SPH Modeling of One-Dimensional Nonrectangular andNonprismatic Channel Flows with Open Boundaries

Tsang-Jung Chang1 and Kao-Hua Chang2

Abstract: In this study, the authors solve the shallow water equations (SWE) with smoothed particle hydrodynamics (SPH) for one-dimensional (1D) nonrectangular and nonprismatic channel flows with open boundaries. To date, 1D SPH-SWE has been only developedto simulate rectangular prismatic channel flows with closed and open boundaries. However, for practical hydraulic problems, channel crosssections are not always rectangular and prismatic. A general approach is proposed in this study to extend the engineering application rangeof 1D SPH-SWE to nonrectangular and nonprismatic channels with open boundaries by introducing the wetted cross-sectional area and thewater discharge in SWE and combining the method of specified time interval with the inflow/outflow algorithm. Three benchmark studycases, aiming at testing various steady flow regimes in nonrectangular and nonprismatic channels, are adopted to validate the newly proposedapproach. Through the investigation of the convergence analysis and numerical accuracy test of the study cases, the results show that thepresent SPH-SWE approach is capable to model 1D nonrectangular and nonprismatic channel flows with open boundaries. DOI: 10.1061/(ASCE)HY.1943-7900.0000782. 2013 American Society of Civil Engineers.

CE Database subject headings: Hydrodynamics; Shallow water; Channel flow.

Author keywords: Smoothed particle hydrodynamics; Shallow water equations; Method of specified time interval; Open boundaries;Nonrectangular and nonprismatic channel.

Introduction

The shallow water equations (SWE) are widely used to math-ematically describe a wide variety of free surface flows in rivers,estuaries, and coasts (Chaudhry 1993; Cunge et al. 1980). ManyEulerian mesh-based methods are currently available to solveSWE in hydraulic engineering applications. Recently, a pureLagrangian meshless method, smoothed particle hydrodynamics(SPH) (Monaghan 2005; Gomez-Gesteira et al. 2010), is widelyapplied to the numerical formulation of SWE (Wang and Shen1999; Ata and Soulaimani 2005; Rodriguez-Paz and Bonet 2005;De Leffe et al. 2010; Vacondio et al. 2012a; Chang et al. 2011; Kaoand Chang 2012). Compared with Eulerian mesh-based methods,SPH has the following advantages: (1) no convective term (nonlin-ear term), which often causes numerical oscillations, exists in thegoverning equations (Liu and Liu 2003); (2) the wave propagationof free surface can be intrinsically tracked (De Leffe et al. 2010);and (3) the wet-dry interface can be automatically described with-out any special treatment (Vacondio et al. 2012a). Hence, SPH-SWE has recently been receiving more and more attention. Somesignificant achievements are summarized. Wang and Shen (1999)firstly investigated inviscid dam-break flows using SPH-SWE. Ataand Soulaimani (2005) derived a new artificial viscosity for thedam-break problem with wet bed. Rodriguez-Paz and Bonet (2005)

proposed a variational SPH-SWE formulation to maintain the massand momentum conservation. De Leffe et al. (2010) adopted ananisotropic kernel with variable smoothing length and performedSPH-SWEmodeling of shallow-water coastal flows. Vacondio et al.(2012a) presented the modified virtual boundary particle method todeal with various complex shapes of solid boundaries. Chang et al.(2011) and Kao and Chang (2012) extended SPH-SWE to inves-tigate shallow-water dam-break flows in realistic open channels.Additionally, to improve the SPH-SWE solutions, Vacondio et al.(2012) developed a particle splitting procedure to enhance the res-olution in small-depth problems and studied dam break cases withanalytic and experimental results. Vacondio et al. (2013) appliedtwo corrections for balancing discontinuous bed slopes includingconservative and nonconservative approaches to handle bottom dis-continuity problems.

The aforementioned references of SPH-SWE are all related toclosed boundary conditions. The implementation of SPH-SWE inopen boundary conditions remains difficult because of the Lagran-gian nature of SPH. Vacondio et al. (2012b) pioneeringly intro-duced the characteristic boundary method into SPH-SWE tosimulate rectangular prismatic channel flows with open boundaries.They adopted the Riemann invariants to determine proper boundaryconditions to overcome the well-posed problems (Anderson 1995).Their work enabled 1D SPH-SWE to deal with open boundaries inrectangular prismatic channels. However, for natural channels, theshape of channel cross sections is not always rectangular and pris-matic. The Riemann invariants cannot be formulated in the casesof nonrectangular channels except triangular channels (Sanders2001). It is also not suitable to simulate nonprismatic channel flowswith the water depth and the water velocity in SWE (Vacondioet al. 2012b). To extend the engineering application range of 1DSPH-SWE, the authors adopt the wetted cross-sectional area andthe water discharge in SWE to reflect irregular channel shapein nonprismatic channels. The authors also solve the character-istic equations and to establish the open boundary conditions in

1Professor, Dept. of Bioenvironmental Systems Engineering, NationalTaiwan Univ., Taipei 10617 Taiwan. E-mail: tjchang@ntu.edu.tw

2Postdoctoral Researcher, Dept. of Bioenvironmental SystemsEngineering, National Taiwan Univ., Taipei 10617 Taiwan (correspondingauthor). E-mail: f94622026@ntu.edu.tw

Note. This manuscript was submitted on October 1, 2012; approved onMay 23, 2013; published online on May 25, 2013. Discussion period openuntil April 1, 2014; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Hydraulic Engineering,Vol. 139, No. 11, November 1, 2013. ASCE, ISSN 0733-9429/2013/11-1142-1149/$25.00.

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http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000782http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000782http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000782http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000782http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000782

nonrectangular channels with the method of specified time intervals(Chaudhry 1993; Sturm 2010).

This paper is organized as follows: the core of SPH andSPH-SWE formulation are briefly introduced. Then the newlyproposed approach for open boundaries is presented. Finally, threebenchmark study cases, representing various steady flows innonrectangular and nonprismatic channels, are adopted to validatethe newly proposed approach. In these three study cases, threecombinations of inflow/outflow boundary conditions and channelcross sections illustrate the ability of the newly proposed approachon various channel cross sections and flow conditions.

SPH-SWE Methodology

SPH Formulation

SPH is a pure interpolation method. Any physical quantity ofparticle i (i) is approximated by the weighted summation as

i Xj

mjjAj

Wijxi xjj; hi 1

where mj ( AjVj Ajx0) is the mass of particle j; Vj ( x0at the initial state) is the volume of particle j; Aj is the wetted cross-section area of particle j;x0 is the initial particle spacing; xi is theposition of particle i; rij ( jxi xjj) is the distance between par-ticle i and particle j;Wijxi xjj; hi ( Wi) is the kernel functionof particle I; and the cubic spline function (Liu and Liu 2003;Chang et al. 2012b) is chosen as the kernel function, and hi isthe smoothing length of particle i ( 1.3x0).

Two SPH formulas of the operation of the gradient or diver-gence used in common are listed in the following (Liu and Liu2003). Eq. (2) is symmetric with respect to i and j to be utilizedto perform the divergence operation. The gradient operation is per-formed by Eq. (3), which is asymmetric with respect to i and j:

i 1AiXj

mjj i Wijxi xjj; hi 2

i AiXj

mj

iA2i

jA2j

Wijxi xjj; hi 3

Shallow Water Equations

The traditional SPH-SWE solves the water depth and the watervelocity to provide flow simulations, which is only suitable forrectangular and prismatic channel flows. The wetted cross-sectionalarea and the water discharge are introduced in SPH-SWE herein toextend it to nonrectangular and nonprismatic channel flows. TheLagrangian form of SWE in terms of the wetted cross sectionand the water discharge can be written in Eq. (4) (the continuityequation) and Eq. (5) (the momentum equation):

DADt

A Q=Ax 4

DQDt

Q Q=Ax gAdwx gAS0 Sf 5

where D=Dt is the total time derivative [D=D =t u=2x;Q is the water discharge; u is the water velocity ( Q=A); A is thewetted cross-sectional area; dw is the water depth and it is generallya function of A [specifically, A Bdw for a rectangular cross

section and A dw (Bmdw) for a trapezoidal cross section];B is the bottom width, m is the side slope; S0 is the bed slope;Sf is the friction slope ( n2Q2=A2R4=3); n is the Manning rough-ness coefficient; R is the hydraulic radius; and g is the gravitationalacceleration.

SPH-SWE Implementation

Evaluation of Wetted Cross-Sectional AreaCommonly, two numerical approaches can solve the wetted cross-sectional area in SPH-SWE. One is solving the continuity equationof Eq. (4), and another is using the weighted summation formula asshown in Eq. (1). The present model chooses the latter approach tocalculate the wetted cross-section area of each particle i. A variablesmoothing length scheme is applied to obtain better accuracy ofSWE solutions. Thus, the smoothing length of particle i is con-nected to the wetted cross-section area (Ata and Soulaimani 2005;Rodriguez-Paz and Bonet 2005; De Leffe et al. 2010; Vacondioet al. 2012a) with

hi h0;iA0;iAi

1=Dm 6

where A0;i and h0;i are the initial wetted cross-sectional area andsmoothing length for particle i, respectively; and Dm is the numberof space dimensions (Dm 1 in this study).

Because of using the variable smoothing length scheme, theNewton-Raphson iterations are thus performed to solve the wettedcross-sectional area of particle i. The iterative procedure (Ata andSoulaimani 2005; Rodriguez-Paz and Bonet 2005; De Leffe et al.2010; Vacondio et al. 2012a) is as follows:

Ak1i Aki1 Res

ki Dm

Reski Dm ki

7

with

Reski Aki Xj

mjWijxi xjj; hki 8

where Reski is the residual of particle i at the kth iteration; and ki is

defined as

ki Xj

mjrijdWidrij

9

The Newton-Raphson iterative procedure will be stopped as

Resk1iAki

1010 10

Discretized Momentum Equation. The discretized SPH formof the momentum equation [Eq. (5)] is thus given in the followingto evaluate the rate of water discharge of each particle i:

DQiDt

QiAi

Xj

mj

QjAj

QiAi

Wi

gA2iXj

mj

diA2i

djA2j

Wi gAiS0;i Sf;i 11

where Wi 1=2Wijxi xjj; hi Wjjxi xjj; hj.Because the smoothing length is variable, the expression of thegradient of a kernel function (Wi) is a hybrid combination of

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the scatter andthe gather interpretations herein (Hernquist andKatz 1989).

Fig. 1 shows the computational domain. In this study, the com-putational domain is divided into three zones, i.e., the inflow zone(between the inlet boundary and the inflow boundary), the fluidzone (between the inflow boundary and the outflow boundary),and the outflow zone (between the outflow boundary and the outletboundary). In addition, four types of particles are used, includinginflow particles, inner particles, outflow particles, and virtual bedparticles (i.e., rectangular points, circle points, diamond points, andtriangular points of Fig. 1). The former three types are, respectively,within the flow zone, the fluid zone, and the outflow zone. The lasttype is arranged in the computational domain.

As to the bed gradient term S0;i in the right-hand side ofEq. (11), a virtual bed particle that has one specified value of bedslope is introduced (Vacondio et al. 2012a). The volume of a virtualbed particle equals to x0. Then the bed slope of the inflow/inner/outflow particle i is computed by the weighted summation of thebed slope of each virtual bed particle surrounding the inflow/inner/outflow particle i as shown in Eq. (12):

S0;i Xj

Vvbj Svb0;j

~Wjjxi xvbj j; hvbj 12

where the Shepard filter scheme is used (Randles and Libersky1996; Chang et al. 2012a), ~W means the corrected kernelfunction as

~Wjjxi xvbj j; hvbj Wjjxi xvbj j; hvbj P

j Vvbj Wjjxi xvbj j; hvbj 13

The superscript vb indicates the virtual bed particle andhvbj 1.3x0.

Additionally, for the aim of enabling the numerical methodstable, an artificial viscosity is introduced (Ata and Soulaimani2005) as

Yij

Aij cijuij xijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2ij 2

q 14

where c is the celerity ( ffiffiffiffiffiffiffiffiffigHdp ); Hd is the hydraulic depth;Aij Ai Aj=2; cij ci cj=2; uij ui uj; xij xi xj;and is a constant ( 106).

Time-Marching Scheme. To update particle positions andvelocities in time, the leap-frog time-marching scheme (Rodriguez-Paz and Bonet 2005; Vacondio et al. 2012a) is herein used. Becausethe SPH is an explicit method, the time step (t) has to satisfy theCFL condition as

t 0.25 min

x0ui

ffiffiffiffiffiffiffiffiffiffiffigHd;i

p

15

Open Boundary Treatment

Because the Riemann invariants can only be formulated for rec-tangular and triangular channels (Sanders 2001), the use of theRiemann invariants at the inflow/outflow boundaries to seek openboundary conditions (Vacondio et al. 2012b) has limited applica-tions. As a result, a more general approach, which adopts the wet-ted cross-sectional area and the water discharge in SWE andcombines the method of specified time interval with the inflow/outflow algorithm, is proposed to model 1D nonrectangular andnonprismatic channels with open boundaries. The supercriticalflow problems require two variables (the wetted cross-sectionalarea and the water discharge) at the inflow boundaries while thesubcritical flow problems need only one variable (the wettedcross-sectional area or the water discharge) at both inflow and out-flow boundaries. Hence, the method of specified time interval isused to solve the characteristic equations and obtain the remainingvariable at the boundaries. The time step is specified according tothe CFL condition.

Characteristic Equations

The characteristic equations (Chaudhry 1993; Cunge et al. 1980)are classified into the positive and negative forms based on the di-rections of characteristic lines. Eqs. (16) and (17) are the positivecharacteristic equations and Eqs. (18) and (19) are the negativecharacteristic equations:

CDdwDt

cgDuDt

cS0 Sf 16

along

C dxdt

u c 17

and

C DdwDt

cgDuDt

cS0 Sf 18

along

C dxdt

u c 19

Fig. 1. Sketch of the computational domain

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Method of Specified Time Interval

The method of specified time interval (Chaudhry 1993; Sturm2010) is often used to evaluate the boundary conditions for explicitnumerical methods. Fig. 2 shows the sketch of the method of speci-fied time interval. The characteristic equations of Eqs. (16) through(19) can be discretized by the finite difference approximations ofthe derivatives to obtain the discretized characteristic equations,i.e., Eqs. (20) through (23). In this method, Eqs. (20) through (23)are solved to seek the unknown values of the variables at the inflow/outflow boundaries:

CuS uR gcR dw;S dw;R gS0;R Sf;Rt 20

C xS xRt

uR cR 21

CuP uL gcL dw;P dw;L gS0;L Sf;Lt 22

C xP xLt

uL cL 23

where the subscripts P and S represent at the inflow and outflowboundaries, respectively, and the subscripts L and R both representinside the fluid zone.

As the flow is subcritical at the inflow/outflow boundaries, thevalues of the water velocity and the water depth are prescribedat the inflow and outflow boundaries, respectively. Hence, Eq. (20)is solved together with Eq. (21) to provide the value of thewater depth at the inflow boundary. Similarly, the value of the watervelocity at the outflow boundary is given by solving Eqs. (22)through (23). The detailed procedure of solving the discretizedcharacteristic equations is presented as follows.

Point S at the outflow boundary is on the trajectory of the pos-itive characteristic line RS as shown as Fig. 2, the water depth andthe water velocity at point S can be evaluated by solving Eq. (20)only if all of the variables at point R are known. To obtain the valuesof all the variables at point R, the computational domain isseparated into ghost cells whose length equals to the initial particlespacing (x0) (Fig. 2). The linear interpolation of the water veloc-ity between point C and point D is performed and that combinesEq. (21) to derive Eq. (24):

uD uRuD uC

xS xRx0

uR cRtx0

24

In a similar interpolation, the linear relationships of the celerityand the water depth between point C and point D are presented inEqs. (24) and (25):

cD cRcD cC

uR cRtx0

25

dw;D dw;Rdw;D dw;C

uR cRtx0

26

To solve Eqs. (24) and (25) together, the water velocity uR andthe celerity cR can be found, and then the water depth dw;R can alsobe determined by Eq. (26).

In the same way, for point P of Fig. 2 on the trajectory of thenegative characteristic line LP, the linear interpolations betweenpoint A and point B are performed and the discretized negativecharacteristic equations of Eq. (22) and (23) are utilized to givethe water depth and the water velocity at point P.

Inflow/Outflow Boundary Conditions

Four kinds of boundary conditions can be specified according tothe local flow conditions (the Froude number) at the inflow/outflowboundaries:1. Subcritical outflow condition: As the subcritical flow occurs at

the outflow boundary (Line RS of the Fig. 2), the water depthis prescribed, and the water discharge is determined by usingEq. (28), which is derived from combining Eq. (20) with thecontinuity equation of Eq. (27):

QS ASuS 27

QS ASuR gcR dw;S dw;R gS0;R Sf;Rt

28

2. Subcritical inflow condition: When the subcritical flow occursat the inflow boundary (Line LP of the Fig. 2), the water dis-charge is prescribed, and the water depth is calculated throughsolving Eq. (30) using the Newton-Raphson iterations.Eq. (30) is derived from combining Eq. (22) with the conti-nuity equation of Eq. (29):

QP APuP 29

fdw;P APuL

gcL

dw;P dw;L

gS0;L Sf;Lt 0 30

3. Supercritical outflow condition: If the supercritical flow oc-curs at the outflow boundary, there is not necessary to specifythe boundary condition there. Hence, the water depth and thewater discharge are extrapolated from the fluid zone.

4. Supercritical inflow condition: Asthe supercritical flow occursat the inflow boundary, the water depth and the waterdischarge are prescribed.

Inflow/Outflow Algorithms

The inflow/outflow algorithm developed by Federico et al. (2012)is applied to solve open boundary problems in this study. The de-tailed procedure includes the following steps: (1) When an inflowparticle (rectangular points of Fig. 1) moves across the inflowboundary, it will become an inner particle (circle points of Fig. 1).The momentum equation then controls the behavior of an innerparticle. At the same time, the new inflow particle will be producedin the inflow zone and the inflow boundary conditions such asthe water discharge (Qp) and the water depth (dw;p) at the inflow

Fig. 2. Sketch of the method of specified time interval

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boundary are assigned to the inflow particle. (2) An inner particle isrecognized as an outflow particle (diamond points of Fig. 1) as itmoves across the outflow boundary. In addition, the outflow boun-dary conditions such as the water discharge (Qs) and the waterdepth (dw;s) at the outflow boundary are assigned to the outflowparticle. An outflow particle will be deleted as it crosses the outletboundary. Therefore, the total number of particles within thecomputation process will remain a constant as flows reach thesteady state.

Results and Discussion

In this section, three benchmark study cases (Macdonald 1995;Macdonald et al. 1997; Khalifa 1980) validate the newly proposedapproach. Three combinations of inflow/outflow open boundaryconditions and channel cross sections illustrate the ability of theproposed approach on various channel shapes and flow conditions.To investigate the convergence analysis and numerical accuracytest, the L2 relative error norm based on the variable , as shownin Eq. (31), is introduced:

L2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP simulatedi analytic=measuredi 2P analytic=measuredi 2

vuut 31

where simulatedi and analytic=measuredi are the simulated physical

quantity and the analytic or measured data at the ith particle,respectively.

Rectangular Prismatic Channel

The purpose of the first study case is to investigate the perfor-mance of the proposed approach on rectangular prismatic channelflows given by MacDonald (1995). The channel is 100-m longand 10-m wide and the Manning roughness coefficient of thechannel is 0.03. The bed slope is nonuniform and the bed eleva-tion profile of the channel is plotted in Fig. 3(a). This study casehas the analytic solution. The flow is supercritical at the inflowboundary and at the outflow boundary, with the given water dis-charge 20 m3 s1 and the water depth 0.7506 mat the inflowboundary.

Convergence AnalysisIn this study case, four initial particle numbers of 100, 200, 500,1,000 in the computational domain (i.e., the initial particle spacings(x0) are 1, 0.5, 0.2, and 0.1 m, respectively) are considered toconduct the convergence analysis. Through Eq. (31), the valuesof L2dw (L2 relative error norm based on the water depth) andL2Q (L2 relative error norm based on the water discharge) aresummarized in Table 1. Both L2dw and L2Q decrease as theinitial particle number increases. Thus, the proposed approachcan converge to the analytic solution. In addition, the convergencerates are 1.19 and 1.5, respectively, for the water depth and thewater discharge. A convergence criterion is defined as the differ-ence in L2dw or L2Q between two initial particle numbers lessthan 0.005. The initial particle number of 500 is adequate hereinand the simulated results of using 500 particles are given in thisstudy case.

Numerical AccuracyFig. 3(b) compares the numerical accuracy of the simulated resultand the analytic solution of the water depth profile in the channel.Fig. 3(c) demonstrates the spatial variation of the simulated Froudenumber of the channel and Fig. 3(d) further shows the simulatedand analytic water discharges of the channel. From Figs. 3(a and c),

the flow is supercritical at the inflow boundary and changes tosubcritical since the channel bed slope reduced, which results in ahydraulic jump occurring at x 100=3 m as shown as Fig. 3(b).After the hydraulic jump, the flow returns to supercritical due to theeffect of sharp variation in the channel bed slope.

For the numerical accuracy test, as shown in Fig. 3(b), thesimulated and analytic water depth profiles of the entire channelare well consistent. In Table 1, L2dw is only 0.013. Furthermore,Fig. 3(d) shows the simulated and analytic water discharges of thechannel. L2Q is about 0.0035 (Table 1). An oscillation near thehydraulic jump is detected in Fig. 3(d). This is the main source of

Fig. 3. (a) Bed elevation profile of the rectangular prismatic channel;(b) profiles of the simulated and analytic water depths; (c) spatial var-iation of the simulated Froude number of the channel; (d) simulated andanalytic water discharges of the channel

Table 1. L2dw and L2Q of Various Initial Particle Numbers (N) in theRectangular Prismatic Channel

N (x0) L2dw Difference (dw) L2Q Difference (Q)(1) 100 (1.0 m) 0.026 1 2 0.006 0.0040 1 2 0.0004(2) 200 (0.5 m) 0.020 2 3 0.007 0.0036 2 3 0.0001(3) 500 (0.2 m) 0.013 3 4 0.002 0.0035 3 4 0.0001(4) 1000 (0.1 m) 0.011 0.0034

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L2Q in this case. Such a discontinuity in water depths near thehydraulic jump results in disordered particles that lead to the waterdischarge oscillation. Overall, the simulated results of water dis-charges are accurately predicted.

Trapezoidal Prismatic Channel

To test the performance of the newly proposed approach on non-rectangular channel flows, the trapezoidal prismatic channel prob-lem given by MacDonald et al. (1997) was selected as the secondstudy case. This study case aims to demonstrate that as dealing withinflow/outflow boundary conditions, the proposed approach canimprove the limitation of the Riemann invariants that can onlybe formulated for rectangular and triangular channels in the char-acteristic boundary method given by Vacondio et al. (2012b). Thenonuniform bed-slope channel is 1,000-m long and its width andperimeter are 10 2dw m and 10 2

ffiffiffi2

pdw m, respectively. The

Manning roughness coefficient of the channel is 0.02. The flowsat the inflow and outflow boundaries are both subcritical, and thewater discharge 20 m3 s1 and the depth 1.35 m are specified as theinflow and outflow boundary conditions, respectively.

Convergence AnalysisIn the convergence analysis, four initial particle numbers of 100,200, 500, 1,000 in the computational domain [i.e., the initial particlespacings (x0) are 10, 5, 2, and 1 m, respectively] are adoptedin this case. Table 2 shows the values of L2dw and L2Q for thefour initial particle numbers. The simulated results approach to theanalytic solution as the initial particle number increases. The con-vergence rates for the water depth and the water discharge are 0.78and 1.77, respectively. The differences in L2dw and L2Q areboth less than 0.005 between 500 particles and 1,000 particles.Thus, the simulated results of using 500 particles are given in thisstudy case.

Numerical AccuracyFigs. 4(a and c) depict the bed elevation and simulated Froude num-ber profiles of the channel, respectively. The bed slope is nonuni-form and the flow changes from subcritical to supercritical, andback to subcritical. A hydraulic jump occurs at the location ofx 600 m. The simulated and analytic water depth profiles ofthe channel are both shown in Fig. 4(b). The agreement betweenthe simulated and analytic results is quite satisfactory. For thenumerical accuracy test, L2dw is only 0.009 (Table 2). In addi-tion, as shown in Fig. 4(d), the simulated water discharges are com-pared with the analytic solution. In Table 2, L2Q is 0.0035. Themain source of L2Q is also from an oscillation occurs near thehydraulic jump in Fig. 4(d). To sum up, the proposed approach issignificantly capable of modeling nonrectangular channel flows.

Rectangular Nonprismatic Channel

The third study case is to investigate the performance of the pro-posed approach on nonprismatic channel flows. The experimentalmodel of flows in a rectangular divergent channel conducted by

Khalifa (1980) is adopted. The channel is horizontal, 2.5-m longand its width is variable as shown as Fig. 5. The flow is supercriticalat the inflow boundary, and it returns to subcritical at the outflowboundary because of the channel expansion. As to the inflow/outflow boundary conditions, the discharge 0.0263 m3 s1 andthe depth 0.088 m are given as the inflow boundary conditionsand the depth 0.195 m is specified as the outflow boundarycondition. The Manning roughness coefficient is set as 0.015.

Convergence AnalysisIn this case, three initial particle numbers, including 25(x0 0.1 m), 50 (x0 0.05 m), and 100 (x0 0.025 m)particles, are given in the computational domain to perform theconvergence analysis. The values of L2dw and L2Q for the threeinitial particle numbers in the rectangular nonprismatic channel arereported in Table 3. No obvious difference in L2dw exists as theinitial particle number is more than 50. However, L2Q decreasesas the initial particle number increases. The proposed approach ob-viously can be convergent in the rectangular nonprismatic channel.Based on the differences in L2dw and L2Q of Table 3, thesimulated results of the initial particle number of 50 are appropriateto describe the flow phenomena.

Table 2. L2dw and L2Q of Various Initial Particle Numbers (N) in theTrapezoidal Prismatic Channel

N (x0) L2dw Difference (dw) L2Q Difference (Q)(1) 100 (10 m) 0.020 1 2 0.006 0.0059 1 2 0.0018(2) 200 (5 m) 0.014 2 3 0.005 0.0041 2 3 0.0006(3) 500 (2 m) 0.009 3 4 0.002 0.0035 3 4 0.0001(4)1000 (1 m) 0.007 0.0034

Fig. 4. (a) Bed elevation profile of the trapezoidal prismatic channel;(b) profiles of the simulated and analytic water depths; (c) spatial var-iation of the simulated Froude number of the channel; (d) simulated andanalytic water discharges of the channel

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Numerical AccuracyThe channel width profile of the entire rectangular nonprismaticchannel is described in Fig. 5. As shown in Fig. 5, the channelexpansion starts at x 0.65 m, resulting in a transition from super-critical flow to subcritical flow. Thus, a hydraulic jump occursaround x 1 m. Fig. 6(a) presents the comparison between thesimulated and measured water depth profiles. In Table 3, L2dwis about 0.068. The simulated and measured water depths are quiteconsistent. Fig. 6(b) shows the comparison between the simulatedand measured water discharges. L2Q is 0.0199 (Table 3). Again,an oscillation in the simulated water discharges near the hydraulicjump is found in Fig. 6(b). The simulated water discharges are in-fluenced by the rapid variation of water depth in the hydraulic

jump. The result shows that the proposed approach can deal withnonprismatic channel flows as well.

Conclusions

The authors propose a new SPH-SWE approach that combines themethod of specified time interval with the inflow/outflow algorithmto model 1 D nonrectangular and nonprismatic channels with openboundaries. Unlike the traditional 1 D SPH-SWE approach, thenew approach adopts the wetted cross-section area and the waterdischarge in SWE to simulate flows in prismatic and nonprismaticchannels. The authors introduce the method of specified time in-terval to extend the SPH-SWE application to 1 D nonrectangularchannel flows. Three study cases, aiming at various steady flowregimes in nonrectangular and nonprismatic channels, are adoptedto validate the newly proposed approach. The convergence analysisis performed to study the appropriate initial particle number in eachcase, and the numerical accuracy test is next conducted. The simu-lated results show good agreement with the analytic and measuredresults. Although retaining the water discharges is difficult for lo-cations near the water depth discontinuity, the simulated results stillgive satisfactory results of water discharges. Thus, the present SPH-SWE approach has been proved its capability in modeling 1 D non-rectangular and nonprismatic channel flows with open boundaries.

References

Anderson, J. D. (1995). Computational fluid dynamics: The basics withapplications, McGraw-Hill, New York.

Ata, R., and Soulaimani, A. (2005). A stabilized SPH method for inviscidshallow water flows. Int. J. Numer. Meth. Fluid., 47(2), 139159.

Cunge, J. A., Holly, F. M., and Verwey, A. (1980). Practical aspects ofcomputational river hydraulics, Pitman Advanced Publishing Program,London.

Chang, K. H., Kao, H. M., and Chang, T. J. (2012b). Lagrangian modelingof particle concentration distribution in indoor environment with differ-ent kernel functions and particle search algorithms. Build. Environ.,57, 8187.

Chang, T. J., Chang, K. H., Kao, H. M., and Chang, Y. S. (2012a).Comparison of a new kernel method and a sampling volume methodfor estimating indoor particulate matter concentration with Lagrangianmodeling. Build. Environ., 54, 2028.

Chang, T. J., Kao, H. M., Chang, K. H., and Hsu, M. H. (2011). Numericalsimulation of shallow-water dam break flows in open channels usingsmoothed particle hydrodynamics. J. Hydrol., 408(12), 7890.

Chaudhry, M. H. (1993). Open-channel flow, Prentice-Hall, New Jersey.De Leffe, M., Le Touze, D., and Alessandrini, B. (2010). SPH modeling of

shallow-water coastal flows. J. Hydraul. Res., 48(supp 1), 118125.Federico, I., Marrone, S., Colagrossi, A., Aristodemo, F., and Antuono, M.

(2012). Simulating 2D open-channel flows through an SPH model.Eur. J. Mech. B Fluids, 34, 3546.

Gomez-Gesteira, M., Rogers, B. D., Dalrymple, R. A., and Crespo, A. J. C.(2010). State-of-the-art of classical SPH for free-surface flows.J. Hydraul. Res., 48(supp 1), 627.

Hernquist, L., and Katz, N. (1989). TREESPH: A unification of SPH withthe hierarchical tree method. Astrophys. J. Suppl. Ser., 70, 419446.

Khalifa, A. M. (1980). Theoretical and experimental study of the radialhydraulic jump. Ph.D. thesis, Univ. of Windsor, Windsor, ON.

Kao, H. M., and Chang, T. J. (2012). Numerical modeling of dambreak-induced flood and inundation using smoothed particle hydrodynamics.J. Hydrol., 448449, 232244.

Liu, G. R., and Liu, M. B. (2003). Smoothed particle hydrodynamics:A meshfree particle method, World Scientific, Singapore.

Macdonald, I. (1995). Test problems with analytic solutions for steadyopen channel flow. Numerical Analysis Rep. 6/94, Univ. of Reading,Dept. of Mathematics, Reading, UK.

Table 3. L2dw and L2Q of Various Initial Particle Numbers (N) in theRectangular Nonprismatic Channel

N (x0) L2dw Difference (dw) L2Q Difference (dw)(1) 25 (0.1 m) 0.073 1 2 0.005 0.0385 1 2 0.0186(2) 50 (0.05 m) 0.068 2 3 0.000 0.0199 2 3 0.0040(3) 100 (0.025 m) 0.068 0.0159

Fig. 6. (a) Comparison between the simulated and measured waterdepth profiles; (b) comparison between the simulated and measuredwater discharges of the nonprismatic channel

Fig. 5. Channel width profile of the entire rectangular nonprismaticchannel

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vers

ity o

n 10

/19/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

http://dx.doi.org/10.1002/(ISSN)1097-0363http://dx.doi.org/10.1016/j.buildenv.2012.04.017http://dx.doi.org/10.1016/j.buildenv.2012.04.017http://dx.doi.org/10.1016/j.buildenv.2012.02.006http://dx.doi.org/10.1016/j.jhydrol.2011.07.023http://dx.doi.org/10.1080/00221686.2010.9641252http://dx.doi.org/10.1080/00221686.2010.9641242http://dx.doi.org/10.1086/191344http://dx.doi.org/10.1016/j.jhydrol.2012.05.004

Macdonald, I., Baines, M. J., Nichols, N. K., and Samuels, P. K. (1997).Analytic benchmark solutions for open channel flows. J. Hydraul.Eng., 123(11), 10411045.

Monaghan, J. J. (2005). Smoothed particle hydrodynamics. Rep. Prog.Phys., 68(8), 17031759.

Randles, P. W., and Libersky, L. D. (1996). Smoothed particle hydrody-namics: Some recent improvements and applications. Comput.Method. Appl. M., 139(14), 375408.

Rodriguez-Paz, M., and Bonet, J. (2005). A corrected smooth particlehydrodynamics formulation of the shallow-water equations. Comput.Struct., 83(1718), 13961410.

Sanders, B. F. (2001). High-resolution and non-oscillatory solution ofthe St. Venant equations in non-rectangular and non-prismaticchannels. J. Hydraul. Res., 39(3), 321330.

Sturm, T. W. (2010). Open channel hydraulics, McGraw-Hill, New York.

Vacondio, R., Rogers, B. D., and Stansby, P. K. (2012). Accurate particlesplitting for smoothed particle hydrodynamics in shallow water withshock capturing. Int. J. Numer. Methods Fluids, 69(8), 13771410.

Vacondio, R., Rogers, B. D., and Stansby, P. K. (2012a). Smoothedparticle hydrodynamics: approximate zero-consistent 2-D boundaryconditions and still shallow-water tests. Int. J. Numer. Methods Fluids,69(1), 226253.

Vacondio, R., Rogers, B. D., Stansby, P. K., and Mignosa, P. (2012b). SPHmodeling of shallow flow with open boundaries for practical floodsimulation. J. Hydraul. Eng., 138(6), 530541.

Vacondio, R., Rogers, B. D., Stansby, P. K., and Mignosa, P. (2013). Acorrection for balancing discontinuous bed slopes in two-dimensionalsmoothed particle hydrodynamics shallow water modeling. Int. J.Numer. Methods Fluids, 71(7), 850872.

Wang, Z., and Shen, H. T. (1999). Lagrangian simulation of one-dimensional dam-break flow. J. Hydraul. Eng., 125(11), 12171220.

JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2013 / 1149

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/19/

13. C

opyr

ight

ASC

E. F

or p

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righ

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eser

ved.

http://dx.doi.org/10.1061/(ASCE)0733-9429(1997)123:11(1041)http://dx.doi.org/10.1061/(ASCE)0733-9429(1997)123:11(1041)http://dx.doi.org/10.1088/0034-4885/68/8/R01http://dx.doi.org/10.1088/0034-4885/68/8/R01http://dx.doi.org/10.1016/S0045-7825(96)01090-0http://dx.doi.org/10.1016/S0045-7825(96)01090-0http://dx.doi.org/10.1016/j.compstruc.2004.11.025http://dx.doi.org/10.1016/j.compstruc.2004.11.025http://dx.doi.org/10.1080/00221680109499835http://dx.doi.org/10.1002/fld.v69.8http://dx.doi.org/10.1002/fld.v69.1http://dx.doi.org/10.1002/fld.v69.1http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000543http://dx.doi.org/10.1002/fld.v71.7http://dx.doi.org/10.1002/fld.v71.7http://dx.doi.org/10.1061/(ASCE)0733-9429(1999)125:11(1217)