Propagation of solitary waves through signi cantly …authors.library.caltech.edu/355/1/SHIjfm98.pdfJ. Fluid Mech. (1998), vol. 362, pp. 157{176. Printed in the United Kingdom c 1998

J. Fluid Mech. (1998), vol. 362, pp. 157–176. Printed in the United Kingdom

c© 1998 Cambridge University Press

157

Propagation of solitary waves throughsignificantly curved shallow water channels

By A I M I N S H I1, M I C H E L L E H. T E N G2

AND T H E O D O R E Y. W U3

1 Environmental Fluid Dynamics Research, Jamesburg, NJ 08831, USA2 Department of Civil Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USA

3 Engineering Science, California Institute of Technology, Pasadena, CA 91125, USA

(Received 20 September 1996 and in revised form 18 November 1997)

Propagation of solitary waves in curved shallow water channels of constant depthand width is investigated by carrying out numerical simulations based on the gener-alized weakly nonlinear and weakly dispersive Boussinesq model. The objective is toinvestigate the effects of channel width and bending sharpness on the transmissionand reflection of long waves propagating through significantly curved channels. Ournumerical results show that, when travelling through narrow channel bends includingboth smooth and sharp-cornered 90◦-bends, a solitary wave is transmitted almostcompletely with little reflection and scattering. For wide channel bends, we find that,if the bend is rounded and smooth, a solitary wave is still fully transmitted with littlebackward reflection, but the transmitted wave will no longer preserve the shape ofthe original solitary wave but will disintegrate into several smaller waves. For solitarywaves travelling through wide sharp-cornered 90◦-bends, wave reflection is seen tobe very significant, and the wider the channel bend, the stronger the reflected waveamplitude. Our numerical results for waves in sharp-cornered 90◦-bends revealed asimilarity relationship which indicates that the ratios of the transmitted and reflectedwave amplitude, excess mass and energy to the original wave amplitude, mass andenergy all depend on one single dimensionless parameter, namely the ratio of thechannel width b to the effective wavelength λe. Quantitative results for predictingwave transmission and reflection based on b/λe are presented.

1. IntroductionSince Scott Russell first discovered in 1834 that a solitary wave propagates without

changing its shape and speed in a straight channel of uniform depth and width, theproperty of solitary waves in straight channels has been extensively studied by manyauthors. These studies include the discovery by Zabusky & Kruskal (1965) that twocolliding solitary waves pass through each other without losing their identity (forwhich feature the term ‘soliton’ was coined), the analytical proof by Benjamin (1972)on the robust stability of solitary waves, and the theoretical, numerical, and experi-mental investigations of transmission and reflection of solitary wave propagating instraight channels of variable depth and width by Peregrine (1967), Madsen & Mei(1969), Johnson (1973), Shuto (1974), Goring (1978), Miles (1979), Chang, Melville& Miles (1979), Wu (1981), Schember (1982), Kirby & Vengayil (1988), Teng & Wu(1992, 1994), among others.

158 A. Shi, M. H. Teng and T. Y. Wu

In nature and in engineering applications, rivers, harbours, and canals often havewinding turns in direction, therefore it is important to understand how water wavespropagate through curved channels. Thus far, only a few studies have been focusedparticularly on investigating the evolution of solitary waves propagating in curvedchannels, yet there have been useful results obtained from several related studies.

Rostafinski (1972) obtained analytical solutions based on the linear wave equationsto describe the propagation of extremely long acoustic waves in smoothly curvedrectangular ducts whose width was at least two orders of magnitude smaller than thewavelength. His theoretical results showed that long waves in a narrow curved line (inthe limit that the duct width approaches zero) propagate as if the line was straight,while in relatively wider channels, the propagation of long waves was found to beprofoundly influenced by the sharpness of the channel bend. He discovered that thedistribution of the tangential fluid velocity in the transverse direction inside a narrowcircular-arc bend is the same as that of a potential vortex. He also found that thephase velocity would increase proportionally to the sharpness of the bend. To studyshort periodic surface waves propagating in wide (with channel width about five timesthe wavelength) smoothly curved water channels, Kirby, Dalrymple & Kaku (1994)applied the parabolic approximation based on the linear wave equations, and theirresults revealed that through wide channel bends, both wave reflection and diffractionare important. In fact, the reflection of short waves was so strong that growth of aMach stem at the outer wall of the channel bend was observed. Another interestingstudy was carried out by Webb & Pond (1986) who investigated theoretically (bysolving the linear wave equations through separation of variables and expressingthe solution with harmonic basis functions) the propagation of linear periodic longwaves under the influence of the Coriolis force through smoothly curved channelbends. Several parameters, including the ratio of channel width to wavelength (whichranged from 0.03 to 0.5 in Webb & Pond’s study), bending angle, bending sharpness,and Coriolis force, were found to affect the transmission and reflection of longwaves through smoothly curved channel bends. Their analytical solution indicatedthat increasing the bend sharpness would give rise to a higher reflection coefficient.However, as the authors noted, the analytical method failed in very sharp bends suchas sharp-cornered right-angled bends.

To model nonlinear waves, Katopodes & Wu (1987) developed a finite elementscheme to solve the weakly nonlinear and weakly dispersive Boussinesq equationsfor simulation of undular bores and solitary waves propagating through channelcontractions and expansions. Their study was focused on examining the accuracy andefficiency of the finite element scheme and the proper boundary conditions withoutdiscussing in detail the transmission and reflection of solitary waves through channelbends. Recently, Shi & Teng (1996) carried out numerical simulations based on Wu’s(1981) generalized Boussinesq (gB) equations to study the evolution of nonlinearsolitary wave through narrow curved water channels. They found that a solitarywave is almost completely transmitted with little reflection or scattering even whenpropagating through significantly curved narrow channels including those with 180◦-bends. This result was seen to be consistent with the result found by Rostafinski(1972) for long acoustic waves in smoothly curved narrow ducts. In Shi & Teng’s(1996) study, all the channels investigated had the ratio of channel width over waterdepth equal to one.

The present study is an extension of previous ones to investigate the propagationof solitary waves, which are shallow water long waves with well-balanced nonlinearand dispersive effects, through significantly curved rectangular channels of uniform

Propagation of solitary waves through curved water channels 159

depth and width, with particular emphasis on examining wave propagation throughsharp-cornered right-angled bends. Both wide and narrow right-angled bends areinvestigated by varying the channel width systematically. In addition, the propagationof solitary waves through wide smoothly curved channel bends is simulated, and theresults are compared with those in sharp-cornered right-angled bends. The objectiveis to examine the effects of channel width, bending sharpness and wavelength ontransmission and reflection of solitary waves through significantly curved channels.The present results are compared with the results obtained from the previous studieson long acoustic waves and periodic water waves in smoothly curved ducts or channelsbased on linear theory.

The numerical simulation in the present study is based on Wu’s (1981) generalizedweakly nonlinear and weakly dispersive Boussinesq equations by using the same finitedifference scheme developed by Wang, Wu & Yates (1988, 1992). In our numericalsimulation, the flow is assumed inviscid and incompressible, and the effects of surfacetension and Coriolis force are excluded.

2. Governing equationsThe generalized Boussinesq (gB) equations derived by Wu (1981, (41)–(42)) are a

set of weakly nonlinear and weakly dispersive wave equations that can be used todescribe the generation and propagation of shallow water long waves over variabletopographies. This set of equations is generalized in the sense that it can modelgeneral wave problems where water depth may vary both in space and in time, andwaves may travel in different directions simultaneously. In addition, compared withthe traditional Boussinesq equations involving surface elevation and two horizontalvelocity components (Katopodes & Wu 1987), Wu’s gB equations are written interms of surface elevation and velocity potential to reduce the number of unknownsand have been shown to be more efficient numerically. For the particular problemconsidered in the present study, namely the evolution of solitary wave propagating incurved channels of constant depth and width, the gB equations can be written for acurvilinear coordinate system as

ζt +1

L2[(1 + ζ)φx]x + [(1 + ζ)φy]y +

k

L(1 + ζ)φy = 0, (1)

φt +1

2L2φ2x +

1

2φ2y + ζ − 1

3L2φxxt −

1

3φyyt −

1

3

k

Lφyt = 0, (2)

where ζ is the free-surface elevation, φ the depth-averaged velocity potential, and x isthe boundary-fitted coordinate tangent to the inner channel wall and y is normal to xas shown in figure 1. All the variables are in non-dimensional form, with length scaledby water depth h, time by (h/g)1/2 and velocity by (gh)1/2, where g is the accelerationdue to gravity. The typical channel bend curvature is k = 1/R, where R is the localradius of the inner channel wall, and the scale factor L is given by L = 1 + ky.The subscripts in the equations indicate the specific differentiations. In the gB model,equation (1) is the continuity equation, and equation (2) is derived from the integralof the Euler equation by using perturbation expansion to the second order.

The proper boundary conditions applied in the present study are the unperturbedcondition at the far upstream and far downstream boundaries, and no-penetrationcondition at the channel walls, namely

ζ = 0, φx = 0, and φy = 0 at x = ±∞; φy = 0 at y = 0, b, (3)

where b is the channel width.


yyyx

b

R

Figure 1. Coordinate system.

It is known that a solitary wave travels in a straight channel of uniform depth andwidth without changing its shape and speed. This stationary solitary wave solutionto the Boussinesq equations is given by (Teng 1997)

ζ(x, t) =α sech2β(x− x0 − ct)

1 + α tanh2β(x− x0 − ct), (4)

c =

{6(1 + α)2

α2(3 + 2α)[(1 + α)ln(1 + α)− α]

}1/2

(5)

where β = [3α/4(1 + 0.68α)]1/2 and x0 is the initial solitary wave position. Theexpression (5) for wave speed c is directly adopted from the original exact solutionin closed form of the gB equations first solved in Teng & Wu (1992, pp. 227–229)(see also Yates 1995), while the wave profile (4) is an empirical formula (Yates 1995;Teng 1997) that gives one of the best approximations to the exact solitary waveprofile based on the Boussinesq equations (Teng & Wu 1992) whose closed form isnot known. This stationary solitary wave solution will be used to test the accuracyof our numerical scheme, and it will also be used as the initial condition in all ournumerical simulations presented in this paper.

To examine the mass and energy conservation laws, we define the excess mass Me

of the wave system as

Me =

∫ +∞

−∞

∫ b

0

ζL dy dx, (6)

and the total mechanical energy as

E =

∫ +∞

−∞

∫ b

0

1

2

[1

L2φ2x + φ2

y + ζ2

]L dy dx. (7)

Multiplying (1) with L and integrating the resulting equation over x and y, we have

dMe

dt+

∫ b

0

∫ +∞

−∞

1

L[(1 + ζ)φx]x dxdy +

∫ +∞

−∞

∫ b

0

[L(1 + ζ)φy]ydydx = 0

which gives

dMe

dt= −

∫ b

0

1

L(1 + ζ)φx

∣∣∣+∞−∞

dy −∫ +∞

−∞L(1 + ζ)φy

∣∣∣b0dx.

By applying the boundary conditions (3), we readily show that dMe/dt = 0, whichindicates that excess mass is conserved exactly based on the gB model. For an inviscid


wave system without external forcing, the total mechanical energy must also beconserved. Combining the results from (1) and (2), we can show that conservationof mechanical energy is also satisfied by the gB model accurately to the higherorder consistent with the order of accuracy of the initial perturbation expansion inderiving the gB model from the Euler equations. This proof involves straightforwardbut lengthy algebraic manipulations which are omitted from our present paper forbrevity. (Related derivations and discussions on energy conservation based on asection-mean Boussinesq long-wave model of the same class as the gB model can befound in Teng 1990.) Conservation of mass and mechanical energy is used as oneof the criteria in our computation to monitor the accuracy and correctness of ournumerical simulation.

For nonlinear long waves propagating through curved channels, there are severalparameters that may affect wave transmission and reflection. These parameters includewave amplitude α, wavelength λ, channel width b, and bending sharpness a which wasdefined by Rostafinski (1972, 1976) as a = 1 + kb. For solitary waves, we define theeffective wavelength λe as the wavelength within which the wave elevation everywhereis larger than 1% of its amplitude α. From (4), this λe is given by

λe =2

βln

(1 + 0.01α)1/2 + 0.991/2

(0.01 + 0.01α)1/2, (8)

which predicts that λe=22.2, 13.2, and 10.6 for solitary waves of amplitude α = 0.1, 0.3,and 0.5, respectively. In our numerical simulation, the effects of the aforementionedparameters are investigated.

3. Numerical schemeTo solve the gB equations (1)–(2) numerically, we adopt a predictor–corrector finite

difference scheme developed by Wang et al. (1988, 1992). Specifically, Euler forward isused for time derivatives and central difference is applied for spatial differentiations.We notice that since the gB equations have mixed spatial and temporal derivativeterms in both horizontal directions, it can be quite complicated to solve this set ofequations by using direct matrix inversion. In the present study, we adopt an iterativescheme which is relatively simple and numerically efficient as demonstrated in Wanget al. (1988, 1992). The detailed numerical scheme is given as follows:

predictor: ζ∗i,j = ζni,j + ∆tEni,j ,

φ∗i,j = F∗i,j + ∆t

(1 +

2

3L2j (∆x)2

+2

3(∆y)2

)−1

Gni,j ,

(9)

corrector: ζn+1i,j = ζni,j + 1

2∆t(E∗i,j + En

i,j),

φn+1i,j = Fn+1

i,j + 12∆t

(1 +

2

3L2j (∆x)2

+2

3(∆y)2

)−1

(G∗i,j + Gni,j),

(10)

where

Emi,j = − 1

L2j (∆x)2

(1 + ζmi,j)(φmi+1,j − 2φmi,j + φmi−1,j)−

1

4L2j (∆x)2

(ζmi+1,j

−ζmi−1,j)(φmi+1,j − φmi−1,j)−

1

(∆y)2(1 + ζmi,j)(φ

mi,j+1 − 2φmi,j + φmi,j−1)

− 1

4(∆y)2(ζmi,j+1 − ζmi,j−1)(φ

mi,j+1 − φmi,j−1)−

k

2∆yLj(1 + ζmi,j)(φ

mi,j+1 − φmi,j−1),


Fmi,j = φni,j +

(1 +

2

3L2j (∆x)2

+2

3(∆y)2

)−1 [1

3L2j (∆x)2

(φmi+1,j + φmi−1,j

−φni+1,j − φni−1,j) +1

3(∆y)2(φmi,j+1 + φmi,j−1 − φni,j+1 − φni,j−1)

+k

6Lj∆y(φmi,j+1 − φmi,j−1 − φni,j+1 + φni,j−1)

],

Gmi,j = − 1

8L2j (∆x)2

(φmi+1,j − φmi−1,j)2 − 1

8(∆y)2(φmi,j+1 − φmi,j−1)

2 − ζmi,j .

Here ∆x, ∆y and ∆t represent spatial and temporal increments, i, j are indicesdenoting the spatial grid points in the x- and y-directions, respectively, and n, ∗, andn+ 1 represent different time levels of computation. The dummy variable m can be nor ∗ for E and G, and ∗ or n+ 1 for F .

We note that in (9) and (10), the values of φ at new time levels appear on both theleft-hand side and the right-hand side of the equations. This indicates that (9) and (10)need to be solved iteratively. In our computation, at each step the values of φ at theprevious step are used as the initial guess for the new step, and (9) and (10) are iterateduntil the wave field is converged, which is judged by max|ζ(new step)−ζ(old step)| < δin our computations. Here δ is a small positive number that is usually chosen to be nolarger than 0.1% of the |ζ| value itself. The accuracy and efficiency of the numericalscheme used in the present study will be further discussed in the next section.

4. Numerical resultsBefore simulating the propagation of solitary waves in curved channels, we first

tested the numerical scheme on a solitary wave travelling in a straight channel ofuniform depth and width. The initial condition was given by (4) with amplitudeα = 0.5. After the solitary wave had travelled for 50 non-dimensional time units orabout 60 water depths, our numerical results show that the amplitude of the solitarywave increased by only 2.1%, and the changes in excess mass and mechanical energywere 0.0% and 2.1%, respectively. The error in wave speed was found to be 0.05%.These results thus validate the reasonably high accuracy of the numerical scheme.

4.1. Solitary waves propagating through sharp-cornered 90◦-bends

In our numerical simulations, all the water channels investigated here have a singlesharp-cornered 90◦-bend and a total length of 200 water depths along the innerchannel wall, with the upstream and downstream legs of the channel each being 100water depths in length. The origin of the x-coordinate is set at the far upstreamend of the channel. In each case, the channel width and water depth are uniformthrough out the channel, but the dimensionless channel width (i.e. the ratio of channelwidth to water depth) is varied in order to investigate the effect of channel width ontransmission and reflection of solitary waves through sharp-cornered 90◦-bends. Forall the cases in this section, the initial position of the solitary waves is at x0 = 75.

The numerical results of a solitary wave of initial amplitude α = 0.3 propagatingthrough a narrow channel with width b = 1 and with a sharp-cornered 90◦-bendare shown in figure 2(a–c). From the results, we can see that the solitary wave isalmost completely transmitted with only a small amount of reflection and scattering.It is seen that the transmitted wave is purely one-dimensional with a uniform wavecrest across the channel. A detailed comparison between the transmitted and the


0.4

1.00

140

120

100

80

601.0

(a)

t = 4.4

0.4

1.00

140

120

100

80

601.0

t =21.2

0.4

1.00

140

120

100

80

601.0

t =22.1

0.4

1.00

140

120

100

80

601.0

t =22.9

0.4

1.00

140

120

100

80

601.0

t =24.3

0.4

1.00

140

120

100

80

601.0

t = 44.1

ζ

ζ

ζ

ζ

ζ

ζ

x

y

(b)1.0

096 98

102

104

0 1.0

0.002

0.02

0.10

0.20

0.30

0.20

0.10

0.02

100

t = 22.9

0.4

0.3

0.2

0.1

0

–0.1110 120 130 140

Initial wave profile at t = 0

Transmittedat t = 44.1

x

Wav

e el

evat

ion,

ζ

(c)

Figure 2. Numerical results of a solitary wave of initial amplitude α = 0.3 and x0 = 75.0 propagatingthrough a narrow sharp-cornered 90◦-bend with b = 1; (a) wave elevation at different time instants;(b) contour lines of the wave field at time t = 22.9; (c) comparison between the transmitted and theoriginal wave profiles along the channel centreline.

initial wave profiles (figure 2c) further confirms the nearly complete transmissionand preservation of the original solitary wave shape. Our numerical results showthat, in this case, 99.9% of the total excess mass and 98.2% of the total energy aretransmitted.

The numerical results for a solitary wave of α = 0.3 travelling through two widerchannels with b = 5 and 10 are shown in figures 3 and 4. We observe that theamplitude of the leading reflected wave becomes much stronger in the wider channels.


0

140

(a)

0.4

0.5

0140

120

100

80

60

5.0

t = 4.4

0.4120

100

80

60

5.0

t = 21.2

0.4

5.0

0

140

120

100

80

60

5.0

t = 26.5

0.4

5.0

0

140

120

100

80

60

5.0

t =29.1

0.4

5.0

0140

120

100

80

60

5.0

t = 48.5

ζ

ζ

ζ

ζ

ζ

xy

(b)

5.0

090 95

105

1100 5.0

0.002 0.04

–0.0

4 0.00

0.10

0.02

0.002

t = 26.5

0.4

0.3

0.2

0.1

0

–0.1110 120 130 140



x

Wav

e el

evat

ion,

ζ

(c)

0.5

0.10

0.200.300.40

0.50

100

Figure 3. As figure 2 but with b = 5.

In our study, we have also simulated solitary waves of initial amplitude α = 0.3propagating in channels of width b = 3 and 7, and the detailed numerical resultsshowing the variation of the transmitted and reflected wave amplitudes (calculatedat the channel centerline) with channel width b are presented in figure 5(a, b). In thisfigure, the leading transmitted and reflected waves are shifted in x in order to plot allwave peaks at x′ = 0 of the shifted x′ coordinate for the convenience of comparisonbetween waves in different channels. These results show clearly that the reflected (ortransmitted) wave amplitude increases (or decreases) as channel width increases.

To further investigate the effect of channel width on wave transmission and re-flection and to analyse the dominant dimensionless parameters that govern the phe-


nomenon of long waves propagating through sharp-cornered 90◦-bends, we havesimulated the propagation of solitary waves of different initial amplitudes α = 0.1and 0.5 through both narrow and wide channel bends. Our results show that thefundamental wave features with α = 0.1 and 0.5 are very similar to those in theprevious cases with α = 0.3. Most importantly, we found that when the ratios αT/αand αR/α, where αT , αR and α represent the amplitude of the leading transmitted,reflected and the initial waves along the channel centreline, are plotted (in figure 6)against the dimensionless parameter b/λe, the results from all the cases with differentinitial wave amplitudes fall along the same curve, indicating that the ratio of channelwidth b to wavelength λe is the key similarity parameter in long-wave transmissionand reflection through sharp-cornered 90◦-bends.

Based on the numerical results presented in figure 6, empirical formulas for pre-dicting the leading transmitted and reflected wave amplitudes through sharp-corneredright-angled bends with 0 < b/λe < 1 are obtained as follows:

αT

α=

{1, 0 < b/λe < 0.20.28(b/λe)

−0.72, 0.2 6 b/λe < 1.0,(11)

αR

α=

{1.19(b/λe)

0.9, 0 < b/λe < 0.4;0.58(b/λe)

0.14, 0.4 6 b/λe < 1.0,(12)

where the averaged relative difference between the numerical results and the empiricalapproximation based on the four formulas are found to be 3%, 4%, 12% and 1%,respectively.

For solitary waves propagating through wide sharp-cornered channel bends, we alsoobserve that the leading reflected wave appears to be very much like a solitary waveexcept that there is an oscillatory wave region immediately following the wave tail.Owing to the negative wave elevation which tends to cancel the excess mass possessedby the leading positive wave, the net reflected excess mass is found to be smallerthan what the leading wave amplitude may suggest. For example, in the case of thesolitary wave with α = 0.3 travelling through the wide channel bend with b = 10, eventhough the reflected wave amplitude is seen to be much stronger than the transmittedwave amplitude, the transmitted excess mass MeT is calculated to be 97.2% of theoriginal total excess mass, indicating that only 2.8% of the original mass is actuallybeing reflected. In the same case, 52.1% of the initial energy is transmitted while47.9% is reflected. In our calculations, the reflection region is considered to includethe upstream leg of the channel and the square region inside the bend, while thetransmission region covers the downstream leg of the channel starting from x = 100.The final transmitted and reflected mass and energy are calculated at a time instantwhen the leading transmitted wave has travelled sufficiently far downstream of thechannel bend. After that time, the transmitted and reflected mass and energy showedno significant change in time. The complex features of the reflected waves travellingin the opposite direction in comparison with the transmitted waves is the result ofcombined wave actions that include wave reflection due to the change in travellingdirection and to the (transverse) expansion and contraction of the channel widthat the bend, and due to wave diffraction around the sharp corner. Since oscillatorywave trains can have large energy but possess little net excess mass, wave energyseems to be a better measure of the intensity of the reflected waves in this case.Figure 7(a, b) shows the detailed numerical results of the ratio of transmitted andreflected excess mass MeT , MeR and energy ET , ER to the original excess mass Me

and energy E, respectively, of all the cases studied here. Again, we find that the


0

140

(a)

0.4

10.0

120

100

80

60

10.0

t = 4.4

ζ

0

140

0.4

10.0

120

100

80

60

10.0

t = 27.3

ζ

0

140

0.4

10.0

120

100

80

60

10.0

t = 30.9

ζ

0

140

0.4

10.0

120

100

80

60

10.0

t = 35.3

ζ

0

140

0.4

10.0

120

100

80

60

10.0

t = 39.7

ζ

0

140

0.4

10.0

120

100

80

60

10.0

t = 44.1

ζ

0

140

0.4

10.0

120

100

80

60

10.0

t = 57.3

ζ

yx

Figure 4 (a). For caption see facing page.

relative transmitted and reflected mass and energy for all amplitudes depend on onedimensionless parameter, namely the ratio of channel width b to wavelength λe.

4.2. Solitary waves propagating through smoothly curved 90◦-bends

For solitary waves travelling through smoothly curved channels, the first case weinvestigated is a solitary wave of initial amplitude α = 0.3 at position x0 = 85


(b)

5.0

090

110

0 10

0.0

–0.0

20.02

0.001

t = 30.9

0.4

0.3

0.2

0.1

0

–0.1120 130 140



x

Wav

e el

evat

ion,

ζ

(c)

0.1

0.20

–0.04

0.0–0.02

100

0.10.2

0.30.4

0.5

Figure 4. As figure 2 but with b = 10.

propagating through a smoothly curved narrow (b = 1) 90◦-bend with constantradius R = 10 and bending sharpness a = 1.1. The numerical results for this case arepresented in figure 8(a–c) showing the evolution of the solitary wave at different timeinstants through the 90◦-bend. These results show that during its passage through asmooth bending narrow channel, a solitary wave exhibits the conspicuous featuresthat its crest turns radially straight and tilts higher outward against the outer wallin keeping balance with the centrifugal force. After exiting from the channel bend,the solitary wave quickly regains its original shape and speed, leaving no trace ofreflected waves behind.

The results for solitary waves of initial amplitude α = 0.3 and position x0 = 85propagating through wider smooth channel bends with b = 5, R = 10, a = 1.5 andb = 10, R = 10, a = 2 are shown in figures 9 and 10. These results show that the crestof the solitary wave in a wide channel bend is no longer radially straight, and thetransmitted wave no longer preserves the original shape of a solitary wave. Instead,the transmitted wave is seen to have disintegrated into several smaller waves whoseelevation varies across the channel. However, even though the transmitted wave nolonger preserves its original shape, it is interesting to note that, in all cases withsmooth channel bends studied here, the waves are transmitted almost completely,with no significant wave reflection detected. This feature is different from that forwaves propagating through sharp-cornered wide channel bends.

To examine the flow variations in the transverse direction, plots of wave elevationζ and longitudinal fluid velocity u across the channel inside the channel bends for


0.4

0.2

0

–0.210 0 –10 –20

Unperturbedwater surfaceb =10

7531

Reflected

(b)

Ele

vati

on, ζ

x′

0.4

0.2

0

–0.210 0 –10 –20

Initial solitonat t = 0

b =1357

10

(a)E

leva

tion

, ζ

Transmitted

Figure 5. Numerical results of transmitted and reflected wave amplitudes with channel width bfor a solitary wave of initial amplitude α = 0.3 travelling through sharp-cornered 90◦-bends: (a)transmitted waves; (b) reflected waves.

all three cases discussed above and an additional case with α = 0.3, b = 1 andR = 2 (which gives a = 1.5) are shown in figure 11(a, b). The wave elevation ζ andvelocity u shown here are taken around midway inside the channel bends where theflow variations in the transverse direction appear to be the most significant. Fora solitary wave travelling through the two smooth narrow bends with b = 1, thewave-induced fluid velocity u is found to decrease inversely with the local radiusr = R+y, the same as the velocity distribution of an inviscid vortex confined betweentwo concentric cylindrical walls. This result is consistent with the result reported byRostafinski (1972) for long acoustic waves propagating in smoothly curved narrowducts. In contrast, for waves travelling through wide channel bends with b = 5 and10, we found that the tangential fluid velocity is no longer proportional to the inverseof the local radius; instead, it increases from the inner wall to the outer wall, showinga trend similar to the typical velocity distribution in a wide river bend.

Another issue of interest is the effect of channel bends on wave speed. To calculatethe averaged wave speed throughout the passage of a solitary wave in a curvedchannel, we define the averaged wave speed as the total longitudinal distance travelledalong the channel centreline divided by the corresponding travel time. For all casespresented in §4.1 and §4.2, we calculated the average wave speed from the initial waveposition, which is about one wavelength upstream of the channel bend, to wherethe tail of the leading transmitted wave has travelled out of the bend. To examine


1.0

0.5

0.1

0.05

(b)

b/λe

0.05

0.1 0.5 1.0

αRα

1.0

0.5

0.1

0.05

(a)

0.05

0.1 0.5 1.0

αTα α = 0.1

α = 0.3α = 0.5

Figure 6. Plot of the ratio of (a) transmitted and (b) reflected wave amplitude αT and αR to theinitial amplitude α vs. the ratio of channel width b to wavelength λe with logarithmic axes.

100

0.2

(b)

b/λe

20

0.4 0.6 0.8

ET

/E(%

) 80

60

40

0 1.0

100

0.2

(a)20

0.4 0.6 0.8

MeT

/Me(

%) 80

60

40

0 1.0

α = 0.1α = 0.3α = 0.5

Figure 7. Numerical results of transmitted (a) excess mass MeT and (b) mechanical energy ET fora solitary wave of initial amplitude α travelling through sharp-cornered 90◦-bends, where Me andE are initial excess mass and mechanical energy, and b and λe are channel width and effectivewavelength, respectively.


(a)

0.4ζ

t = 8.8

1.00

140

120100

800

1.0

0.4ζ

t = 20.3

1.00

140

120100

800

1.0

0.4ζ

t = 23.7

1.00

140

120100

800

1.0

0.4ζ

t = 30.9

1.00

140

120100

800

1.0

0.4ζ

t = 48.5

1.00

140

120100

800

1.0

0.4

0.3

0.2

0.1

0

–0.1130 140 150

x



Wav

e el

evat

ion,

ζ (c)(b)1

095 100

115.8

1200 1

0.001

0.02

0.10.2

0.3

0.20.1

0.020.00.0

0.0

0.0

0.0

yx

t = 23.7

Figure 8. Numerical results of a solitary wave with initial amplitude α = 0.3 and x0 = 85.0 travellingthrough a narrow smoothly curved 90◦-bend with b =1, R = 10, and a = 1.1: (a) wave elevation atdifferent time instants; (b) contour lines of the wave field at time t = 23.7; (c) comparison betweenthe transmitted and the original wave profiles along the channel centreline.

the effect of channel bend on wave speed, the numerical results for wave speedin curved channels are compared with the speed calculated by using (5) for wavestravelling in a straight channel, with α in (5) taken as the average of the initial waveamplitude and the amplitude of the leading transmitted wave. Detailed results forwave speed in both the smoothly curved channels and channels with sharp-corneredbends are presented in tables 1 and 2. In these tables, x0, α, x, αT , α, c(5), c, and diff.%represent, respectively, the initial wave position, the initial wave amplitude, the finalwave position, the amplitude of the leading transmitted wave, the averaged amplitude


(a)

0.4ζ

t = 8.8

5

0

140

120100

800

5

0.4ζ

t = 20.3

5

0140

120100

800

5

0.4ζ

t = 23.7

5

0

140

120100

800

5

0.4ζ

t =27.2

5

0

140

120100

800

5

0.4ζ

t = 48.5

5

0

140

120100

800

5

0.4

0.3

0.2

0.1

0

–0.1130 140 150

x

Initial wave profile at t = 0Transmittedat t = 48.5

Wav

e el

evat

ion,

ζ (c)(b)

5

095 100

115.8

1200 5

0.0001

0.02

0.10.2

0.3

0.2

0.1

0.02

0.0

–0.06

yx

120

0.001

t =23.7

Figure 9. As figure 8 but with b = 5, and a = 1.5.

0.5 (α+ αT ), the wave speed in straight channels based on α calculated from (5), theactual wave speed in curved channels, and the relative difference between c(5) and c.

Our results show that the averaged wave speed for a solitary wave travellingthrough a smoothly curved channel of constant depth is almost the same as that ina straight channel given by (5). Comparing the cases with narrow and wide channelbends with sharpness a ranging from 1.1 to 2, we find that the bend sharpness a haslittle effect on the averaged wave speed in smoothly curved channels. For solitarywaves travelling through sharp-cornered 90◦-bends, the wave speed is seen to beslightly faster than that in a straight channel based on the same wave amplitudeand same travel length. The difference is quite small, except in the case of the wide


(a)

0.4ζ

t = 8.8

50

140

120100

800

5

0.4

ζ

t = 20.3

5

0

140

120100

800

5

0.4ζ

t = 23.7

120100

800

5

0.4ζ

t = 30.9

140

120100

800

5

0.4ζ

t = 66.2

10

0

140

120100

8005

0.4

0.3

0.2

0.1

0

–0.1140 150

x

Initial wave profile at t = 0Transmittedat t = 66.2

Wav

e el

evat

ion,

ζ (c)

(b)

10

095 100

115.8

1200 5

0.0001

0.02

0.1

0.2

0.3

0.4

0.1

0.020.0

–0.02

yx

0.001t =23.7

5160

10

10

10

10

1010

510

140

05

10

05

10

160

0.0

Figure 10. As figure 8 but with b = 10, and a = 2.0.

channel with b = 10. However, in this case, the amplitude of the transmitted wavehas decreased so much compared with the initial wave amplitude that it may not bemeaningful to compare the wave speed with the one calculated by using (5) based onthe averaged wave amplitude.

In our computations, the typical values we used for ∆x, ∆y and ∆t are 0.1, 0.1and 0.05, respectively, and the convergence criterion δ is set to δ = 0.001|ζ|. We havealso tested the conditionally stable scheme by using ∆x = ∆y = 0.2, ∆t = 0.05 and∆x = ∆y = 0.05, ∆t = 0.05 and obtained the same results. (The minimum iterationnumber is different for different ∆x, ∆y and ∆t.) The conservation of excess mass andmechanical energy was monitored in all our computations, and the maximum errorsin mass and energy conservation in all our simulations of waves propagating through


0 0.2

0.2

0.4F

luid

vel

ocit

y, u

0.4 0.6 0.8 1.0

y/b

b =1, R = 2

b =1, R =10

b =10, R =10

b =5, R = 10

(b)

0 0.2

0.2

0.4

Wav

e el

evat

ion,

ζ0.4 0.6 0.8 1.0

b =1, R = 10

b =1, R = 2

b =10, R =10

b =5, R = 10

(a)

Figure 11. Numerical results showing the variations of (a) wave elevation ζ and (b) longitudinalfluid velocity u across the channel width inside the smoothly curved 90◦-bends about halfwayaround the bend. The amplitude of the initial solitary wave is α = 0.3.

Width b Sharpness a x0 α x αT α c(5) c Diff.%

1 1.1 85 0.3 139 0.299 0.3 1.134 1.133 0.11 1.5 85 0.3 139 0.299 0.3 1.134 1.131 0.35 1.5 85 0.3 136 0.276 0.288 1.129 1.122 0.610 2.0 85 0.3 149 0.196 0.248 1.113 1.093 1.8

Table 1. Wave speed c in smoothly curved channels

Width b x0 α x αT α c(5) c Diff.%

1 75 0.3 125 0.294 0.297 1.133 1.154 1.95 75 0.3 125 0.181 0.240 1.109 1.142 3.010 75 0.3 133 0.102 0.201 1.093 1.183 8.2

Table 2. Wave speed c in channels with sharp-cornered 90◦-bends

both smooth and sharp-cornered bends were 1.2% for Me and 4.1% for E. A typicalcase in our numerical simulation uses about 46 minutes of CPU time on a Cray c90.

5. DiscussionOur present numerical results revealed very similar behaviours of solitary waves

travelling through narrow curved channels and long acoustic waves in narrow bendingducts studied by Rostafinski (1972). The similarity lies in that a long wave is transmit-ted almost completely in a narrow curved channel with little backward reflection, andthe tangential fluid velocity inside a smooth circular-arc bend is proportional to theinverse of the local radius in the transverse direction, same as the velocity distributionof a potential vortex. Rostafinski’s (1972) analytical solution based on the linear waveequations predicted that wave speed would increase as bending sharpness increases.Even though this effect is not apparent in our present results for solitary wavesin smoothly curved channels, our results indicated that the wave speed is slightly


faster through a sharp-cornered 90◦-bend than that in a straight channel. The consis-tency between our numerical results for long water waves in narrow curved channelsand Rostafinski’s analytical solution for long acoustic waves in narrow curved ductsfurther validates the correctness of the numerical scheme and our simulation results.

In the present study, new results are obtained for solitary waves travelling in widechannels. As we noted, the wave behaviour through narrow smoothly curved andsharp-cornered channel bends is very similar: in other words, for narrow curved chan-nels, the bending sharpness is not an important parameter. However, the transmissionand reflection of solitary waves through wide smoothly curved and sharp-corneredbends are seen to be very different. We observe that when travelling through widesharp-cornered 90◦-bends, wave reflection is very significant, whereas through a widesmoothly curved channel bend, the initial wave is fully transmitted with no backwardreflection. This result can be useful in predicting and controlling wave transmissionand reflection in hydraulic engineering applications.

The maximum channel width we examined in the present study is ten times thewater depth. Waves in much wider channels are not simulated because the gB modelis a depth-averaged shallow water model and may not be applicable to modellingwaves in extremely wide channel bends where secondary flow may occur.

Although there is no backward reflection of a solitary wave travelling through awide smooth channel bend, there seems to be a noticeable lateral reflection from theouter channel wall inside the bend which causes the initial wave to disintegrate intoseveral smaller waves whose elevation varies across the channel. This reflection fromthe outer wall in wide channels is qualitatively consistent with the effect of channelwalls on short waves as reported by Kirby et al. (1994). However, since long andshort waves have different characteristics, their behaviour in curved channel bendsmay not be compared quantitatively.

It will be interesting and valuable to conduct laboratory experiments to fur-ther investigate the phenomenon of solitary waves propagating through curved andbranching channels. A sharp-cornered right-angled channel is currently being set upin the hydraulics lab at the University of Hawaii. Quantitative measurements ofwave transmission and reflection are being planned. We are particularly interested inexamining the similarity relationships (11), (12) between the relative transmitted orreflected wave amplitude and the ratio of channel width to wavelength, which arerevealed by our present numerical results.

Another problem of interest is the propagation of solitary waves in branchingchannels. In Shi & Teng (1996), solitary waves travelling through a T-shaped narrowbranching channel were studied. Extending the study to include wide branchingchannels with general branching angles is currently being pursued.

6. ConclusionThe generalized nonlinear and dispersive Boussinesq model (Wu 1981) was com-

puted numerically to simulate the propagation of solitary waves in channels withsmoothly curved and sharp-cornered 90◦-bends. Our numerical results show thatwhen travelling through narrow channel bends, including both smoothly curved andsharp-cornered bends, a solitary wave is transmitted almost completely with littlereflection and scattering. The longitudinal fluid velocity inside a smoothly curvedcircular-arc narrow bend is found to vary inversely with the local radius across thechannel, the same as the velocity distribution of a potential vortex and also consistent


with the results for long acoustic waves in smoothly curved narrow ducts (Rostafinski1972).

For solitary waves travelling through wide smoothly curved channels, our resultsshow that the initial wave is fully transmitted without backward reflection. However,in this case, the transmitted wave no longer preserves the shape of the original solitarywave but disintegrates into several smaller waves due to lateral reflection from theouter channel wall. In addition, the longitudinal fluid velocity inside a wide smoothbend is no longer proportional to the inverse of the local radius; instead, it increasesfrom the inner wall to the outer wall. We also found that, the speed of solitarywaves propagating through smoothly curved channels is the same as that in a straightchannel.

For a solitary wave travelling through a wide channel with a sharp-cornered 90◦-bend, the backward reflection is seen to be very significant, and the wider the channelbend, the stronger the reflected wave amplitude. A similarity relationship is revealedby our numerical results: the relative transmitted and reflected wave amplitude, excessmass and energy, scaled by the original amplitude, mass and energy, all depend onone single dimensionless parameter, namely the ratio of the channel width b tothe effective wavelength λe. This result not only helped to reveal the key similarityparameter (i.e. b/λe) that governs the transmission and reflection of solitary wavesthrough sharp-cornered bends, but also helped us to obtain simple power laws forpredicting the transmitted and reflected wave amplitudes based on b/λe. Regardingthe wave speed, solitary waves are observed to travel slightly faster in a channel witha sharp-cornered bend than in a straight channel.

The authors wish to thank Professor Allen Chwang at the Hong Kong University,Professor Philip Liu at Cornell University and Professor Harry Yeh at the Universityof Washington for providing us with their valuable comments. The authors appreciategreatly the helpful discussions with Professors C. C. Mei, R. Grimshaw, C.-S. Yih, S. J.Lee, and S. H. Kwon at the 2nd International Conference on Hydrodynamics in HongKong. We are also grateful to the reviewers for their helpful comments, especially forthose on the effect of wavelength and on the important results by Rostafinski (1972,1976) and by Kirby et al. (1994). Computations in the present study were performedon Cray c90 at the San Diego Supercomputer Center (SDSC) which is supported bythe US National Science Foundation.

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