diﬀusive wave solutions for open channel ﬂows with uniform...

Advances in Water Resources 29 (2006) 1000–1019

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Diffusive wave solutions for open channel flowswith uniform and concentrated lateral inflow

P. Fan *, J.C. Li

Institute of Mechanics, Chinese Academy of Sciences, No. 15 Beisihuanxi Road, Beijing 100080, China

Received 4 October 2004; received in revised form 31 August 2005; accepted 31 August 2005Available online 13 October 2005

Abstract

The diffusive wave equation with inhomogeneous terms representing hydraulics with uniform or concentrated lateral inflow intoa river is theoretically investigated in the current paper. All the solutions have been systematically expressed in a unified form interms of response function or so called K-function. The integration of K-function obtained by using Laplace transform becomesS-function, which is examined in detail to improve the understanding of flood routing characters. The backwater effects usuallyresulting in the discharge reductions and water surface elevations upstream due to both the downstream boundary and lateral infloware analyzed. With a pulse discharge in upstream boundary inflow, downstream boundary outflow and lateral inflow respectively,hydrographs of a channel are routed by using the S-functions. Moreover, the comparisons of hydrographs in infinite, semi-infiniteand finite channels are pursued to exhibit the different backwater effects due to a concentrated lateral inflow for various channeltypes.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Diffusive wave; Analytical solution; Laplace transformation; Backwater effect; Flood routing

1. Introduction

The quasi-linear hyperbolic Saint-Venant equationsystem (Saint-Venant, 1871) is generally used to describeunsteady flows in a river. Since rare analytical solution isavailable in most practical situations, the resolution ofthem mainly relies on numerical approaches based oneither the characteristic lines [1], finite difference schemes[2,6,15], or finite volume method [29,13].

Within the framework of the original Saint-Venant

equations, river hydraulics may be classified as dynamic,gravity, diffusion or kinematic wave models, corre-sponding to different forms of the momentum equation,respectively [10,33,8]. Dynamic wave model retains allthe terms of the momentum equation, whereas gravity

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* Corresponding author. Tel.: +86 10 62554125; fax: +86 1062561284.

E-mail address: [email protected] (P. Fan).

wave model neglects the effects of bed slope and viscousenergy loss and describes flows that are dominated byinertia. As a matter of fact, the acceleration terms inthe Saint-Venant equations can be neglected in mostpractical applications of flood routing in natural chan-nels. The system is thus reduced to a single parabolicequation known as the diffusive wave model [25]. Ifthe water depth gradient term is further omitted, thekinematic wave equation is acquired. The criteria fordemarcating kinematic wave and diffusive wave havebeen fully discussed recently [22,23,26–28,20]. Cappela-ere [5] devised a new method to improve the accuracyof the general nonlinear diffusive wave approach. Final-ly, solutions of the full diffusion wave model can be sim-ulated by the Muskingum–Cunge (M–C) method [24]. Itis shown that the solutions of the diffusion wave equa-tion following the M–C approach always give accurateresults once a straightforward condition is imposed [3].In Barry�s study, the results are found to be of the

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Nomenclature

B river width [m]cos(Æ) cosine functionC celerity of diffusive wave [m/s]D diffusive coefficient for diffusive wave [m2/s]erf(Æ) error functionexp(Æ) exponent functionf(x, t) source term in Eq. (3)Fs(x, t) defined in Eq. (97)Fr the Froud numberg the gravitational acceleration [m/s2]h water depth [m]H(Æ) Heaviside functionK0(x,n, t) defined in Eq. (8)Ku(x, t) kernel function relating to the upstream in-

flowKd(x, t) kernel function relating to the downstream

outflowKl(x, t) kernel function relating to the lateral inflowL channel lengthn iteration numberN(n) defined in Eq. (63)p Laplace variableq the discharge per unit width [m2/s]ql the lateral inflow per unit area of the conflu-

ent surface [m/s]Q discharge [m3/s]Q0, Qu, Qd, Ql discharge for initial condition, up-

stream, downstream boundary conditionsand lateral inflow respectively

~Qu; ~Qd Laplace transform of Qu and Qd respectivelyR defined in Eq. (73)S0 the bed slopeSf the friction slopeSu(x, t) S-function relating to the upstream inflowSd(x, t) S-function relating to the downstream out-

flowSl(x, t) S-function relating to the lateral inflowsin(Æ) sinusoidal functiont time [s]t0 duration for the pulse inflow input [s]T0 flood durationul the velocity component of lateral inflow in

the mainstream flow direction [m/s]v the mean cross-sectional velocity [m/s]x spatial coordinate [m]x0, x1, x2 coordinates of the confluent point, two

sides of the confluent region, respectively [m]X(x), T(t) defined in Eq. (10)u defined in Eq. (68)~u Laplace transform of ud(Æ) Dirac delta functionn, u, g, s dummy integration variable

* the convolution operationL Laplace transformation operationL�1 inverse Laplace transformation operation

P. Fan, J.C. Li / Advances in Water Resources 29 (2006) 1000–1019 1001

second-order accuracy if the Courant number happensto be 1/2. Moreover, the accuracy criteria for the linear-ized diffusion routing problem are discussed by Bajrach-arya [4]. Both the spatial weighting factor and the timestep are selected judiciously in this study to obtainguaranteed second-, third-, and fourth-order accurateschemes.

As an approximation, the diffusive wave model forunsteady flow routing has been used more frequentlyin recent years, since it may yield satisfactory solutionsand reasonably account for the downstream backwatereffect. As is well known, the diffusive wave equationcan be solved analytically if celerity and diffusivity areassumed constant. Up to now, a number of analyticalsolutions for the diffusion equation have been derivedunder the hypotheses of negligible inhomogeneous termor without uniformly distributed lateral inflow [17,11,19,28]. Therefore, the application of previous solutionsis very limited and not applicable in most of the practi-cal circumstances for river networks system. Earlieranalytical works scarcely considered the influences of

various inflows. Hayami [17] initially presented the fa-mous Hayami solution with the assumptions that thechannel reach is semi-infinite only with an upstreamboundary condition specified and the inhomogeneousterm is omitted in the governing equations. Conse-quently, the disadvantage of Hayami model is obviousin that the backwater effects of downstream boundaryand lateral inflow are ignored. Dooge et al. [11], Doogeand Napiorkowski [12] provided a solution of a finitechannel, in which backwater effects of the lateral inflowwere also neglected. Tawatchai and Shyam [30] derivedanalytical results in terms of water depth for a finitechannel with a concentrated lateral inflow, which wasapplied to route the floods in a hypothetical rectangularchannel with different upstream, downstream, and lat-eral boundary conditions and also showed good resultswhen applied to simulate flood flow conditions in 1980and 1981 in the Lower Mun River, in Northeast Thai-land. An excellent means was hence provided by thismodel to analyze individual or overall effects of theboundary conditions and much less effort and time were

1002 P. Fan, J.C. Li / Advances in Water Resources 29 (2006) 1000–1019

required at a particular station. Chung et al. [7] usednumerical inverse Laplace transform to study the back-water effect of the downstream boundary in a finitechannel. Based on the Hayami�s solution, Moussa [19]developed a method in order to solve the diffusive waveequation with a lateral inflow or outflow uniformly dis-tributed over a channel reach. Unfortunately, this solu-tion is still based on the hypothesis of a semi-infinitechannel and thus unable to study the influences ofdownstream boundary and concentrated lateral inflow.Singh [28] presented more overall solutions, except forcases with a concentrated lateral inflow. On the otherhand, the characteristics of response functions havenot been fully discussed. Moramarco et al. [18] obtainedthe analytical solutions by linearizing the full Saint-Venant equations with uniform lateral inflow. Fanet al. [14] yielded a solution for an infinite channel witha concentrated lateral inflow and compared it with thenumerical solutions. It can be seen that more generalsolutions of the diffusive wave model with source termsin various types of channel have not been given thus far.And different backwater effects from the downstreamboundary and the lateral inflow need further systematicexploration as well.

In the present paper, analytical methods are devel-oped to derive general solutions in a unified form ofexpression in terms of kernel functions applicable forinfinite, semi-infinite and finite channels with concen-trated and uniform lateral inflows. At the same time,the hydrographs reflecting flood routing process in dif-ferent cases are carefully examined with S-functionsgiven.

Fig. 1. Sketches for the four kinds of channel types, in which case 1,case 2, case 3 and case 4 represents an infinite channel, a semi-infinitechannel with upstream boundary, a semi-infinite channel with down-stream boundary and a finite channel, respectively.

2. General solution of the diffusive wave equation

The classic Saint-Venant equations for one-dimen-sional, unsteady, open-channel flow in a rectangularchannel can be written in the form,

ohotþ oq

ox¼ ql

Bð1Þ

oqotþ o

oxq2

hþ gh2

2

� �¼ ghðS0 � SfÞ þ ql

ul

Bð2Þ

where h is the water depth, q the discharge per unitwidth, B the river width, t the time, x the distance alongthe flow direction. ql denotes the lateral inflow per unitwidth of the confluent region. S0 is the bed slope, Sf

the friction slope, g the gravitational acceleration, ul

the velocity component of lateral inflow in the main-stream flow direction.

Neglecting the inertia term as Fr2� 1 and eliminat-ing variable h from Eq. (1) and (2), a single diffusivewave equation with a source term in terms of dischargeQ = qB can be rendered as (see Appendix E),

oQotþ C

oQox¼ D

o2Qox2þ f ðx; tÞ ð3Þ

in which,

C ¼ 1

BdQdh

ð4Þ

D ¼ Q2BSf

� Q2BS0

ð5Þ

where C and D are the diffusion wave celerity and diffu-sion coefficient, respectively, Q the discharge. The inho-mogeneous term in Eq. (3) looks like,

f ðx; tÞ ¼ Cql � Doql

oxð6Þ

The characteristic scale for flood diffusion in a chan-nel is found to be proportional to

p(DT0), where T0 is

the flood duration. Accordingly, a channel can be spec-ified as finite, infinite or semi-infinite by comparingscales of the channel length L and of the flood diffusionp

(DT0). Supposing that the distance between upstreamwater source and downstream sink (such as lake or estu-ary) L � pðDT 0Þ, the channel is usually regarded as fi-nite and hence the boundary conditions at two endsshould be imposed. In contrast, a channel is assumedsemi-infinite or infinite provided that L�p(DT0), inwhich only one or neither boundary condition need con-sideration (see Fig. 1). In the following text, we are con-cerned with diffusive waves in above-mentioned threekinds of channel forms with uniformly distributed orconcentrated lateral inflows.

2.1. Diffusive waves in infinite channels

In an infinite channel, the upstream and downstreamboundary conditions are usually ignored and the initialcondition is specified as,

Qðx; tÞjt¼0 ¼ Q0ðxÞ; �1 < x < þ1 ð7ÞThe solution of Eq. (3) in infinite space can readily beacquired (see Appendix A) as,

Fig. 2. Sketches for the channel with the two kinds of lateral inflow, inwhich case 1 and case 2 describe uniform and concentrated lateralinflows, respectively.


Qðx; tÞ ¼Z þ1

�1Q0ðnÞK0ðx; n; tÞdn

þZ þ1

�1f ðn; tÞ � K0ðx; n; tÞdn ð8Þ

in which,

K0ðx; n; tÞ ¼1

2ffiffiffiffiffiffiffiffipDtp exp �ðx� n� CtÞ2

4Dt

!ð9Þ

where * denotes the convolution operation.In most circumstances, the temporal and spatial dis-

tributions of lateral inflow discharge ql(x, t) are usuallyindependent. Thus ql(x, t) can be separated into a prod-uct of two factors as follows [19],

qlðx; tÞ ¼ X ðxÞT ðtÞ ð10Þwhere X(x) is the discharge distribution function inspace, T(t) implies the total discharge of the lateral in-flow. Note that ql(x, t) has the dimension of dischargeper width [m2/s]. Accordingly the dimension of X(x) is[1/m] and the dimension of T(t) is identical to that ofdischarge [m3/s].

By substituting Eq. (10) into Eq. (6), the source termin the original diffusive wave equation (3) can be repre-sented as,

f ðx; tÞ ¼ T ðtÞ C � X ðxÞ � Dd

dxX ðxÞ

� �ð11Þ

The second term at the right hand of Eq. (8) is trans-formed into,Z 1

�1f ðn; tÞ � K0ðx; n; tÞdn

¼ T ðtÞ �Z 1

�1C � X ðnÞ � D

d

dnX ðnÞ

� �K0ðx; n; tÞdn

ð12Þ

Similar to the definition of the kernel function in Hay-

ami�s solution [19], the kernel function (simplified toK-function in this paper) Kl(x, t) relating to the lateralinflow discharge can be defined as,

K lðx; tÞ ¼Z 1

�1C � X ðnÞ � D

d

dnX ðnÞ

� �K0ðx; n; tÞdn ð13Þ

satisfying,Z 1

�1f ðn; tÞ � K0ðx; n; tÞdn ¼ T ðtÞ � K lðx; tÞ ð14Þ

In reality, the lateral discharge into a channel is clas-sified in two categories. One for the system of streamand aquifer [19,21] and the other describes channels withtributaries. The lateral inflow uniformly distributesalong a river reach for the former, whereas the lateral in-flow concentrates in the tributary for the latter (seeFig. 2). As a result, the distribution function X(x) canbe put in the following forms:

(i) For lateral inflows uniformly distributed in theregion [x1,x2], the distribution function X(x) canbe written as,

X ðx; x1; x2Þ ¼Hðx� x1Þ � Hðx� x2Þ

x2 � x1

¼1

x2�x1x 2 ½x1; x2

0 x 62 ½x1; x2

(ð15Þ

in which Heaviside function satisfies,

Hðx� x1Þ ¼1 x P x1

0 x < x1

�ð16Þ

(ii) For a lateral inflow concentrated at point x0

(assuming the width of the tributary negligible)the distribution function X(x) satisfies,�
X ðxÞ ¼ dðx� x0Þ ¼
1 x ¼ x0

0 x 6¼ x0

ð17Þ

in which d(x) is the Dirac function.

2.1.1. Diffusive waves with a uniform lateral inflow

By substituting Eqs. (15) and (9) into Eq. (13) wehave expression of the K-function Kl(x, t) for the lateralinflow uniformly distributed in region [x1,x2] as follows:

K lðx; tÞ ¼C

x2 � x1

Z x2

x1

K0ðx; n; tÞdn

þ Dx2 � x1

½K0ðx; x2; tÞ � K0ðx; x1; tÞ

¼ �C2ðx2 � x1Þ

erfx� x2 � Ct

2ffiffiffiffiffiDtp

� ��

� erfx� x1 � Ct

2ffiffiffiffiffiDtp

� ��þ

ffiffiffiffiDp

2ffiffiffiffiffiptpðx2 � x1Þ

exp �ðx� x2 � CtÞ2

4Dt

!"

� exp �ðx� x1 � CtÞ2

4Dt

!#ð18Þ

in which erf(x) is the Error function.


The K-function is essentially the same as the instanta-neous unit hydrograph (IUH), the integration of whichis customarily called the storage function [9,32]. As thehydrograph at any location of a channel can be routeddirectly by the storage function, more interests in thepresent paper hence are laid on the storage function(hereafter simply called S-function) than IUH. For in-stance, if the discharge process and S-function for a lat-eral inflow input are defined as Ql(x, t) and Sl(x, t)respectively, the lateral inflow is approximated by histo-grams as [30],

T ðtÞ ¼ T j; j� 1 < t 6 j; j ¼ 1; . . . ; n ð19Þ

Then the discharge process induced by the lateral inflowinput satisfies,

Qlðx; tÞ ¼ T 1Slðx; tÞ þXn

j¼1

Slðx; t � jÞðT jþ1 � T jÞ ð20Þ

where the definition of Sl(x, t) for the lateral inflow iswritten as below,

Slðx; tÞ ¼Z t

0

K lðx; sÞds ¼ 1 � K lðx; tÞ ð21Þ

It shows from the above expression that S-functionrepresents the flood routing process with a continuousunit input. With parameters set as C = 1 m/s, D =10,000 m2/s (the values of C and D remain unchangedhenceforth unless they are specified additionally), x1 =5 km and x2 = 10 km, Sl(x, t) are illustrated inFig. 3(a), while the limits of which are depicted inFig. 3(b). Fig. 3(a) shows that, (a) Sl(x, t) is negativefor x < x1 and approximates to zero as time tends toinfinity; Each curve has a minimum which diminishesas x approaches to x1; (b) Sl(x, t) increases with t forx > x2 and decreases with x; Each curve has the com-mon limit, unit; (c) The limits of Sl(x, t) rise with thecoordinates for x1 < x < x2. Consequently, we may lead

Fig. 3. (a) Sl(x, t) for an infinite channel with a lateral inflow uniformly dist

to the conclusion that the lateral inflow not only propa-gates downstream but also results in the dischargereduction upstream, which is usually called backwatereffect. In the confluent region, Sl(x, t) rises with the coor-dinates. Conversely, Sl(x, t) reduces with distance out ofthe confluent region, which is apparently characteristicof diffusive waves. With a continuous unit input, theflood waves die out and the steady flow state recoversas time approaches to infinity.

Using characters of the Laplace transformation gives,

L½K0ðx; n; tÞjp¼0 ¼1

Cexp

Cðx� n� jx� njÞ2D

� �

¼1C x > n

1C exp Cðx�nÞ

D

� x < n

8<: ð22Þ

in which L represents the Laplace transformation opera-tor. Accordingly, the limit of Sl(x, t) can be calculated inthe following way:

limt!1

Slðx; tÞ¼Z 1

0

K lðx;sÞds¼ L½K lðx; tÞjp¼0

¼ Cx2� x1

Z x2

x1

L½K0ðx;n; tÞjp¼0 dn

þ Dx2� x1

½L½K0ðx;x2; tÞjp¼0�L½K0ðx;x1; tÞjp¼0

¼1 x> x2

x�x1

x2�x1x1 < x< x2

0 x< x1

8><>: ð23Þ

It is clearly seen that the limits accords with what hasshown in Fig. 3(b).

2.1.2. Diffusive waves with a concentrated lateral inflow

The expression of the K-function Kl(x, t) for a lateralinflow concentrated at x0 can be attained by substitutingEqs. (17) and (9) into Eq. (13),

ributed in the region [5 km, 10 km]; (b) limits of Sl(x, t) shown in (a).

Fig. 4. Hydrographs for an infinite channel with a pulse input from alateral inflow concentrated at x0 = 10 km.


K lðx; tÞ ¼ CK0ðx; x0; tÞ þ DoK0ðx; n; tÞ

on

n¼x0

¼ x� x0 þ Ct

4ffiffiffiffiffiffiffiffiffiffipDt3p exp �ðx� x0 � CtÞ2

4Dt

!ð24Þ

Then the S-function is,

Slðx; tÞ ¼Z t

0

x� x0 þ Cs

4ffiffiffiffiffiffiffiffiffiffipDs3p exp �ðx� x0 � CsÞ2

4Ds

!ds

¼

12

1� erf x�x0�Ct2ffiffiffiffiDtp

� h i!t!1 1; x > x0

12erf C

ffitp

2ffiffiffiDp

� !t!1 1

2; x ¼ x0

12

erfðx0�xþCt2ffiffiffiffiDtp Þ � 1

h i!t!1 0; x < x0

8>>>><>>>>:

ð25Þ

Comparing Eq. (25) to Eq. (23), they are different inthat the limit of Sl(x, t) is 1/2 at the confluent pointx = x0.

Usually, the discharge reductions and water surfaceelevations upstream due to the downstream constric-tions such as the reservoir and piers are termed as thedownstream backwater effect. This effect generating asteady and spatially gradually varying water surfaceprofile gives rise to serious impacts on the sedimentstransport and flood prevention in a river [30,31]. Actu-ally, a backwater effect may be generated as well bythe lateral inflow for a river, which is usually ignoredhowever. In the current paper, the negative minimumvalues of Sl(x, t) is used to scale the backwater effectdue to the lateral inflow. The following two equationsevaluate the zero point of the first order and the secondorder derivatives of Sl(x, t).

o

oterf

x0 � xþ Ct

2ffiffiffiffiffiDtp

� �� t¼x0�x

C

¼ � x0 � x� Ct

4ffiffiffiffiffiffiffiDt3p

� �e� x0�xþCt

2ffiffiffiDtp

� 2t¼x0�x

C

¼ 0 ð26Þ

o2

ot2erf

x0 � xþ Ct

2ffiffiffiffiffiDtp

� �� t¼x0�x

C

¼ 8Dt½3ðx0 � xÞ � Ct � ðx0 � xþ CtÞðx0 � x� CtÞ2

16ffiffiffiffiffiffiffiffiffiD3t7p

24

e� x0�xþCt

2ffiffiffiDtp

� 235t¼x0�x

C

> 0 ð27Þ

As a result, the minimum value of Sl(x, t) for x < x0 is,

Min½Slðx; tÞ ¼1

2erf

x0 � xþ Ct

2ffiffiffiffiffiDtp

� �� 1

� �t¼x0�x

C

¼ � 1

21� erf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCðx0 � xÞ

D

r ! ð28Þ

which indicates that the minimum value of Sl(x, t) de-creases with jx0 � xj. Therefore, we may conclude thatthe backwater effect is intensified as moving close tothe confluent point and reaches maximum right at theconfluent point x = x0.

2.1.3. Hydrographs for a concentrated lateral

inflow input

Traditionally used for flood routing process study byhydrologists, the hydrographs due to different inputsincluding lateral inflow, upstream inflow and down-stream outflow are investigated as well in the currentpaper. The lateral inflow distributed along the riverreach uniformly or even varying in space very probablybecomes a more interesting problem and needs furtherstudy. Hence, only hydrographs due to a concentratedlateral inflow are considered here. A pulse input froma concentrated lateral inflow T(t) can be set as follows:

T ðtÞ ¼1 t 6 t0

0 t > t0

�; t0 ¼ 7200 s ð29Þ

in which T(t) is unit for t < t0 and zero for t > t0

(t0 = 2 h). Hydrograph for the concentrated lateral in-flow input can be governed by,

Qlðx; tÞ ¼Z t

0

T ðsÞK lðx; t � sÞds

¼Slðx; tÞ t 6 t0

Slðx; tÞ � Slðx; t � t0Þ t > t0

�ð30Þ

If neglecting the initial condition, hydrograph for aninfinite channel with a concentrated lateral inflow inputis illustrated in Fig. 4. It shows that the flood quantityfrom a tributary not only propagates in the downstreamdirection but also causes a backwater effect resulting inthe discharge reduction in the upstream direction. Both


the discharges propagated in the downstream directionand the backwater effect in the upstream direction atten-uate with the distance from the confluent point.

2.1.4. Sensitivity analysis of parameters C and D

The graphs for various celerity C and diffusivity D areplotted in Figs. 5–7, where C varies from 0.5 to 2 m/s andD from 10,000 to 100,000 m2/s, respectively. In Fig. 5hydrographs are compared at x = 0 km (x � x0 < 0),showing that the backwater effect decreases with C andincreases with D. Fig. 6 shows that the flood peak in-creases with C and reduces with D for x = 15 km(0 < x � x0 < Ct0 = 7.2 km). In contrast, for x = 30 kmand x � x0 > Ct0 in Fig. 7, the flood peak rises with both

Fig. 5. Comparisons of hydrographs at x = 0 km for an infinite channel withfixed at 10,000 m2/s and C ranges from 0.5 to 2 m/s; (b) C is fixed at 1 m/s

Fig. 6. Comparisons of hydrographs at x = 15 km for an infinite channel withfixed at 10,000 m2/s and C ranges from 0.5 to 2 m/s; (b) C is fixed at 1 m/s

C and D. All of these conclusions (summarized in Table1) can also be explained by Eq. (25).

2.2. Diffusive waves in semi-infinite channels with

upstream boundary

For a semi-infinite channel with upstream boundary,the boundary condition of the original diffusion equa-tion is specified as,

Qðx; tÞjx¼0 ¼ QuðtÞ; 0 6 t < þ1 ð31Þ

together with the initial condition,

Qðx; tÞjt¼0 ¼ Q0ðxÞ; 0 6 x < þ1 ð32Þ

a pulse input from a lateral inflow concentrated at x0 = 10 km; (a) D isand D ranges from 10,000 to 100,000 m2/s.

a pulse input from a lateral inflow concentrated at x0 = 10 km; (a) D isand D ranges from 10,000 to 100,000 m2/s.

Fig. 7. Comparisons of hydrographs at x = 30 km for an infinite channel with a pulse input from a lateral inflow concentrated at x0 = 10 km; (a) D isfixed at 10,000 m2/s and C ranges from 0.5 to 2 m/s; (b) C is fixed at 1 m/s and D ranges from 10,000 to 100,000 m2/s.

Table 1Effects of increasing parameter C and D on characters of flood wave

Flood character Only C increases Only D increases

Upstream Downstream Upstream Downstream

x � x0 < Ct0 x � x0 > Ct0 x � x0 < Ct0 x � x0 > Ct0

Backwater effect Decrease IncreaseFlood peak Increase Increase Decrease Increase

Upstream and downstream denotes the direction of diffusive waves.Note: Reverse condition occurs when C and D decrease.


The solution in the semi-infinite space can be derived(see Appendix B) as,

Qðx; tÞ ¼ QuðtÞ � Kuðx; tÞ þZ 1

0

Q0ðnÞK0ðx; n; tÞdn

þZ 1

0

f ðn; tÞ � K0ðx; n; tÞdn ð33Þ

where,

Kuðx; tÞ ¼x

2ffiffiffiffiffiffiffipDp

t3=2exp �ðx� CtÞ2

4Dt

!ð34Þ

K0ðx; n; tÞ ¼1


4Dt

!"

� expCxD

� �exp �ðxþ nþ CtÞ2

4Dt

!#ð35Þ

Obviously Ku(x, t) is the K-function for the upstreamboundary inflow.

2.2.1. Diffusive waves with a upstream inflow

Note that the K-function Ku(x, t) is essentially thesame as the solution obtained by Hayami [17]. TheS-function for Ku(x, t) is written as below,

Suðx; tÞ ¼Z t

0

Kuðx; sÞds

¼Z t

0

xþ Cs

4ffiffiffiffiffiffiffiffiffiffipDs3p exp �ðx� CsÞ2

4Ds

!ds

þ expCxD

� �Z t

0

x� Cs

4ffiffiffiffiffiffiffiffiffipDsp exp �ðxþ CsÞ2

4Ds

!ds

¼ 1

21� erf

x� Ct

2ffiffiffiffiffiDtp

� �� þ 1

2exp

CxD

� �

1� erfxþ Ct

2ffiffiffiffiffiDtp

� �� !t!1 1 ð36Þ

which shows the unit limit of Su(x, t) as time approach-ing to infinity. It can be concluded from Fig. 8(a) as well.The curves in Fig. 8(a) representing Su(x, t) for a semi-infinite channel with upstream boundary have the com-mon limit, unit, and reduce along with the channelreach.

2.2.2. Diffusive waves with a uniform lateral inflowFor a lateral inflow uniformly distributed in region

[x1,x2], the K-function Kl(x, t) can be obtained by substi-tuting Eq. (35) into Eq. (18) as follows:

Fig. 8. S-functions for a semi-infinite channel with the upstream boundary at x = 0 km; (a) Su(x, t) for a upstream inflow; (b) Sl(x, t) for a lateralinflow uniformly distributed in region [5 km, 10 km].


K lðx; tÞ ¼C

x2 � x1

Z x2

x1

K0ðx; n; tÞdn

þ Dx2 � x1

½K0ðx; x2; tÞ � K0ðx; x1; tÞ

¼ �C2

erf x�x2�Ct2ffiffiffiffiDtp

� � erf x�x1�Ct

2ffiffiffiffiDtp

� x2 � x1

þ C2

expCxD

� � erf xþx2þCt2ffiffiffiffiDtp

� � erf xþx1þCt

2ffiffiffiffiDtp

� x2 � x1

þffiffiffiffiDp

2ffiffiffiffiffiptp

exp � ðx�x2�CtÞ24Dt

� � exp � ðx�x1�CtÞ2

4Dt

� x2 � x1

�ffiffiffiffiDp

2ffiffiffiffiffiptp exp

CxD

� �

exp � ðxþx2þCtÞ2

4Dt

� � exp � ðxþx1þCtÞ2

4Dt

� x2 � x1

ð37Þ

Since Eq. (37) has not a more concise expression,therefore, the S-function can hardly find an analyticalexpression. Fig. 8(b) plots Sl(x, t) exhibiting similarcharacteristics to that in Fig. 3(a).

2.2.3. Diffusive waves with a concentrated lateral inflowThe K-function for a lateral inflow concentrated at x0

can be derived by substituting Eq. (35) into Eq. (24),


on

n¼x0

¼

x

2ffiffiffiffiffiffiffipDt3p exp � ðx�CtÞ2

4Dt

� ¼ Kuðx; tÞ x0 ¼ 0

x�x0þCt

4ffiffiffiffiffiffiffipDt3p exp � ðx�x0�CtÞ2

4Dt

� þ xþx0�Ct

4ffiffiffiffiffiffiffipDt3p

exp CxD

� �exp � ðxþx0þCtÞ2

4Dt

� x0 > 0

8>>>>><>>>>>:

ð38Þ

Integrating the above equation gives,

Slðx; tÞ ¼

x0 ¼ 0 Suðx; tÞ

x0 > 0

12

1� erf x�x0�Ct2ffiffiffiffiDtp

� h iþ 1

2exp Cx

D

� � 1� erf xþx0þCt

2ffiffiffiffiDtp

� h ix > x0

12erf C

ffitp

2ffiffiffiDp

� þ 1

2exp Cx

D


2ffiffiffiffiDtp

� h ix ¼ x0

12

erf jx�x0jþCt2ffiffiffiffiDtp

� � 1

h iþ 1

2exp Cx

D


2ffiffiffiffiDtp

� h ix < x0

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

ð39ÞIt is clearly seen that the above function has the same lim-it as Eq. (25). Comparing Eqs. (39) and (25) leads to dif-ferences of Sl(x, t) for the cases that the channel is infiniteand semi-infinite with upstream boundary as follows:

DSlðx; tÞ ¼1

2exp

CxD

� �1� erf

xþ x0 þ Ct

2ffiffiffiffiffiDtp

� �� ð40Þ

in which DSl(x, t) stands for the influence of the upstreamboundary on the flow due to lateral inflow input. AsDSl(x, t) is positive, the discharge induced by the lateralinflow input is always enlarged by the upstream bound-ary. For a certain flood duration t, if x0� 2

p(Dt), we

have DSl(x, t)! 0 as x!1. Physically, it means thatif the distance between the upstream boundary and theconfluent point is long enough, influence of the upstreamboundary vanishes.

2.2.4. Hydrographs for upstream boundary and

concentrated lateral inflow inputs

By using the similar method as indicated in Eq. (30)(Sl(x, t) in that equation should be replaced by Su(x, t)here), hydrographs with a pulse input from the upstream

Fig. 9. Hydrographs for a semi-infinite channel with the upstream boundary at x = 0 km; (a) a pulse input from the upstream inflow; (b) a pulseinput from the lateral inflow concentrated at x0 = 10 km.

Fig. 10. Comparisons of hydrographs for the cases that the channel isinfinite and semi-infinite with upstream boundary at x = 0 km respec-tively; a pulse input from the lateral inflow concentrated at x0 = 5 km.


boundary are plotted in Fig. 9(a), showing the floodrouting processes clearly from the upstream inflow input.

In the meantime, Fig. 9(b) shows the hydrograph witha pulse input from a concentrated lateral inflow in asemi-infinite channel, of which the backwater effect fromthe lateral inflow appears similar to that in Fig. 4. Thedifferences are illustrated in Fig. 10, in which the hydro-graphs from a concentrated lateral inflow are comparedfor an infinite channel and semi-infinite channel with up-stream boundary respectively. It can be inferred that theupstream boundary of the semi-infinite channel enlargesthe discharges from the lateral inflow, meanwhile lessensthe backwater effect of the lateral inflow.

2.3. Diffusive waves in semi-infinite channels with

downstream boundary

For a semi-infinite channel with downstream bound-ary, the boundary condition and initial condition are asfollows:

Qðx; tÞjx¼0 ¼ QdðtÞ; �1 < x 6 0 ð41ÞQðx; tÞjt¼0 ¼ Q0ðxÞ; �1 < x 6 0 ð42Þ

The solution in the semi-infinite space can be expressed(see Appendix C) as,

Qðx; tÞ ¼ QdðtÞ � Kdðx; tÞ þZ 0


þZ 0


where,

Kdðx; tÞ ¼jxj


t3=2exp �ðjxj þ CtÞ2

4Dt

!ð44Þ

and K0(x,n, t) is identical to Eq. (35).

2.3.1. Diffusive waves with downstream outflow

Integrating the K-function Kd(x, t) gives the S-func-tion for the downstream outflow Sd(x, t) as follows:

Sdðx; tÞ ¼Z t

0

jxj � Cs

4ffiffiffiffiffiffiffiffiffiffipDs3p exp �ðjxj þ CsÞ2

4Ds

!ds

þ exp �CjxjD

� �Z t

0

jxj þ Cs

4ffiffiffiffiffiffiffiffiffipDsp

exp �ðjxj � CsÞ2

4Ds

!ds

¼ 1

21� erf

jxj þ Ct

2ffiffiffiffiffiDtp

� ��

þ 1

2exp �Cjxj

D

� �1� erf

jxj � Ct

2ffiffiffiffiffiDtp

� �� ð45Þ

Fig. 11. (a) Sd(x, t) for a semi-infinite channel with downstream boundary at x = 0 km; (b) limits of Sd(x, t) in (a).


which is plotted in Fig. 11(a). Sd(x, t) are always posi-tive, implying that the discharge in a channel rises dueto an outflow at the downstream boundary. Moreover,the induced discharge reduces as moving away fromthe downstream boundary. Fig. 11(b) illustrates the lim-it of the curves in Fig. 11(a), which satisfies exp(Cx/D).By using Laplace transformation, the limit of Sd(x, t) isexpressed as,

limt!1

Sdðx; tÞ ¼ L½Kdðx; tÞjp¼0

¼ expCx2D

� �exp � jxjffiffiffiffi

Dp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip þ C2

4D

s0@

1Ap¼0

¼ expCxD

� �ð46Þ

Fig. 12. (a) Sl(x, t) for a semi-infinite channel with a lateral inflow uniformlyx = 0 km; (b) limits of Sl(x, t) in (a).

which is found in agreement with the diagrams inFig. 11(b).


Since K0(x,n, t) is identical to Eq. (35), the K-functionKl(x, t) with a lateral inflow uniformly distributed in re-gion [x1,x2] has the same expression as Eq. (37). The S-function Sl(x, t) now is plotted in Fig. 12(a), where thecurves show different characters from those inFig. 3(a) or Fig. 8(b). It indicates an important fact thatdue to the different integral regions, the expression ofSl(x, t) for a semi-infinite channel with upstream bound-ary differs from that with downstream boundary, even ifthe K-functions in each case are the same. Fig. 12(a) alsodemonstrates that (a) for x < x1 Sl(x, t) are negative anddecreases with time; Each curve has a negative limitwhich diminishes to zero as x tends to infinity; (b) for

distributed in region [�10 km, �5 km], the downstream boundary at


x > x2 Sl(x, t) increases at first and decreases with timesubsequently; Each curve has a positive limit whichdiminishes to zero at the downstream boundaryx = 0 km; (c) for x1 < x < x2 the limits of Sl(x, t) increasewith the coordinate. Hence it shows that the down-stream boundary exerts different influence on the flowof the lateral inflow from the upstream boundary. Theupstream boundary always amplifies the discharge dueto the lateral inflow input, while the downstream bound-ary reduces the discharge.

The Laplace transformation of K0(x,n, t) gives,

L½K0ðx; n; tÞ ¼1

2ffiffiffiffiDp exp

Cðx� nÞ2D

� �1ffiffiffiffiffiffiffiffiffiffiffiffiffi

p þ C2

4D

q

exp � jx� njffiffiffiffiDp


4D

s0@

1A

24

� exp � jxþ njffiffiffiffiDp


4D

s0@

1A35 ð47Þ

And the limit of Sl(x, t) for lateral inflow uniformly dis-tributed in region [x1,x2] can be given as,

limt!1

Slðx; tÞ ¼

1� exp CxD

� �x > x2

x�x1

x2�x1� exp Cx

D

� �x1 6 x 6 x2

� exp CxD

� �x < x1

8>><>>: ð48Þ

As compared with Eq. (23), we may find that the differ-ence between them happens to be exp(Cx/D), namely,the limit of Sd(x, t) in Eq. (46), which implies that Eq.(48) exhibits the superposition of effects of lateral inflowand downstream boundary. Fig 12(b) plots the limit ofSl(x, t) and displays the strongest backwater effect occur-ring at x = x1.


By using similar approach to Eq. (38), the K-functionKl(x, t) for lateral inflow concentrated at x0 can be ob-tained as follows:


on

n¼x0

¼

x

2ffiffiffiffiffiffiffipDt3p exp � ðx�CtÞ2

4Dt

� ¼ �Kdðx; tÞ x0 ¼ 0

x�x0þCt

4ffiffiffiffiffiffiffipDt3p exp � ðx�x0�CtÞ2

4Dt

� þ xþx0�Ct

4ffiffiffiffiffiffiffipDt3p exp Cx

D

� � exp � ðxþx0þCtÞ2

4Dt

� x0 < 0

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð49Þ

the integration of which leads to the S-function asbelow,

Slðx; tÞ ¼

x0 ¼ 0 � Sdðx; tÞ

x0 < 0

12

1� erf jx�x0j�Ct2ffiffiffiffiDtp

� h iþ 1

2exp Cx

D

� � erf jxþx0j�Ct

2ffiffiffiffiDtp

� � 1

h ix > x0

12erf C

ffitp

2ffiffiffiDp

� þ 1

2exp Cx

D


2ffiffiffiffiDtp

� � 1

h ix ¼ x0

12

erf jx�x0jþCt2ffiffiffiffiDtp

� � 1

h iþ 1

2exp Cx

D


2ffiffiffiffiDtp

� � 1

h ix < x0

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:

ð50Þand the limitation turns out,

limt!1

Slðx; tÞjx0<0 ¼1� exp Cx

D

� �x > x0

12� exp Cx

D

� �x ¼ x0

� exp CxD

� �x < x0

8><>: ð51Þ

It is clearly seen that the K-functions Kl(x, t) for a semi-infinite channel with downstream boundary Eq. (49) isthe same as that with the upstream boundary Eq. (38).However, Eq. (50) differs from Eq. (25) significantly.The difference of the two Sl(x, t) for x0 < 0 is,

DS0lðx; tÞ ¼1

2exp

CxD

� �erfjxþ x0j � Ct

2ffiffiffiffiffiDtp

� �� 1

� �ð52Þ

There is a slight difference between the above equationand Eq. (40), which shows the downstream boundarygenerates different influences on the flow of the lateralinflow input from the upstream boundary. DS0lðx; tÞ is al-ways negative indicating that the discharge induced bythe lateral inflow input is reduced by the downstreamboundary. Furthermore, this effect appears to vanish ifthe distance between the downstream boundary andthe confluent point is long enough.

2.3.4. Hydrographs for the downstream boundary

outflow and concentrated lateral inflow

Hydrographs for a pulse outflow from downstreamboundary are plotted in Fig. 13(a). It shows that an out-flow from the downstream boundary also enlarges thedischarges of a semi-infinite channel. Actually, if a reser-voir is regarded as the downstream boundary of a chan-nel in the upland, the increasing of outflow from thisreservoir is equivalent to the rise of discharge and henceleads to the descending of the water stage.

Hydrographs for a pulse input from a concentratedlateral inflow in a semi-infinite channel are shown inFig. 13(b), of which the implications are analogous tothat revealed in Fig. 4. The differences between thetwo cases are shown in Fig. 14. We may conclude thatthe downstream boundary exerts an opposite influenceon the flow of a lateral inflow input as compared tothe upstream boundary cases. In other words, the

Fig. 13. Hydrographs for a semi-infinite channel with downstream boundary at x = 0 km; (a) a pulse input from the downstream outflow; (b) a pulseinput from the lateral inflow concentrated at x0 = �10 km.

Fig. 14. Comparisons of hydrographs for the channel is infinite andsemi-infinite with downstream boundary at x = 0 km respectively; apulse input from the lateral inflow concentrated at x0 = �5 km.


backwater effect of the lateral inflow is intensified by thedownstream boundary.

2.4. Diffusive waves in finite channels

Boundary conditions for a finite channel are usuallydescribed as,

Qðx; tÞjx¼0 ¼ QuðtÞ; Qðx; tÞjx¼L ¼ QdðtÞ ð53Þ

and the initial condition is,

Qðx; tÞjt¼0 ¼ Q0ðxÞ ð54Þ

Then, the solution of a finite channel can be obtained(see Appendix D) as,

Qðx; tÞ ¼ QuðtÞ � Kuðx; tÞ þ QdðtÞ � Kdðx; tÞ

þZ L

0


þZ L

0


in which,

Kuðx; tÞ ¼ expCx2D� C2t

4D

� �

Xþ1n¼1

2npD

L2exp � n2p2D

L2t

� �sin

npxL

ð56Þ

Kdðx; tÞ ¼ expCðx� LÞ

2D� C2t

4D

� �

Xþ1n¼1

2npD

L2ð�1Þnþ1 exp � n2p2D

L2t

� �sin

npxL

ð57Þ

K0ðx; n; tÞ ¼ expCðx� nÞ

2D� C2t

4D

� �2

L

Xþ1n¼1

exp � n2p2D

L2t

� �sin

npnL

sinnpxL

ð58Þ

2.4.1. Diffusive waves with the upstream inflow and

downstream outflow

Both K-function Ku(x, t) and Kd(x, t) with the up-stream inflow and downstream outflow are found tobe identical to those obtained by Dooge et al. [11] andSingh [28]. Subsequently, the S-functions Su(x, t) andSd(x, t) are derived in the following way:


Suðx; tÞ ¼Z t

0

Kuðx; sÞds

¼ expCx2D

� �Xþ1n¼1

8npD2

ðC2L2 þ 4n2p2D2Þ

1� exp �C2L2 þ 4n2p2D2

4DL2t

� �� sin

npxL

ð59Þ

Sdðx; tÞ ¼Z t

0

Kdðx; sÞds

¼ expCðx� LÞ

2D

� �Xþ1n¼1

ð�1Þnþ18npD2

ðC2L2 þ 4n2p2D2Þ

1� exp �C2L2 þ 4n2p2D2

4DL2t

� �� sin

npxL

ð60Þ


Substituting Eq. (58) into Eq. (18) gives the K-func-tion Kl(x, t) for the lateral inflow uniformly distributedin region [x1,x2],

K lðx; tÞ ¼ expCx2D� C2t

4D

� �2

L

Xþ1n¼1

NðnÞ exp � n2p2D

L2t

� �sin

npxL

ð61Þ

integration of which leads to,

Slðx; tÞ ¼Z t

0

K lðx; sÞds

¼ expCx2D

� �2

L

Xþ1n¼1

4DL2NðnÞ4n2p2D2 þ C2L2

1� exp � 4n2p2D2 þ C2L2

4DL2t

� �� sin

npxL

ð62Þ

where,

NðnÞ ¼ 2D

ðx2 � x1Þð1þ ð2DC

npL Þ

2Þ

2DC

npL

� �2 � 1

2sin

npx2

L

� exp �Cx2

2D

� ��"

� sinnpx1

L

� exp �Cx1

2D

� ��

þ 2DC

npL

cosnpx1

L

� exp �Cx1

2D

� ��

� cosnpx2

L

� exp �Cx2

2D

� ��#ð63Þ


The K-function Kl(x, t) and S-function Sl(x, t) withthe lateral inflow concentrated at x0 can be obtainedby substituting Eq. (57) into Eq. (24), as follows:


on

n¼x0

¼ expCðx� x0Þ

2D� C2t

4D

� �Xþ1n¼1

exp � n2p2D

L2t

� �

CL

sinnpx0

Lþ 2npD

L2cos

npx0

L

� �sin

npxL

ð64Þ

Slðx; tÞ ¼ expCðx� x0Þ

2D

� �Xþ1n¼1

4DL2

4n2p2D2 þ C2L2

1� exp � 4n2p2D2 þ C2L2

4DL2t

� ��

CL

sinnpx0

Lþ 2npD

L2cos

npx0

L

� �sin

npxL

ð65Þ

For the extreme instance that confluent point is atx0 = 0, Kl(x, t) can be simplified into,

K lðx; tÞ ¼ expCx2D� C2t

4D

� �

Xþ1n¼1

2npD

L2exp � n2p2D

L2t

� �sin

npxL

¼ Kuðx; tÞ ð66Þ

and for x0 = L, simplified as,

K lðx; tÞ ¼ expCðx� LÞ

2D� C2t

4D

� �

Xþ1n¼1

2npD

L2ð�1Þn exp � n2p2D

L2t

� �sin

npxL

¼ �Kdðx; tÞ ð67Þ

The above two equations are similar to Eqs. (38) and(49), which indicates that the lateral inflow seems topropagate downstream when a lateral inflow concen-trates at the upstream boundary and the backwater ef-fect occurs when it concentrates at the downstreamboundary.

2.4.4. Hydrographs for a concentrated lateral inflow

inputThe hydrograph of diffusive waves with a lateral in-

flow input concentrated at x0 = 10 km in a finite channelof the length 20 km is plotted in Fig. 15, in which thewhole flood process is clearly exhibited. Fig. 16 com-pares the hydrograph at x = 15 km in three cases: thechannels are as long as 20 km, 50 km and semi-infinitechannel without downstream boundary, respectively. Itshows that the hydrograph of lateral inflow input is

Fig. 16. Comparisons of hydrographs plotted at x = 15 km, for thecases that a finite channel with length 20 km, 50 km and a semi-infinitechannel with upstream boundary, respectively; a pulse input from thelateral inflow concentrated at x0 = 10 km.

Fig. 15. Hydrographs for a finite channel with upstream and down-stream boundary at x = 0 km and x = 20 km respectively; a pulseinput comes from the lateral inflow concentrated at x0 = 10 km.


lessened by the downstream boundary for a channelwith length 20 km compared to a semi-infinite channel

Table 2Solutions obtained for various types of channel

Channel types Solutions

Upstream boundary inflow Downstream bou

I

SI-U Eq. (36)—Su(x, t)

SI-D Eq. (45)—Sd(x, t)

F Eq. (59)—Su(x, t) Eq. (60)—Sd(x, t)

Note: �I� means infinite; �SI-U� means semi-infinite with upstream boundary; �

and that the effect of the downstream boundary vanishesprovided the channel length exceeds 50 km.

3. Conclusions

In the present paper, we have systematically extendedthe results by Hayami [17], Dooge et al. [11], Moussa[19] and Singh [28] and expressed the general solutionsof the diffusive wave model in a unified form in termsof K-function applicable for infinite, semi-infinite and fi-nite channels with concentrated and uniform lateral in-flows (as summarized in Table 2).

By using Laplace transform method, the integrationof K-function, S-function, is obtained, which representsthe flood routing process with a continuous unit input.

(1) Characteristics of Sl(x, t), S-function for an infi-nite channel with uniform lateral inflow in the region[x1,x2] are analyzed in detail. (a) Sl(x, t) are negativefor x < x1 and approximates to zero as time tends toinfinity; Each curve has a minimum which diminishesas x approaches to x1; (b) Sl(x, t) increases with t forx > x2 and decreases with x; Each curve has the com-mon limit, unit; (c) The limits of Sl(x, t) rise with thecoordinates for x1 < x < x2.

(2) Su(x, t) and Sd(x, t) representing S-function for asemi-infinite channel with upstream inflow or down-stream outflow are further examined, respectively. BothSu(x, t) and Sd(x, t) are positive, implying the upstreaminflow as well as downstream outflow enhance the dis-charge in a channel. The limits for Su(x, t) and Sd(x, t)are unit and exp(Cx/D), respectively. It shows that theboundary conditions at two ends of a channel exert dif-ferent influences on the Sl(x, t), namely, the dischargefrom the lateral inflow input. The upstream boundaryalways amplifies the discharge, while the downstreamboundary diminishes it. As time approaches to infinity,the limits of Sl(x, t) in an infinite channel are the sameas that in a semi-infinite channel with upstream bound-ary. However, comparing the limits of Sl(x, t) in an infi-nite channel to that in a semi-infinite channel with

ndary outflow Inputs from lateral inflow

Uniform Concentrated

Eq. (18)—Kl(x, t) Eq. (24)—Kl(x, t)Eq. (25)—Sl(x, t)



Eq. (61)—Kl(x, t) Eq. (64)—Kl(x, t)Eq. (62)—Sl(x, t) Eq. (65)—Sl(x, t)

SI-D� means semi-infinite with downstream boundary; �F� means finite.


downstream boundary, we have a difference exp(Cx/D)which happens to be the limit of Sd(x, t). It exhibitsthe superposition of effects of lateral inflow and down-stream boundary to some extent.

The discharge reduction in the upstream directiondue to both downstream boundary and lateral inflowis usually called as backwater effect. In the presentpaper, hydrographs are used to observe the reduced dis-charge and hence to analyze the backwater effect. Thebackwater effect of the lateral inflow in an infinite chan-nel is firstly investigated, which appears to have a max-imum at the confluent point and then reduce as movingaway from the confluent region in the upstream direc-tion. Furthermore, we find that the backwater effect oflateral inflow is apparently affected by the boundaryconditions. It shows that the upstream boundary allevi-ates the backwater effect of lateral inflow, while thedownstream boundary enhances it.

Acknowledgements

The financial support from the National Natural Sci-ence Foundation of China under the grant 10332050 isappreciated. The authors also acknowledge the helpfulcomments provided by the anonymous reviewers.

Appendix A

By using transformation,

Qðx; tÞ ¼ expCx2D� C2t

4D

� �uðx; tÞ ð68Þ

the convection–diffusion equation (3) can be convertedinto a pure diffusion equation,

ouot¼ D

o2uox2þ exp � Cx

2Dþ C2t

4D

� �f ðx; tÞ ð69Þ

and the initial condition for an infinite channel Eq. (7) istransformed into,

uðx; tÞjt¼0 ¼ uðx; 0Þ ¼ exp � Cx2D

� �Q0ðxÞ ð70Þ

By using Laplace transform, Eq. (69) can be con-verted into,

o2 ~uðx; pÞox2

� pD

~uðx; pÞ ¼ Rðx; pÞD

; �1 < x < þ1 ð71Þ

in which,

L½uðx; tÞ ¼ ~uðx; pÞ; L½f ðx; tÞ ¼ ~f ðx; pÞand L½QuðtÞ ¼ ~QuðpÞ ð72Þ

Rðx; pÞ ¼ � exp � Cx2D

� �~f x; p � C2

4D

� �� uðx; 0Þ ð73Þ

Solution of this second order ordinary differentialequation is,

~uðx; pÞ ¼ �ffiffiffiffiDp

2ffiffiffipp

Z x

�1

Rðn; pÞD

exp �ffiffiffiffipD

rðx� nÞ

� �dn

�ffiffiffiffiDp

2ffiffiffipp

Z 1

x

Rðn; pÞD


rðn� xÞ

� �dn

ð74Þ

Hence using the inverse Laplace transformationyields,

uðx; tÞ ¼ � 1

2ffiffiffiffiffiffiffiffipDtp

Z 1

�1L�1½Rðn;pÞ � exp �ðx� nÞ2

4Dt

!dn

ð75ÞSubstituting the above equation into Eq. (68) gives,

Qðx; tÞ ¼Z 1


þZ 1


where,


2D� C2t

4D

� �

1

2ffiffiffiffiffiffiffiffipDtp exp �ðx� nÞ2

4Dt

!

¼ 1


4Dt

!ð77Þ

Appendix B

For semi-infinite channels with upstream boundary,the initial and boundary conditions are proposed asEqs. (32) and (31) respectively. By using the same trans-formation as Eq. (68), the initial and boundary condi-tions are transformed into,

uðx; tÞjt¼0 ¼ uðx; 0Þ ¼ exp � Cx2D


uðx; tÞjx¼0 ¼ uð0; tÞ ¼ expC2t4D

� �QuðtÞ ð79Þ

By using Laplace transform, Eq. (69) can be con-verted into,

o2 ~uðx; pÞox2

� pD

~uðx; pÞ ¼ Rðx; pÞD

; 0 6 x < þ1 ð80Þ

with boundary condition into,

~uð0; pÞ ¼ ~Qu p � C2

4D

� �ð81Þ


in which,

L½uðx; tÞ ¼ ~uðx; pÞ; L½f ðx; tÞ ¼ ~f ðx; pÞand L½QuðtÞ ¼ ~QuðpÞ ð82Þ

Rðx; pÞ ¼ � exp � Cx2D

� �~f x; p � C2

4D

� �� uðx; 0Þ ð83Þ

Solution of this second order ordinary differentialequation is,

~uðx; pÞ ¼ ~Qu p � C2

4D

� �exp �

ffiffiffiffipD

rx

� �

þffiffiffiffiDp

2ffiffiffipp

Z 1

0

Rðn; pÞD


rðnþ xÞ

� �dn

�ffiffiffiffiDp

2ffiffiffipp

Z x

0

Rðn; pÞD


rðx� nÞ

� �dn

�ffiffiffiffiDp

2ffiffiffipp

Z 1

x

Rðn; pÞD


rðn� xÞ

� �dn

ð84ÞUsing the inverse Laplace transformation gives,

uðx; tÞ ¼ QuðtÞ expC2t4D

� �� x

2ffiffiffiffiffiffiffiffiffiffipDt3p exp � x2

4Dt

� ��

þ 1


Z 1

0

L�1½Rðn;pÞ � exp �ðxþ nÞ2

4Dt

!dn

� 1


Z 1

0

L�1½Rðn;pÞ � exp �ðx� nÞ2

4Dt

!dn

ð85ÞSolution of the original equation is therefore ob-

tained by substituting the above equation into Eq. (68)as follows:

Qðx; tÞ ¼ QuðtÞ � Kuðx; tÞ þZ 1

0


þZ 1

0


where,

Kuðx; tÞ ¼ expCx2D� C2t

4D

� �x


t3=2exp � x2

4Dt

� �

¼ x


t3=2exp �ðx� CtÞ2

4Dt

!ð87Þ


2D� C2t

4D

� �1


exp �ðx� nÞ2

4Dt

!� exp �ðxþ nÞ2

4Dt

!" #

¼ 1

2ffiffiffiffiffiffiffiffipDtp exp � ðx� n� CtÞ2

4Dt

!"

� expCxD

� �exp �ðxþ nþ CtÞ2

4Dt

!#ð88Þ

Appendix C

For a semi-infinite channel with downstream bound-ary, the governing equation, initial and boundary condi-tions are defined as Eqs. (3), (42) and (41). Solution forEq. (80) in region (�1, 0] is,

~uðx; pÞ ¼ ~Qd p � C2

4D

� �exp

ffiffiffiffipD

rx

� �

þffiffiffiffiDp

2ffiffiffipp

Z 0

�1

Rðn; pÞD

exp

ffiffiffiffipD

rðxþ nÞ

� �dn

þffiffiffiffiDp

2ffiffiffipp

Z x

0

Rðn; pÞD

exp

ffiffiffiffipD

rðx� nÞ

� �dn

�ffiffiffiffiDp

2ffiffiffipp

Z x

�1

Rðn; pÞD


rðx� nÞ

� �dn

ð89Þ

By applying the inverse Laplace transformation wehave,

uðx; tÞ ¼ QdðtÞ expC2t4D

� �� x

2ffiffiffiffiffiffiffiffiffiffipDt3p exp � x2

4Dt

� ��

þ 1


Z 0

�1L�1½Rðn;pÞ � exp �ðxþ nÞ2

4Dt

!dn

� 1


Z 0

�1L�1½Rðn;pÞ � exp �ðx� nÞ2

4Dt

!dn

ð90Þ

Thus substituting the above equation into Eq. (68) givesthe solution as,

Qðx; tÞ ¼ QdðtÞ � Kdðx; tÞ þZ 0


þZ 0


where,

Kdðx; tÞ ¼ expCx2D� C2t

4D

� �jxj


t3=2exp � x2

4Dt

� �

¼ jxj2ffiffiffiffiffiffiffipDp

t3=2exp �ðjxj þ CtÞ2

4Dt

!ð92Þ

and K0(x,n, t) identical to Eq. (88).

Appendix D

For a finite channel the initial and boundary condi-tions are proposed as Eqs. (54) and (53). By taking thesame transformation as Eq. (68), the initial and bound-ary conditions are converted into,

uðx; tÞjt¼0 ¼ exp � Cx2D



uðx; tÞjx¼0 ¼ exp C2t4D

� QuðtÞ;

uðx; tÞjx¼L ¼ exp � CL2Dþ C2t

4D

� QdðtÞ

ð94Þ

The boundary conditions of this problem are stillinhomogeneous. Considering such transformation,

/ðx; tÞ ¼ uðx; tÞ � L� xL

expC2t4D

� �QuðtÞ

� xL

exp �CL2Dþ C2t

4D

� �QdðtÞ ð95Þ

we have a problem with homogeneous boundary condi-tions. The governing equation is accordingly as,

o/ot¼ D

o2/ox2þ F sðx; tÞ ð96Þ

in which,

F sðx; tÞ ¼ exp � Cx2Dþ C2t

4D

� �f ðx; tÞ

� C2

4DL� x

Lexp

C2t4D

� �QuðtÞ

� C2

4DxL

exp �CL2Dþ C2t

4D

� �QdðtÞ

� L� xL

expC2t4D

� �dQuðtÞ

dt

� xL

exp �CL2Dþ C2t

4D

� �dQdðtÞ

dtð97Þ

and the initial and boundary conditions are,

/ðx; tÞjt¼0 ¼ /ðx; 0Þ ð98Þ/ðx; tÞjx¼0 ¼ 0; /ðx; tÞjx¼L ¼ 0 ð99Þ

where,

/ðx; 0Þ ¼ exp � Cx2D

� �Q0ðxÞ �

L� xL

Quð0Þ

� xL

exp �CL2D

� �Qdð0Þ ð100Þ

Solution of this problem can be obtained by the sep-aration of variables method,

/ðx; tÞ ¼X

n

exp � npL

� 2

Dt� ��

� F nðtÞ þ /ndðtÞ��

sinnpxL

� ð101Þ

in which the Fourier coefficients are,

F nðtÞ ¼2

L

Z L

0

F sðx; tÞ sinnpxL

� dx ð102Þ

/n ¼2

L

Z L

0

/ðx; 0Þ sinnpxL

� dx ð103Þ

By substituting Eq. (101) into Eq. (95) we have,

uðx; tÞ ¼X

n

exp � npL

� 2

Dt� ��

� ½F nðtÞ þ /ndðtÞ

sinnpxL

�

þ L� xL

expC2t4D

� �QuðtÞ

þ xL

exp �CL2Dþ C2t

4D

� �QdðtÞ ð104Þ

Finally substituting the above equation to Eq. (68) gives,

Qðx; tÞ ¼ QuðtÞ � Kuðx; tÞ þ QdðtÞ � Kdðx; tÞ

þZ L

0


þZ L

0


in which,

Kuðx; tÞ ¼ expCx2D�C2t

4D

� �Xþ1n¼1

2npD

L2

exp �n2p2D

L2t

� �sin

npxL

ð106Þ

Kdðx; tÞ ¼ expCðx� LÞ

2D�C2t

4D

� �Xþ1n¼1

2npD

L2ð�1Þnþ1

exp �n2p2D

L2t

� �sin

npxL

ð107Þ

K0ðx;n; tÞ ¼ expCðx� nÞ

2D�C2t

4D

� �2

L

Xþ1n¼1

exp �n2p2D

L2t

� �

sinnpn

Lsin

npxL

ð108Þ

Appendix E

In Fig. 17, a channel with a tributary with a rectangu-lar cross section is illustrated, in which the width of mainstem and tributary are B and Bl, respectively. The dis-charge in the main stem and tributary are denoted by Q

and Ql, respectively. The confluent angle is h. L representsthe length of the confluent area. The cell with lateral in-flow is considered, length of which is dx. In a time intervaldt, the net volume of the water for this cell, namely, thedifference between the outflow at the right border of thecell and inflow at the left border, should be (oQ/ox)dxdt.The lateral inflow is (Ql/Bl)sinhdx dt. The increment ofthe water for the cell is (oh/ot)Bdx dt. Hence, in line withthe conservation law of the water, there satisfies,

ohot

Bdxdt þ oQox

dxdt ¼ Ql

Bl

sin hdxdt ð109Þ

Fig. 17. Sketch for a channel with a tributary.


As Q = qB, then the continuity equation can be read-ily obtained as following:

ohotþ oq

ox¼ ql

Bð110Þ

or,

oAotþ oQ

ox¼ ql ð111Þ

where Q denotes the discharge, A is the wetted surface,ql = (Ql/Bl)sinh. Similarly, the net momentum for thiscell in a time interval dt should be [o(Qq/h)/ox]dxdt.The velocity component of the lateral inflow in the flowdirection of the mainstream is ul = (Ql/Bl)/cosh/h. Thus,the momentum of the lateral inflow in a time interval dt

is (Ql/Bl)sinh(Ql/h/Bl)/coshdxdt. The impulse in theflow direction made by the water weight, the fric-tional resistance and water pressures are BghS0 dxdt,�BghSf dxdt, and [o(gh2/2)/ox]Bdxdt, respectively.The increment of the momentum for the cell is (oQ/ot)dxdt. Therefore, according to the conservation lawof the momentum, we have,

oQot

dxdt þ o

oxQ

qhþ B

gh2

2

� �dxdt

¼ BghðS0 � SfÞdxdt þ Ql

Bl

sin hQl

hBl

cos hdxdt ð112Þ

which can be simplified into,

oqotþ o

oxq2

hþ gh2

2

� �¼ ghðS0 � SfÞ þ

ql

Bul ð113Þ

The above equation is the momentum equation of con-servative form. As q = hv, the above equation can bealso rewritten into,

hovotþ v

ohotþ 2hv

ovoxþ v2 oh

oxþ gh

ohox

¼ ghðS0 � SfÞ þql

Bul ð114Þ

Likewise, the continuity equation can be transformedinto,

ohotþ v

ohoxþ h

ovox¼ ql

Bð115Þ

Therefore we can obtain the momentum equation of thenon-conservative form as below,

ovotþ v

ovoxþ g

ohox¼ gðS0 � SfÞ þ

qlðul � vÞBh

ð116Þ

which is identical to that described by Singh [28].Let v0, h0 denote the typical velocity and flow depth,

and we can acquire the non-dimensional variables asfollowing:

�x ¼ xh0

; �t ¼ v0th0

; �v ¼ vv0

; �h ¼ hh0

;

�q ¼ qv0h0

; �ql ¼ql

v0h0

; �ul ¼ul

v0

ð117Þ

Then Eqs. (110), (113) and (116) can be transform to

o�ho�tþ oð�h�vÞ

o�x¼ �ql

Bð118Þ

v20

gh0

o�qo�tþ o

o�x�q2

�h

� �� ql

B�ul

� �þ �h

o�ho�x¼ �hðS0 � SfÞ ð119Þ

v20

gh0

o�vo�tþ v

o�vo�x� �qlð�ul � �vÞ

B�h

� �þ o�h

o�x¼ ðS0 � SfÞ ð120Þ

With assumption of sub-critical flow condition, Fr2 =(v0)2/(gh0)� 1, the momentum equation can be simpli-fied as,

ohox¼ ðS0 � SfÞ ð121Þ

in which the terms of inertia are negligible. In addition,the relation of Chezy is known as,

Sf ¼Q2

K2ð122Þ

where the conveyance of the channel is taken as,

KðhÞ ¼ CR1=2h A ð123Þ

in which C is the Chezy coefficient, and Rh is the hydrau-lic radius. The momentum equation can also be writtenas [16],

ohox� S0 þ

Q2

K2¼ 0 ð124Þ

and the continuity equation can be rewritten to

ohotþ 1

BoQox¼ ql

Bð125Þ

Differentiating the above equation with respect to x,yields,

o2h

oxotþ 1

Bo

2Qox2¼ 1

Boql

oxð126Þ

Differentiating the equation of motion with respect to t,and assuming oS0/ot = 0, yields,

o2hoxotþ 2Q

K2

oQot� 2Q2

K3

oKot¼ 0 ð127Þ


According to the continuity equation, we have,

oKot¼ dK

dhohot¼ 1

BdKdh

ql �oQox

� �ð128Þ

Eliminating the term o2h/oxot from Eqs. (126) and (127)renders a single equation,

oQotþ Q

BKdKdh

oQox� K2

2BQo2Qox2¼ Q

BKdKdh

ql �K2

2BQoql

ox

ð129ÞThis is the equation of diffusive wave, with only onedependent variable, Q. The celerity and the diffusioncoefficient are respectively,

C ¼ QBK

dKdh¼ 1

BdQdh

ð130Þ

D ¼ K2

2BQ¼ Q

2BSf

ð131Þ

Therefore, the diffusive wave equation can now be writ-ten as,

oQotþ C

oQox� D

o2Q

ox2¼ Cql � D

oql

oxð132Þ

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