Journal of Hydrology 519 (2014) 1010–1019

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

A new approach to model weakly nonhydrostatic shallow water flowsin open channels with smoothed particle hydrodynamics

http://dx.doi.org/10.1016/j.jhydrol.2014.08.0300022-1694/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: Dept. of Bioenvironmental Systems Engineering,National Taiwan University, Taipei 106, Taiwan. Fax: +886 2 23635854.

E-mail address: hmkao@narlabs.org.tw (H.-M. Kao).

Tsang-Jung Chang a,b, Kao-Hua Chang a,b, Hong-Ming Kao a,c,⇑a Dept. of Bioenvironmental Systems Engineering, National Taiwan University, Taipei 106, Taiwanb Center for Weather Climate and Disaster Research, National Taiwan University, Taipei 106, Taiwanc Taiwan Typhoon and Flood Research Institute, National Applied Research Laboratories, Taipei 100, Taiwan

a r t i c l e i n f o

Article history:Received 6 June 2014Received in revised form 14 August 2014Accepted 15 August 2014Available online 27 August 2014This manuscript was handled by GeoffSyme, Editor-in-Chief

Keywords:Smoothed particle hydrodynamicsWeakly nonhydrostatic pressureBoussinesq equationsPredictor–corrector scheme

s u m m a r y

A new approach to model weakly nonhydrostatic shallow water flows in open channels is proposed byusing a Lagrangian meshless method, smoothed particle hydrodynamics (SPH). The Lagrangian form ofthe Boussinesq equations is solved through SPH to merge the local and convective derivatives as thematerial derivative. In the numerical SPH procedure, the present study uses a predictor–correctormethod, in which the pure space derivative terms (the hydrostatic and source terms) are explicitly solvedand the mixed space and time derivatives term (the material term of B1 and B2) is computed with animplicit scheme. It is thus a convenient tool in the processes of the space discretization compared to otherEulerian approaches. Four typical benchmark problems in weakly nonhydrostatic shallow water flows,including solitary wave propagation, nonlinear interaction of two solitary waves, dambreak flow propa-gation, and undular bore development, are selected to employ model validation under the closed andopen boundary conditions. Numerical results are compared with the analytical solutions or publishedlaboratory and numerical results. It is found that the proposed approach is capable of resolving weaklynonhydrostatic shallow water flows. Thus, the proposed SPH approach can supplement the lack of theSPH–Boussinesq researches in the literatures, and provide an alternative to model weakly nonhydrostaticshallow water flows in open channels.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Due to the requirement of less computational cost compared tothe Navier–Stokes equations, the shallow water equations are themost common choice to mathematically describe many hydraulicengineering problems in rivers and floodplains (Cunge et al.,1980; Morris, 2000; Chaudhry, 2008). The shallow water equationsassume that the water pressure is only dependent on the total flowdepth, resulting in a hydrostatic pressure distribution over the flowdomain and the vertical motion is small enough to be neglected(Cunge et al., 1980; Chaudhry, 2008). So far many numericalresearches (Morris, 2000) have reported that the shallow waterequations are reasonably suitable for the representation of gradu-ally varied flows in rivers and floodplains, however, they fail toaccurately represent rapidly varied flows. For such violent flows,the ratio of the vertical-to-horizontal scales of motion is no longersmall and the vertical acceleration significantly creates a nonhy-

drostatic pressure distribution that should be incorporated intothe equations used to route these flows.

An appropriate option for the computations of rapidly variedflows is the Boussinesq equations, which are the depth-averagedversion by utilizing the Boussinesq assumption (Chaudhry, 2008).They are the simplest class of mathematical models that expandthe shallow water equations with the Boussinesq terms to captureweakly nonhydrostatic physics such as wave refraction and diffrac-tion. The simulated outcomes based on the Boussinesq equationshave been demonstrated to provide good predictions for a rangeof physical configurations such as dambreak flow transport(Mohapatra and Chaudhry, 2004), undular bore evolution (Favre,1935; Peregrine, 1966), and solitary wave propagation (Devkotaand Imberger, 2009). As a result, the Boussinesq equations havebeen prevailing in simulating weakly nonhydrostatic shallowwater flows. In this study, the Boussinesq equations are adoptedherein to remove the restriction of the hydrostatic assumption inthe shallow water equations. The one-dimensional (1D) Bous-sinesq equations of continuity and momentum in an Eulerian formcan be written as (Chaudhry, 2008).

T.-J. Chang et al. / Journal of Hydrology 519 (2014) 1010–1019 1011

@h@t|{z}

Local

þ u@h@x|ffl{zffl}

Convective

þh@u@x¼ 0 ð1Þ

@u@t|{z}

Local

þ u@u@x|ffl{zffl}

Convective

¼ �g@h@x|fflffl{zfflffl}

Hydrostatic pressure term

þ gðS0 � Sf Þ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}Source term

þ h2

3@

@t@2u@x2

!|fflfflfflfflfflffl{zfflfflfflfflfflffl}

B1

þu@

@x@2u@x2

!|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

B2

� @

@x@u@x

� �2

|fflfflfflfflfflffl{zfflfflfflfflfflffl}B3

266664377775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Boussinesq terms

ð2Þ

where t is the time, x is the space Cartesian coordinate, u is thedepth-averaged velocity in the x-direction, g is the gravitationalacceleration, h is the flow depth, S0 is the bed slope, andSf(= n2u|u|/h4/3) is the friction slope with n is the Manning rough-ness coefficient reflecting the roughness of the bottom. Three mainterms (the hydrostatic, source and Boussinesq terms) are includedin the right-hand side of Eq. (2). The Boussinesq terms, B1, B2, andB3 account for the effect of vertical acceleration. The physicalmeaning of B1 is the local acceleration in the vertical direction(z-direction), and B2 and B3 represent the convective accelerationin the x- and z-direction, respectively. For weakly nonhydrostaticshallow-water flows, the material term of B1 and B2 has majorcontributions compared to the third-derivative space term of B3

(Mohapatra and Chaudhry, 2004). Therefore, it is sufficient to onlyinclude the material term of B1 and B2 in Eq. (2) as

@u@tþ u

@u@x¼�g

@h@xþ gðS0 � Sf Þþ

h2

3@

@t@2u@x2

!þ u

@

@x@2u@x2

!" #ð3Þ

Conventionally, there are a variety of numerical methods thatcan be used to solve the Boussinesq equations, including finite dif-ference methods (Zijlema et al., 2011), finite element methods(Walters, 2005) and finite volume methods (Denlinger andO’Connell, 2008). These Eulerian-based approaches can expendconsiderable efforts to yield satisfactory discretization and toreduce truncation errors for the nonlinear convective term usingsecond-order numerical schemes. Similarly, it is also necessary toemploy third- or higher-order accurate schemes to solve thethird-order Boussinesq terms (Abbott, 1979; Basco, 1989;Chaudhry, 2008). Although such higher-order schemes have effec-tive computations for weakly nonhydrostatic pressure correction,they are computational tediously and may still suffer from thechallenge of grid-resolution problems (Mohapatra and Chaudhry,2004; Devkota and Imberger, 2009). On the other side, some mesh-less methods have been applied to hydrodynamics recently.Among them, smoothed particle hydrodynamics (SPH) has beenproved to have some numerical advantages (Wang and Shen,1999; Ata and Soulaimani, 2005; Rodriguez-Paz and Bonet, 2005;De Leffe et al., 2010; Vacondio et al., 2011; Chang et al., 2011;Kao and Chang, 2012; Vacondio et al., 2012a,b; Chang and Chang,2013), which are adequate to be used in solving the Boussinesqequations. Firstly, SPH is a particle method with Lagrangian nature.Particles move with the flow, and the convective term is mergedinto the material derivative so that the numerical dispersion errorresulting from the convective term can be directly eliminated(Devkota and Imberger, 2009). The Lagrangian form of the 1DBoussinesq equations of continuity and momentum can be rewrit-ten from Eqs. (1) and (3) as

DhDt|{z}

Material

¼ �h@u@x

ð4Þ

DuDt|{z}

Material

¼ �g@h@x|fflffl{zfflffl}

Hydrostatic pressure term

þ gðS0 � Sf Þ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}Source term

þ h2

3DDt

@2u@x2

!|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

Material term of B1 and B2

ð5Þ

where DDt ¼ @

@t þ u @@x

� �represents the material derivative. Obviously,

in Eqs. (4) and (5), the Lagrangian description merges the localand convective derivatives as the material derivative.

In addition, SPH allows water particles freely moving in thecomputational domain without being confined in a fixed mesh.Consequently, SPH can conserve mass exactly and has strong capa-bility to deal with large deformation problems (Liu and Liu, 2003).Therefore, it is a suitable numerical tool to solve the aforemen-tioned issues. Standard SPH is basically designed to formulate theNavier–Stokes equations (Liu and Liu, 2003, 2010). So far only fewstudies have attempted to extend SPH to the shallow water equa-tions. Wang and Shen (1999) investigated inviscid dam-breakflows using SPH. Ata and Soulaimani (2005) proposed the stabiliza-tion term of SPH formulation. Rodriguez-Paz and Bonet (2005) pre-sented a corrected variational SPH formulation for shallow waterflows to conserve both the total mass and momentum. De Leffeet al. (2010) adopted an anisotropic kernel with variable smooth-ing length and performed SPH modeling of shallow-water coastalflows. Vacondio et al. (2011) used the characteristic boundarymethod into SPH formulation to simulate rectangular prismaticchannel flows with open boundaries. Chang et al. (2011), andKao and Chang (2012) applied SPH modeling to investigateshallow-water dambreak flows in realistic open channels andfloodplains. Vacondio et al. (2012a,b) improved SPH for the closedboundary conditions by using virtual particles and introduced aparticle-splitting procedure for addressing the issue of adequateparticle-resolution in small-depth problems. Chang and Chang(2013) developed a new SPH scheme to solve the characteristicequations and to establish the open boundaries in non-rectangularand non-prismatic channel flows with the method of specified timeintervals. Nevertheless, there still lacks research efforts that haveutilized SPH to simulate the Boussinesq equations for investigatingweakly nonhydrostatic shallow water flows.

To fill this gap, this study aims to develop a new SPH approachfor investigating weakly nonhydrostatic shallow water flows.Firstly, the Lagrangian form of the 1D Boussinesq equations isderived. The numerical procedure of how to solve the above equa-tions with SPH is given. Next, a comparison of the numericalresults with the analytical solutions or published laboratory datais examined through four benchmark problems (solitary wavepropagation, nonlinear interaction of two solitary waves,dambreak flow propagation, and undular bore development). Thebenefits and limits of the present SPH modeling are discussed.

2. Numerical method

Eqs. (4) and (5) are the 1D time-dependent hyperbolic system ofpartial differential equations. Generalized analytical solutions arenot feasible for these equations. As a result, a numerical approachshould be adopted to obtain the numerical solutions. In this study,a Lagrangian meshless SPH is adopted to solve Eqs. (4) and (5). Inthis numerical procedure, the pure space derivative terms (thehydrostatic and source terms) and the mixed space and timederivatives term (the material term of B1 and B2) of Eq. (5) aredecomposed. The pure space derivative terms are explicitly solvedthrough the predictor–corrector computational framework to givean intermediate value, and then the mixed space and timederivatives term is applied for computational correction with animplicit time-integration scheme.

1012 T.-J. Chang et al. / Journal of Hydrology 519 (2014) 1010–1019

2.1. SPH formulation

SPH is adopted to discretize the Lagrangian form of the 1DBoussinesq equations by using a finite number of water particlesthat possess material properties, such as volume, depth, and movein space with the velocity of flow. From the mathematical point ofview, SPH is a fully pure interpolation method so that any physicalquantity /i of particle i is approximated numerically by theweighted summation as follows:

h/ii ¼XN

j

mj

hj/jWiðjxi � xjj; liÞ ð6Þ

where h� � �i denotes the SPH approximation, mj is the mass by Vjhj

(Vj = Dx0 and hj are the volume and depth of a fluid carried by par-ticle j, respectively), xi and xj are respectively the positions of parti-cles i and j, li is the smoothing length of particle i to determine thesupport size of the kernel function, Dx0 is the initial particle spac-ing, and N is the number of particles inside a circle support domaincentered at position xi with 2li radius (Liu and Liu, 2003).

In SPH, particles are interpolation points and interact with eachother through a smoothing procedure where a local kernel functionassigns to each point a weight value, according to the mutual posi-tions of the interpolating point. Therefore, the precise value of thephysical quantity /i is different from the interpolation process byusing various kernel functions. Among a variety of kernel functionsdocumented in the references (Liu and Liu, 2003; Chang et al.,2012a), the cubic spline kernel function, which is commonly usedin many SPH simulations (Ata and Soulaimani, 2005; De Leffe et al.,2010; Chang et al., 2011; Kao and Chang, 2012), is adopted. Thecubic spline kernel function W can be written as

WðqÞ ¼ 1l�

23� q2 þ 1

2 q3; q 6 116� ð2� qÞ3; 1 < q 6 20; q > 2

8><>: ð7Þ

in which q = |xi � xj|/l and l is the smoothing length.The particle approximation for the first and second spatial

derivatives of any physical quantity /i used in the present studyis listed as

hr/ii ¼1hi

XN

j

mjð/j � /iÞrWiðjxi � xjj; liÞ ð8Þ

hr2/ii ¼XN

j

mj

hj

2ðxi � xjÞ � rWiðjxi � xjj; liÞjxi � xjj2

ð/i � /jÞ ð9Þ

In this paper, the wall boundary conditions are built byrepulsive wall particles (Monaghan, 2005). Due to the use of therepulsive wall particles, a fluid particle near the wall boundarylacks the entire compact domain. To give a reasonable evaluationof the gradient of a fluid particle, the asymmetric gradient formu-lation, i.e., Eq. (8), performs the gradient operation herein. More-over, a hybrid formulation of Eq. (9), which is derived from a SPHfirst-order kernel derivative and a first-order finite differenceapproximation (Cleary, 1998; Cummins and Rudman, 1999; Shaoand Lo, 2003; Monaghan, 2005; Shao, 2011), is adopted to executethe Laplacian operation.

2.2. SPH implementation for the Boussinesq equations

In this section, Eqs. (4) and (5) are transformed into the particleapproximation equations and SPH is implemented for the numeri-

cal solution. Therefore, the discretized form of the continuity andmomentum equations is given in the following.

2.2.1. Numerical evaluation of flow depthAs solving the flow depth in the Boussinesq equations, it con-

tains two commonly used numerical ways. One is to directly solvethe continuity equation of Eq. (4), and the other is to conduct theweighted summation formula of Eq. (6). Due to the reason of theexact presence of mass conservation (Ata and Soulaimani, 2005;Rodriguez-Paz and Bonet, 2005), the present study selects the lat-ter way to calculate the flow depth of particle i rather than the firstone. Therefore, a variable smoothing length scheme connecting tothe flow depth is applied to obtain better accuracy of the solutionof the Boussinesq equations. The variable smoothing length of eachparticle i starts from Eq. (10)

li ¼ l0;ih0;i

hi

� �1=Dm

ð10Þ

where h0,i and l0,i are the initial water depth and smoothing lengthfor particle i (= 1.2 Dx0 in the present study), respectively, and Dm isthe number of space dimensions, which is equal to 1 in this study.

A Newton–Raphson iterative procedure with a variable smooth-ing length scheme is proposed to calculate the flow depth of parti-cle i. The iterative procedure based on the related references (Ataand Soulaimani, 2005; Rodriguez-Paz and Bonet, 2005) is displayedas follows:

hkþ1i ¼ hk

i 1� Reski Dm

Reski Dm þ ak

i

!ð11Þ

and

Reski ¼ hk

i �XN

j¼1

mjWiðjxi � xjj; lki Þ ð12Þ

aki ¼ �

XN

j¼1

mjrijdWi

drijð13Þ

where rij(= |xi � xj|) is the distance between particles i and j, andResk

i is the residual of particle i at the kth iteration.In order to stop the Newton–Raphson iterative procedure, the

tolerance condition is given by

Reski

hki

���������� 6 e ð14Þ

close to the limit e = 10�10, which is usually reached within a fewiterations.

2.2.2. Discretization of the momentum equationIn the right-hand side of Eq. (5), the hydrostatic pressure,

source, and Boussinesq terms have different physical characteris-tics. The program flowchart of the present SPH approach to modelweakly nonhydrostatic shallow water flows is depicted in Fig. 1. Todiscretize Eq. (5) through SPH, the numerical procedure is decom-posed into two major parts, as shown in Fig. 1. Firstly, the hydro-static and source terms are explicitly solved through thepredictor–corrector computational framework to give an interme-diate value. Next, the intermediate flow field is corrected to obtainthe final solution by considering the material term of B1 and B2. Animplicit scheme for the computation of weakly nonhydrostaticpressure correction is used in this study.

2.2.2.1. Predictor step. In the predictor step, the hydrostatic andsource terms in Eq. (5) are firstly computed using the flow vari-

Fig. 1. Program flowchart of the present SPH approach for modeling weakly nonhydrostatic shallow water flows.

T.-J. Chang et al. / Journal of Hydrology 519 (2014) 1010–1019 1013

ables of known time level by neglecting the material term of B1 andB2 to obtain the predicted velocity as

upi ¼ uk

i þDt2� 1

hi

XN

j¼1mj gðhk

j � hki Þ þ

Yij

� rWij|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Hydrostaticpressureterm

þ g S0;i �n2

i uijuijh4=3

i

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Sourceterm

8>>><>>>:9>>>=>>>;

k

ð15Þ

where i and j are indices for the particle number, respectively, thesuperscript k is a time step index, and the superscript p is apredicted index. h0 is the undisturbed initial water depth basedon the long wave assumption that the wave amplitude is smallcompared to the undisturbed initial water depth. rWij ¼ 0:5�½rWiðjxi � xjj; lk

i Þ þrWjðjxj � xij; lkj Þ� is a hybrid combination of

the scatter and gather interpretations (Hernquist and Katz, 1989)for the gradient expression due to the smoothing length beingvariable.

1014 T.-J. Chang et al. / Journal of Hydrology 519 (2014) 1010–1019

The predicted position of particle is thus given by

xpi ¼ xk

i þDt2

uki ð16Þ

and the flow depth of particle i is updated through Eq. (11) to obtainthe predicted flow depth hp

i . In addition, the variable smoothinglength is updated in the predictor step.

Additionally, for the aim of maintaining the present numericalmethod stable, an artificial viscosity is introduced herein (Ataand Soulaimani, 2005) asY

ij

¼ ��cijðuij � xijÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2ij þ g

q ð17Þ

where �cij ¼ 0:5� ðci þ cjÞ, cið¼ffiffiffiffiffiffiffighi

pÞ is the pressure wave speed,

uij = ui � uj, xij = xi � xj, and g(= 10�6) is a constant to avoid a divisionby zero.

Regarding the source term, it includes the bed slope (S0) and thefriction slope (Sf). The source term is computed through a new setof virtual particles that do not move and are used to describe thebathymetry and the Manning roughness coefficient (Vacondioet al., 2012a). By means of a SPH interpolation method, the bedslope of particle i is read

S0;i ¼X

j

Vvpj Svp

0;jfW jðjxi � xvp

j j; lvpj Þ ð18Þ

where the superscript vp indicates a variable value of virtual parti-cle, the volume (Vvp

j ) of virtual particle j is equal to the initial par-ticle spacing (Dx0) in 1D computation, fW is the corrected kernelfunction defined in Eq. (19) by using the Shepard filter scheme(Randles and Libersky, 1996; Chang et al., 2012b; Vacondio et al.,2012a), and lvp

j ð¼ 1:2Dx0Þ is the smoothing length of virtual particlej.

fW jðjxi � xvpj j; l

vpj Þ ¼

Wjðjxi � xvpj j; l

vpj ÞP

jVvpj Wjðjxi � xvp

j j; lvpj Þ

ð19Þ

For the friction slope, to arbitrarily vary the Manning roughnesscoefficient over the computational domain, the same SPH interpo-lation method is applied. A value of the Manning roughness coeffi-cient is assigned to each virtual particle j, and it is calculated ateach time step at the fluid particle i by means of the followinginterpolation.

ni ¼X

j

Vvpj nvp

jfW jðjxi � xvp

j j; lvpj Þ ð20Þ

where nvpj indicates the Manning roughness coefficient of virtual

particle j located at position xvpj .

2.2.2.2. Corrector step. Again, the corrected variables are obtainedfrom the predicted variables as

uci ¼ uk

i þDt2

1hi

XN

j¼1

mj gðhpj�hp

i Þ þY

ij

!rWij þ g S0;i�

n2i uijuijh4=3

i

!( )p

ð21Þ

and

xci ¼ xk

i þDt2

upi ð22Þ

where the superscript c represents a corrected index.The intermediate flow variables of velocity ~ui and position ~xi are

given as follows:

~ui ¼ 2uci � uk

i ð23Þ

~xi ¼ 2xci � xk

i ð24Þ

and both the intermediate flow depth ~hi and the variable smoothinglength are again updated.

2.2.2.3. Final. When the intermediate flow variables are computed,they are finally modified by the material term of B1 and B2 to eval-uate the weakly nonhydrostatic effect. This material term containsmixed space and time derivatives. To formulate the second-orderspace derivative in this term, the hybrid approach (Cleary, 1998;Cummins and Rudman, 1999; Monaghan, 2005; Shao, 2011) thatconnects the SPH first-order kernel derivative and the first-orderfinite difference approximation through Eq. (9) is herein used toreduce the order of discretization. This formulation can eliminatethe numerical instabilities that arise from the second-order kernelderivative, as particles are disordered (Monaghan, 2005). Then, thematerial term used as a weakly nonhydrostatic correction isapplied as

ukþ1i � ~ui

Dt¼ h2

0

3Dt

XN

j¼1

mj

~hj

2ð~xi � ~xjÞ � rWij

j~xi � ~xjj2 þ gðukþ1

i � ukþ1j Þ

!"

�XN

j¼1

mj

~hj

2ð~xi � ~xjÞ � rWij

j~xi � ~xjj2 þ gðuk

i � ukj Þ

!#ð25Þ

where h0 is the undisturbed initial water depth and g(= 10�6) is aconstant to avoid a division by zero.

To obtain the final velocity of particle i at time level k + 1, ukþ1i ,

an implicit scheme coupled with the SPH formulation for the com-putation of the material term is used. Eq. (25) can be rearranged toread

1� h20

3

XN

j¼1

mj

~hj

2ð~xi � ~xjÞ �rWij

j~xi � ~xjj2 þ g

!ukþ1

i þh2

0

3

XN

j¼1

mj

~hj

2ð~xi � ~xjÞ �rWij

j~xi�~xjj2þgukþ1

j

¼ ~ui �h2

0

3

XN

j¼1

mj

~hj

2ð~xi � ~xjÞ �rWij

j~xi � ~xjj2 þgðuk

i � ukj Þ ð26Þ

Eq. (26) is written in a matrix form and can be efficiently solvedusing the preconditioned conjugate gradient algorithm (Press et al.,1992). It should be noted that the implicit and explicit schemes forthe solutions of the material term have been both tested for theirnumerical stability. It is found that the implicit scheme for the B1

and B2 terms is numerically stable, but the explicit scheme is diver-gent during numerical computations. This is the reason why wechoose the implicit scheme herein.

As ukþ1i is updated through Eq. (26), the new position of particle

i at time level k + 1 is given by

xkþ1i ¼ xk

i þDt2ðukþ1

i þ uki Þ ð27Þ

and thus the final flow depth hkþ1i is corrected through Eq. (11). The

variable smoothing length is also updated in the final step.

2.2.3. Boundary conditionsIn the present SPH approach, the simulations of weakly nonhy-

drostatic shallow water flows in open channels require suitableboundary conditions to guarantee well-posed solutions(Anderson, 1995). Two boundary conditions are herein used, i.e.,the closed boundary condition (the solid wall boundary condition)(Ata and Soulaimani, 2005; Rodriguez-Paz and Bonet, 2005; DeLeffe et al., 2010; Liu and Liu, 2010; Chang et al., 2011; Kao andChang, 2012) and the open boundary condition (the in/out-flowboundary condition) (Vacondio et al., 2011; Federico et al., 2012;Chang and Chang, 2013).

2.2.3.1. Closed boundary condition. As the closed boundary condi-tion is adopted, to prevent the interior fluid particles from pene-trating the closed boundary, the no-penetration condition has to

T.-J. Chang et al. / Journal of Hydrology 519 (2014) 1010–1019 1015

be imposed. The repulsive boundary force generated by virtual par-ticles when a real fluid particle approaches the closed boundary isused. Based on Lennard-Jones’ form (Liu and Liu, 2003), the repul-sive force for the closed boundary condition can be defined as

RFij ¼D r0

jxij j

� n1� r0

jxij j

� n2h i

xij

jxij j2; r0

jxij j

� P 1

0; r0jxij j

� < 1

8><>: ð28Þ

where RFij is the force on fluid particle i due to the neighbor virtualparticle j, the constants of n1 and n2 are usually respectively takenas 12 and 4 (Liu and Liu, 2003), xij ¼ xi � xvp

j is the distance betweenthe two particles, the cutoff distance r0 is usually selected approxi-mately close to the initial particle spacing, and D is the square of thelargest particle velocity in the computational domain.

2.2.3.2. Open boundary condition. For the open boundary condition,flow problems are either subcritical or supercritical, or mixed withboth flow regimes. Generally, at the inflow boundaries, both theflow velocity and water depth should be given as the supercriticalflow occurs, and only the flow velocity should be provided if theflow is subcritical. At the outflow boundaries, the water depthshould be specified if the subcritical flow occurs, and no boundarycondition is needed as the flow is supercritical. In the presentstudy, the in/out-flow algorithm developed by Federico et al.(2012) is applied to treat open boundaries. The detailed numericalprocedure can be found in the related references (Vacondio et al.,2011; Federico et al., 2012; Chang and Chang, 2013).

2.2.4. Time marching schemeAn explicit time integration method is used in both the

predictor and corrector steps. Due to its explicit nature, the timeincrement (Dt) has to satisfy the Courant–Friedrichs–Lewy (CFL)condition (Rodriguez-Paz and Bonet, 2005)

Dt ¼ CFL�minDx0

ui þffiffiffiffiffiffiffighi

p !ð29Þ

where CFL is set as 0.4 in this study.

3. Results and discussion

As shown in the previous section, a new SPH approach to solvethe Lagrangian form of the 1D Boussinesq equations is presentedfor simulating weakly nonhydrostatic shallow water flows. In orderto validate the numerical applicability of the proposed approach,four representative benchmark problems containing weaklynonhydrostatic phenomena, including solitary wave propagation,nonlinear interaction of two solitary waves, dambreak flow propa-gation, and undular bore development, are examined. The firststudy case of solitary wave propagation is to test the suitabilityof the present approach in describing the propagation of finite-amplitude waves under the consideration of the weakly nonhydro-static effect. The second case of phase shift of two solitary waves isfor evaluating the ability of the present approach in modelling non-linear interaction. The third study case of dambreak flow propaga-tion has the aim to assess the oscillatory patterns and the periodsof the secondary waves (undulations) in violent flow regions. Theclosed boundary conditions are applied in this case to confine boththe upstream and downstream ends of the computational domain.Finally, the last study case of undular bore development targets oninvestigating how undulation grows at the head of the bore in aweakly nonhydrostatic system. The open boundary conditions(in/out-flow boundaries) are used to set up the upstream waterdischarge and downstream water depth. Two sets of numericalsimulations are conducted by solving the Boussinesq equations

(Eqs. (4) and (5), herein referred to as the SPH–Boussinesqapproach) and the shallow water equations (Eqs. (4) and (5) with-out the Boussinesq terms, i.e., the SPH–SWE approach) for testingtheir numerical accuracy and physical reasonability.

3.1. Solitary wave propagation

In the first study case, a single solitary wave propagating in ahorizontal frictionless rectangular channel, which is 500 m longwith an undisturbed constant water depth (h0 = 1 m) and unitwidth, is considered. This is a good test case to validate the abilityof the proposed SPH approach in simulating weakly nonhydrostaticshallow water waves. The solitary wave has an amplitude (a0) of0.1 m. The initial wave crest is located at 50 m in the positive xdirection (right direction). As the wave moving only toward thepositive x direction, both the water surface displacement andvelocity profiles are given by the Korteweg-de Vries (KdV) analyticsolutions (Peregrine, 1966) as

gðx; tÞ ¼ a0 � sech2½kðx� x0 � cntÞ� ð30Þ

Uðx; tÞ ¼ gffiffiffiffiffiffiffiffiffiffig=h0

qð31Þ

where g(x, t) is the water surface displacement, t is the propagation

time, k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3a0=4h3

0

qis the wave parameter, x0 is the initial location

of wave crest, cn �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðh0 þ a0Þ

pis the nonlinear wave speed, and

U(x, t) is the flow velocity.To investigate the convergence analysis and numerical accuracy

test, The L2 relative error norm as shown in Eq. (32) is used.

L2ð/Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP/simulated

i � /analytic=measuredi

� 2

P/analytic=measured

i

� 2

vuuuut ð32Þ

where /simulatedi and /analytic=measured

i are respectively the simulatedphysical quantity and the analytic or measured data at the ithparticle.

For the convergence analysis, five initial particle numbers of6,250, 12,500, and 25,000 [i.e., the initial particle spacings (Dx0)are 0.08, 0.04, and 0.02 m, respectively] are adopted in this studycase. The values of L2(h) (L2 relative error norm based on the waterdepth h) and L2(U) (L2 relative error norm based on the watervelocity U) are decreased as the initial particle number increases,so that the proposed approach can converge to the analytic solu-tion. In addition, the convergence rates are 1.39 and 1.01 for thewater depth and the water velocity, respectively. The simulatedresults adequately match with the KdV analytical solutions as thereleased particle number is more than 25,000. As a consequence,to balance sufficient numerical accuracy and acceptable computingtime, 25,000 particles are uniformly placed in the computationaldomain in this study case.

For the numerical accuracy test, Fig. 2a and b compare the sim-ulated wave depth and velocity profiles against the KdV analyticalsolutions for the SPH–Boussinesq and SPH–SWE approaches,respectively. The total propagation time is 100 s. It can be clearlyseen from Fig. 2a that the wave amplitude and shape are almostpreserved within 80 s with the SPH–Boussinesq approach. The val-ues of L2(h) and L2(U) are respectively 0.003 and 0.301 from 0 s to80 s. However, it shows a slight difference in both wave depth andvelocity of the crest compared to the analytical solutions after 80 s.This phenomenon is because that the use of artificial viscosity can-not fully prevent particle disorder after a long time simulation, sothat particle disorder occurs in the vicinity of wave crests. In addi-tion, as shown in Fig. 2b, the wave evolves toward a shock wave

Fig. 2. The comparison of the simulated wave depth and velocity profiles againstthe KdV analytical solutions for (a) the SPH–Boussinesq and (b) SPH–SWEapproaches.

Fig. 3. The evolution of two interacting solitary waves within the time fromt = 100 s to t = 130 s.

Fig. 4. The evolutions of simulated water surface elevations at the gauges G1–G4using the SPH–Boussinesq and SPH–SWE approaches against the measured datagiven by Carmo et al. (1993).

Fig. 5. Spatial snapshot of the water depth profiles at (a) t = 1 s and (b) t = 3 s afterdambreak.

1016 T.-J. Chang et al. / Journal of Hydrology 519 (2014) 1010–1019

using the SPH–SWE approach, giving obvious underestimationcompared to the analytical solutions in both wave depth and veloc-ity of the crest. Therefore, the present SPH–Boussinesq approachcan provide good prediction on solitary wave propagation withoutchanging its shape and velocity.

3.2. Nonlinear interaction of two solitary waves

To test the suitability of the present approach in modelling non-linear interaction of two solitary waves, the second study casegives a phase shift of two solitary waves where a larger wave trailsa smaller wave. For the two initial wave conditions, a larger wave,

located at a distance of 100 m from the beginning of the channel,has an amplitude (a0) of 0.4 m, whereas a smaller wave with anamplitude of 0.1 m is situated at 150 m. These two wave crestsare 50 m away. In this case, there are 70,000 particles(Dx0 = 0.01 m) uniformly placed in the computational domain.The undisturbed constant water depth (h0 = 1 m) is consideredand the two waves are propagated for time duration of 130 s.

Fig. 3 shows the evolution of two interacting solitary waveswithin the time from t = 100 s to t = 130 s. Before the collisionoccurred at t = 120 s, it can be clearly seen from Fig. 3 that theamplitude of the larger wave progressively decreases, while thesmaller wave increases in amplitude leading to a relative phaseshift. After the collision, the amplitude of the larger wave graduallyincreases, and the smaller wave decreases in amplitude. The

T.-J. Chang et al. / Journal of Hydrology 519 (2014) 1010–1019 1017

simulated results of the water depth profiles in Fig. 3 have goodagreement with the numerical simulations given by Devkota andImberger (2009).

3.3. Dambreak flow wave propagation

The third study case is to examine the ability of the presentapproach in investigating the secondary phenomena of undula-tions generated at the wave front under the weakly nonhydrostaticeffect. This dambreak flow problem is also commonly used as abenchmark test through solving the 3D Navier–Stokes equationsfor assessing the nonhydrostatic effect, such as Pu et al. (2013).For this validation, a dambreak flow experiment carried out byCarmo et al. (1993) is chosen. In their experiment, the horizontalrectangular channel is 7.5 m long by 0.3 m wide. The dam islocated in the middle of the channel, dividing the channel intothe upstream reservoir and the downstream channel. The initialwater depth is 0.099 m at the upstream reservoir and is 0.051 min the downstream channel, resulting in the depth ratio of thechannel water to reservoir as 0.515. The Manning roughness coef-ficient of the channel is 0.026. Four water-surface level gauges,located at 2.65, 5.25, 6.25, and 7.25 m from the upstream reservoirhave been used to record wave evolutions in the experiment. Boththe SPH–Boussinesq and SPH–SWE results are thus compared withthe measured data. In this case, a gate is fully opened instanta-neously so that a rarefaction wave is formed and propagatestoward the upstream reservoir. From a physical point of view,

Fig. 6. The simulated water profiles of undular bore development at different timesexperiment (1966) and the finite volume method (FVM) given by Soares-Frazao and Gu

the growth of wave front is due to the free-surface curvature thatis induced extra horizontal pressure gradients as a consequence ofthe vertical acceleration of flow.

Three initial particle numbers of 375, 750, 1,500 [i.e., the initialparticle spacings (Dx0) are 0.02, 0.01, and 0.005 m, respectively]are used to study the convergence in this study case. The conver-gence rate is 1.08 for the water depth. It can be shown that theaccuracy of the proposed approach in this study case is O(Dx1.08).In addition, the simulated results of using 750 particles are givenin this study case. The present SPH approach implements theclosed boundary conditions to confine the still water in bothupstream reservoir and downstream channel. The measured watersurface elevations (Carmo et al., 1993), and the evolutions of sim-ulated water surface elevations given by the SPH–Boussinesq andSPH–SWE approaches at each gauge are depicted in Fig. 4 for com-parison. In Fig. 4, the SPH–Boussinesq approach can provide betterprediction on the physical characteristics of the oscillatory pat-terns and the periods of the secondary waves than the SPH–SWEapproach. The simulated amplitudes of the waves using the SPH–Boussinesq approach are also very close to the experimental data.

Fig. 5 shows the spatial snapshots at the selected times of t = 1and 3 s after dambreak. It can be clearly seen from Fig. 5 that theSPH–Boussinesq and SPH–SWE results of the distances traveledby the wave propagation are almost the same. However, theimportance of dispersion effect is expected to create different wavefronts. When the shallow water equations are used, the step wavefront that is in the absence of numerical dispersion is generated.

with the SPH–Boussinesq and SPH–SWE approaches against the Favre-Peregrineinot (2008).

1018 T.-J. Chang et al. / Journal of Hydrology 519 (2014) 1010–1019

On the contrary, the secondary waves of undulations generated atwave front are reasonably reproduced as solving the Boussinesqequations, because they are dispersive in nature.

3.4. Undular bore development

In the last study case, an undular bore experiment conducted byFavre (1935) and Peregrine (1966) is selected. This case is useful totest the present SPH approach in explaining the physics of thegrowth of undulations in a weakly nonhydrostatic system. In theirexperiment, a flat rectangular channel is 75.58 m long and 0.41 mwide. The initial wave profile is given by

gðx; t0Þ ¼1g

uðx; t0Þffiffiffiffiffiffiffiffigh0

qþ 1

4u2ðx; t0Þ

� �ð33Þ

uðx; t0Þ ¼12

u0

ffiffiffiffiffiffiffiffigh0

q1� tan h

xa

� h ið34Þ

where h0( = 1 m) is the undisturbed initial water depth, u0( = 0.1m/s) is the velocity of the incoming wave, a is a characteristiclength of the bore which describes the shape of the incoming wave(a smaller value of a forms a steeper initial profile), and the waterdepth h = h0 + g(x, t). The SPH–Boussinesq and SPH–SWE resultsare compared with the experiment of Favre and Peregrine and thenumerical results of the hybrid finite volume method with MUSCL4reconstruction given by Soares-Frazao and Guinot (2008).

In this study case, the convergence of the proposed approach isinvestigated by three different spatial resolutions including initialparticle numbers of 800 (Dx0 = 0.1 m), 1,600 (Dx0 = 0.05 m) and3,200 (Dx0 = 0.025 m). The convergence rate is calculated to be1.21 for the water depth. It can be illustrated that the spatial accu-racy is the order of O(Dx1.21). In the followings, the simulatedresults of using 1,600 particles are presented. The open boundaryconditions (in/out flow boundaries) are used to set up theupstream constant water discharge and downstream constantwater depth. Fig. 6 shows the simulated water profiles of undularbore development at different times using the SPH–Boussinesqand SPH–SWE approaches. The outcomes of the experiment ofFavre and Peregrine and the numerical results of Soares-Frazaoand Guinot are also plotted in this figure for comparison. InFig. 6, the SPH–SWE results are only accurate in the early stageof the simulation. As the simulation continues, the bore undulationcannot be produced as shown in Fig. 6d to f. The SPH–SWEapproach can thus only generate a progressive steepening of thebore without any undulation. On the contrary, the SPH–Boussinesqresults allow the growth of undulations at the head of the bore tobe created. Its numerical accuracy during the entire simulationtime is quite satisfactory compared to the Favre-Peregrine experi-ment and the numerical results of Soares-Frazao and Guinot.

Moreover, the advantages of using the SPH–Boussinesqapproach in this study case are further discussed. Fig. 7 is replottedbased on Fig. 6f for more detail illustration of physical meaning. In

Fig. 7. Schematic of the pressure distribution in an undular bore (h0 = the initialundisturbed water depth and u0 = the incoming wave velocity).

Fig. 7, the pressure at point B is less than the hydrostatic pressureand point C is greater than the hydrostatic pressure, resulting inthe change of vertical pressure distribution. Thus, an extra horizon-tal pressure from point C to point B is created to produce a verticalacceleration of flow and lead to an undulation of water surface. It isapparent that only the SPH–Boussinesq approach can simulate thisphenomenon well.

Based on the numerical experience of the four benchmarks, itshould be noted that despite the good prediction of the SPH–Boussinesq approach, it has a time-consuming process of particlesearching in each computational time step. Thus, a parallel versionof the SPH–Boussinesq approach is necessary to enhance the com-putational efficiency for practical engineering applications.

4. Conclusions

This study proposes a new SPH approach to solve the 1D Bous-sinesq equations for investigating weakly nonhydrostatic shallowwater flows. In this approach, a predictor–corrector method isadopted for the time-marching procedure. An explicit time-integration is used for the computation of pure space derivativeterms (the hydrostatic and source terms), and an implicit time-integration scheme is applied for the correction of the mixed spaceand time derivatives term (the material term of B1 and B2). Due tothe Lagrangian nature of SPH, the local and convective derivativesare merged as the material derivative. Therefore, the numericaldispersion error caused by the convective term can be automati-cally removed in the proposed approach. In addition, since thematerial term of B1 and B2 is treated as the second-order spacederivative form, the proposed approach is a convenient numericalscheme in the processes of the space discretization compared toother Eulerian approaches. Four typical benchmarks in weaklynonhydrostatic shallow water flows are used to validate the newlyproposed approach. In an overall sense, this approach is capable ofresolving weakly nonhydrostatic shallow water flows such as soli-tary wave without changing its shape and velocity, two solitarywaves with nonlinear interactions, oscillatory wave occurred inviolent flow regions of dambreak flow, and undulation developedby a mild transitional flow. Their performance of numerical accu-racy is quite satisfactory. Therefore, the present SPH approachcan provide an alternative to model weakly nonhydrostaticshallow water flows in open channels.

Acknowledgement

The authors are grateful for the financial support of this workprovided by National Science Council of Taiwan, R.O.C., underGrant No. NSC 100-2221-E-002-191-MY3.

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