

radic(119909nb minus 119909119901

)2

+ (119910nb minus 119910119901

)2

1198992

=(119910nb minus 119910

119901)

radic(119909nb minus 119909119901

)2

+ (119910nb minus 119910119901

)2

(11)


119880119899+1

119901= 119880119899

119901minus

Δ119905

119860sum

119891

[1198991119865120576119860119891

+ 1198992119866120576119860119891

]119899

+ 119878119899Δ119905 (12)




Yes

No

Start


Read the CFL






Plot results

End


119905 lt Tstop






119880119899+1

119901= 119880119899

119901minus

Δ119905

119860

[

[

sum

119891

((119865119891

)nbsdotout

119860119891

1198991119891

)










ℎ(m

)

119909 (m)

ℎ119906

DAM

0 05 1

ℎ119889


+ sum

119891

((119865119891

)nbsdotin

119860119891

1198991119891

)

+ sum

119891

((119866119891

)nbsdotout

sdot 119860119891

sdot 1198992119891

)

minus sum

119891

((119866119891

)nbsdotin

119860119891

1198992119891

)]

]

+ 119878Δ119905

(13)


(119865119891

)nbsdotout

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

nb minus 119906119899

119901)

(119865119891

)nbsdotin

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

119901minus 119906119899

nb)

(119866119891

)nbsdotout

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

nb minus V119899

119901)

(119866119891

)nbsdotin

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

119901minus V119899

nb)

(14)



)2

+ (119910nb minus 119910119901

)2

)

times (1

max1003816100381610038161003816100381610038161003816119906 + radic119892ℎ

1003816100381610038161003816100381610038161003816

+1

max1003816100381610038161003816100381610038161003816V + radic119892ℎ

1003816100381610038161003816100381610038161003816

)

(15)




3 Results



0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)


solution (119905 = 002 s)

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)

V-Break (119905 = 01 s)


(b)


0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)


solution (119905 = 002 s)

(a)

0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)

V-Break (119905 = 01 s)


(b)


















200

175

150

125

100

75

50

25

0

2001751501251007550250

95m

75m

119909 (m)

119910(m

)



0 while 119885











4 Conclusions



200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200100500 200100500 150

200100500 150200100500 150

150200

200

200

150

150

10 10

1010

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

2150

150

150 5050

0

0100100

10050119909

2150

150

150 5050

0

0100100

10050

150

150

150 5050

0

0 100100

10050119909

2150

150

150 5050

0

0

100

100100

150100

50119909





119909 (m)

119909 (m)

119910(m

)200

1208040

0

160119910

(m)

200

1208040

0

160

20012080400 160

20012080400 160

200

200

120120 8080

40

40

0

002

6

10

160

160

8

4

6

108

4

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

200

200

120120 8080

40

40

0

0160

160

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

(A1)

(A2)

200

100

50

0

150

200100500 150

200

100

50

0

150

200100500 150

200150

500

100

200150

10050

864200

200150

500

100

200150

10050

8

6

04

(B1)

(B2)

150

50100

150100

50

15100100

50

(C)





10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)


(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)


(b)


0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)


119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)


(b)



456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














∬119860

120597119880

120597119905119889119860 + ∬

119860

(nabla sdot ) 119889119860 = ∬119860

119878119889119860 (6)


∬119860

120597119880

120597119905119889119860 + ∮

119897

( sdot ) 119889119897 = ∬119860

119878119889119860 (7)


119889119880119894

119889119905= minus

1

119860119894

sum

119895

( sdot 119894119895

Δ119897119894119895

) + 119878119894 (8)


120597119880

120597119905= minus

1

119860sum

119891

( sdot 119860 119890120576)119891

+ 119878 (9)

= 119867120576

sdot 119890120576

+ 119867120578

sdot 119890120578 (10)

Then

120597119880

120597119905= minus

1

119860sum

119891

119867120576119860119891

+ 119878

int

119905+Δ119905

119905

120597119880

120597119905119889119905 = int

119905+Δ119905

119905

(minus1

119860sum

119891

119867120576119860119891

) 119889119905 + int

119905+Δ119905

119905

119878119889119905

119880119899+1

119901minus 119880119899

119901= minus

1

119860sum

119891

119867120576119860119891

Δ119905 + 119878Δ119905

119891

119875

119890120585

119860119891

119890120578

Δ120585

Δ119909119891

Δ119910119891

119909

119886

119887

119910

nb

119860119910119891

119860119909119891


119867120576

= 1198991119865120576

+ 1198992119866120576

1198991

=(119909nb minus 119909

119901)

radic(119909nb minus 119909119901

)2

+ (119910nb minus 119910119901

)2

1198992

=(119910nb minus 119910

119901)

radic(119909nb minus 119909119901

)2

+ (119910nb minus 119910119901

)2

(11)


119880119899+1

119901= 119880119899

119901minus

Δ119905

119860sum

119891

[1198991119865120576119860119891

+ 1198992119866120576119860119891

]119899

+ 119878119899Δ119905 (12)




Yes

No

Start


Read the CFL






Plot results

End


119905 lt Tstop






119880119899+1

119901= 119880119899

119901minus

Δ119905

119860

[

[

sum

119891

((119865119891

)nbsdotout

119860119891

1198991119891

)










ℎ(m

)

119909 (m)

ℎ119906

DAM

0 05 1

ℎ119889


+ sum

119891

((119865119891

)nbsdotin

119860119891

1198991119891

)

+ sum

119891

((119866119891

)nbsdotout

sdot 119860119891

sdot 1198992119891

)

minus sum

119891

((119866119891

)nbsdotin

119860119891

1198992119891

)]

]

+ 119878Δ119905

(13)


(119865119891

)nbsdotout

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

nb minus 119906119899

119901)

(119865119891

)nbsdotin

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

119901minus 119906119899

nb)

(119866119891

)nbsdotout

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

nb minus V119899

119901)

(119866119891

)nbsdotin

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

119901minus V119899

nb)

(14)



)2

+ (119910nb minus 119910119901

)2

)

times (1

max1003816100381610038161003816100381610038161003816119906 + radic119892ℎ

1003816100381610038161003816100381610038161003816

+1

max1003816100381610038161003816100381610038161003816V + radic119892ℎ

1003816100381610038161003816100381610038161003816

)

(15)




3 Results



0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)


solution (119905 = 002 s)

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)

V-Break (119905 = 01 s)


(b)


0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)


solution (119905 = 002 s)

(a)

0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)

V-Break (119905 = 01 s)


(b)


















200

175

150

125

100

75

50

25

0

2001751501251007550250

95m

75m

119909 (m)

119910(m

)



0 while 119885











4 Conclusions



200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200100500 200100500 150

200100500 150200100500 150

150200

200

200

150

150

10 10

1010

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

2150

150

150 5050

0

0100100

10050119909

2150

150

150 5050

0

0100100

10050

150

150

150 5050

0

0 100100

10050119909

2150

150

150 5050

0

0

100

100100

150100

50119909





119909 (m)

119909 (m)

119910(m

)200

1208040

0

160119910

(m)

200

1208040

0

160

20012080400 160

20012080400 160

200

200

120120 8080

40

40

0

002

6

10

160

160

8

4

6

108

4

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

200

200

120120 8080

40

40

0

0160

160

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

(A1)

(A2)

200

100

50

0

150

200100500 150

200

100

50

0

150

200100500 150

200150

500

100

200150

10050

864200

200150

500

100

200150

10050

8

6

04

(B1)

(B2)

150

50100

150100

50

15100100

50

(C)





10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)


(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)


(b)


0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)


119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)


(b)



456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Yes

No

Start


Read the CFL






Plot results

End


119905 lt Tstop






119880119899+1

119901= 119880119899

119901minus

Δ119905

119860

[

[

sum

119891

((119865119891

)nbsdotout

119860119891

1198991119891

)










ℎ(m

)

119909 (m)

ℎ119906

DAM

0 05 1

ℎ119889


+ sum

119891

((119865119891

)nbsdotin

119860119891

1198991119891

)

+ sum

119891

((119866119891

)nbsdotout

sdot 119860119891

sdot 1198992119891

)

minus sum

119891

((119866119891

)nbsdotin

119860119891

1198992119891

)]

]

+ 119878Δ119905

(13)


(119865119891

)nbsdotout

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

nb minus 119906119899

119901)

(119865119891

)nbsdotin

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

119901minus 119906119899

nb)

(119866119891

)nbsdotout

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

nb minus V119899

119901)

(119866119891

)nbsdotin

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

119901minus V119899

nb)

(14)



)2

+ (119910nb minus 119910119901

)2

)

times (1

max1003816100381610038161003816100381610038161003816119906 + radic119892ℎ

1003816100381610038161003816100381610038161003816

+1

max1003816100381610038161003816100381610038161003816V + radic119892ℎ

1003816100381610038161003816100381610038161003816

)

(15)




3 Results



0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)


solution (119905 = 002 s)

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)

V-Break (119905 = 01 s)


(b)


0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)


solution (119905 = 002 s)

(a)

0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)

V-Break (119905 = 01 s)


(b)


















200

175

150

125

100

75

50

25

0

2001751501251007550250

95m

75m

119909 (m)

119910(m

)



0 while 119885











4 Conclusions



200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200100500 200100500 150

200100500 150200100500 150

150200

200

200

150

150

10 10

1010

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

2150

150

150 5050

0

0100100

10050119909

2150

150

150 5050

0

0100100

10050

150

150

150 5050

0

0 100100

10050119909

2150

150

150 5050

0

0

100

100100

150100

50119909





119909 (m)

119909 (m)

119910(m

)200

1208040

0

160119910

(m)

200

1208040

0

160

20012080400 160

20012080400 160

200

200

120120 8080

40

40

0

002

6

10

160

160

8

4

6

108

4

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

200

200

120120 8080

40

40

0

0160

160

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

(A1)

(A2)

200

100

50

0

150

200100500 150

200

100

50

0

150

200100500 150

200150

500

100

200150

10050

864200

200150

500

100

200150

10050

8

6

04

(B1)

(B2)

150

50100

150100

50

15100100

50

(C)





10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)


(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)


(b)


0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)


119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)


(b)



456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















ℎ(m

)

119909 (m)

ℎ119906

DAM

0 05 1

ℎ119889


+ sum

119891

((119865119891

)nbsdotin

119860119891

1198991119891

)

+ sum

119891

((119866119891

)nbsdotout

sdot 119860119891

sdot 1198992119891

)

minus sum

119891

((119866119891

)nbsdotin

119860119891

1198992119891

)]

]

+ 119878Δ119905

(13)


(119865119891

)nbsdotout

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

nb minus 119906119899

119901)

(119865119891

)nbsdotin

=119865 (119906119899

nb) + 119865 (119906119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(119906 + radic119892ℎ) (119906119899

nb + 119906119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(119906119899

119901minus 119906119899

nb)

(119866119891

)nbsdotout

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

nb minus V119899

119901)

(119866119891

)nbsdotin

=119866 (V119899nb) + 119866 (V119899

119901)

2

minus1

2

1003816100381610038161003816100381610038161003816100381610038161003816

(V + radic119892ℎ) (V119899nb + V119899

119901

2)

1003816100381610038161003816100381610038161003816100381610038161003816

(V119899

119901minus V119899

nb)

(14)



)2

+ (119910nb minus 119910119901

)2

)

times (1

max1003816100381610038161003816100381610038161003816119906 + radic119892ℎ

1003816100381610038161003816100381610038161003816

+1

max1003816100381610038161003816100381610038161003816V + radic119892ℎ

1003816100381610038161003816100381610038161003816

)

(15)




3 Results



0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)


solution (119905 = 002 s)

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)

V-Break (119905 = 01 s)


(b)


0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)


solution (119905 = 002 s)

(a)

0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)

V-Break (119905 = 01 s)


(b)


















200

175

150

125

100

75

50

25

0

2001751501251007550250

95m

75m

119909 (m)

119910(m

)



0 while 119885











4 Conclusions



200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200100500 200100500 150

200100500 150200100500 150

150200

200

200

150

150

10 10

1010

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

2150

150

150 5050

0

0100100

10050119909

2150

150

150 5050

0

0100100

10050

150

150

150 5050

0

0 100100

10050119909

2150

150

150 5050

0

0

100

100100

150100

50119909





119909 (m)

119909 (m)

119910(m

)200

1208040

0

160119910

(m)

200

1208040

0

160

20012080400 160

20012080400 160

200

200

120120 8080

40

40

0

002

6

10

160

160

8

4

6

108

4

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

200

200

120120 8080

40

40

0

0160

160

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

(A1)

(A2)

200

100

50

0

150

200100500 150

200

100

50

0

150

200100500 150

200150

500

100

200150

10050

864200

200150

500

100

200150

10050

8

6

04

(B1)

(B2)

150

50100

150100

50

15100100

50

(C)





10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)


(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)


(b)


0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)


119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)


(b)



456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)


solution (119905 = 002 s)

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Wat

er d

epth

(m)

119909 (m)

V-Break (119905 = 01 s)


(b)


0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)


solution (119905 = 002 s)

(a)

0010203040506070809

1

0 02 04 06 08 1

Velo

city

(ms

)

119909 (m)

V-Break (119905 = 01 s)


(b)


















200

175

150

125

100

75

50

25

0

2001751501251007550250

95m

75m

119909 (m)

119910(m

)



0 while 119885











4 Conclusions



200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200100500 200100500 150

200100500 150200100500 150

150200

200

200

150

150

10 10

1010

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

2150

150

150 5050

0

0100100

10050119909

2150

150

150 5050

0

0100100

10050

150

150

150 5050

0

0 100100

10050119909

2150

150

150 5050

0

0

100

100100

150100

50119909





119909 (m)

119909 (m)

119910(m

)200

1208040

0

160119910

(m)

200

1208040

0

160

20012080400 160

20012080400 160

200

200

120120 8080

40

40

0

002

6

10

160

160

8

4

6

108

4

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

200

200

120120 8080

40

40

0

0160

160

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

(A1)

(A2)

200

100

50

0

150

200100500 150

200

100

50

0

150

200100500 150

200150

500

100

200150

10050

864200

200150

500

100

200150

10050

8

6

04

(B1)

(B2)

150

50100

150100

50

15100100

50

(C)





10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)


(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)


(b)


0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)


119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)


(b)



456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













200

175

150

125

100

75

50

25

0

2001751501251007550250

95m

75m

119909 (m)

119910(m

)



0 while 119885











4 Conclusions



200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200100500 200100500 150

200100500 150200100500 150

150200

200

200

150

150

10 10

1010

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

2150

150

150 5050

0

0100100

10050119909

2150

150

150 5050

0

0100100

10050

150

150

150 5050

0

0 100100

10050119909

2150

150

150 5050

0

0

100

100100

150100

50119909





119909 (m)

119909 (m)

119910(m

)200

1208040

0

160119910

(m)

200

1208040

0

160

20012080400 160

20012080400 160

200

200

120120 8080

40

40

0

002

6

10

160

160

8

4

6

108

4

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

200

200

120120 8080

40

40

0

0160

160

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

(A1)

(A2)

200

100

50

0

150

200100500 150

200

100

50

0

150

200100500 150

200150

500

100

200150

10050

864200

200150

500

100

200150

10050

8

6

04

(B1)

(B2)

150

50100

150100

50

15100100

50

(C)





10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)


(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)


(b)


0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)


119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)


(b)



456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200175

125

75100

5025

0

150

200100500 200100500 150

200100500 150200100500 150

150200

200

200

150

150

10 10

1010

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

200

200

200

150

150

150

50

5050

0

0

0

100

100100

200150

10050

10

550

119909 (m)119910 (m)

119867(m

)

2150

150

150 5050

0

0100100

10050119909

2150

150

150 5050

0

0100100

10050

150

150

150 5050

0

0 100100

10050119909

2150

150

150 5050

0

0

100

100100

150100

50119909





119909 (m)

119909 (m)

119910(m

)200

1208040

0

160119910

(m)

200

1208040

0

160

20012080400 160

20012080400 160

200

200

120120 8080

40

40

0

002

6

10

160

160

8

4

6

108

4

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

200

200

120120 8080

40

40

0

0160

160

Free

surfa

ceel

evat

ion

(m)

119910 (m)119909 (m)

(A1)

(A2)

200

100

50

0

150

200100500 150

200

100

50

0

150

200100500 150

200150

500

100

200150

10050

864200

200150

500

100

200150

10050

8

6

04

(B1)

(B2)

150

50100

150100

50

15100100

50

(C)





10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)


(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)


(b)


0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)


119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)


(b)



456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10987650

00

50

50

50100

100

100

150

150

150

200

200

0 50 100 150 200

200

119909 (m)

119910(m

)

119910(m

)

Wat

er d

epth

(m)

119909 (m)


(a)

119909 (m)

050

100150

200 0

50

100150

200

119910(m

)

0

50

100

150

200

119910(m

)

10864

02

Wat

er d

epth

(m)

050 100 150 200

119909 (m)


(b)


0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)


119909 (m)

(a)

0

2

4

6

8

10

12

0 50 100 150 200

Wat

er d

epth

(m)

Voronoi (V-Break)

119909 (m)


(b)



456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










456789

1011

Wat

er d

epth

(m)


CFLF8LLxF

0 50 100 150 200119909 (m)


119909 (m)

0 8040 120 160 200 0 50 100150

200

119910 (m)

1098

67

5

1234

Wat

er d

epth

(m)


0

50

100

150

200

0 50 100 150 200119909 (m)

119910(m

)


119909 (m)

050

100150

200 0

50

100

150

200

119910(m

)

108642W

ater

dep

th (m

)



Nomenclatures






















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Subscripts









Superscripts



References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







