J. Fluid Mech. (2014), . 739, pp. doi:10.1017/jfm.2013.590 ... · 2 S. L. Wade, B. J. Binder, T. W. Mattner & J. P. Denier corresponds to the onset of a superharmonic (short-wavelength)

J. Fluid Mech. (2014), vol. 739, pp. 1–21. c© Cambridge University Press 2013 1doi:10.1017/jfm.2013.590

On the free-surface flow of very steep forcedsolitary waves

Stephen L. Wade1,†, Benjamin J. Binder1, Trent W. Mattner1

and James P. Denier2

1School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia2Department of Engineering Science, The University of Auckland, Auckland 1142, New Zealand

(Received 29 January 2013; revised 20 August 2013; accepted 1 November 2013)

The free-surface flow of very steep forced and unforced solitary waves is considered.The forcing is due to a distribution of pressure on the free surface. Four types offorced solution are identified which all approach the Stokes-limiting configurationof an included angle of 120◦ and a stagnation point at the wave crests. For eachtype of forced solution the almost-highest wave does not contain the most energy,nor is it the fastest, similar to what has been observed previously in the unforcedcase. Nonlinear solutions are obtained by deriving and solving numerically a boundaryintegral equation. A weakly nonlinear approximation to the flow problem helps withthe identification and classification of the forced types of solution, and their stability.

Key words: channel flow, solitary waves, surface gravity waves

1. IntroductionLarge-amplitude (periodic and solitary) gravity waves have been studied since the

19th century. It was conjectured by Stokes that the limiting case of the tallest wavewould have a corner flow with an included angle of 120◦ (now referred to as theStokes corner flow) (Stokes 1847, 1880; McLeod 1997). This structure was confirmedshortly after by calculations in the early 20th century of the tallest wave by Havelock(1919). In later studies, motivated partly by an application to the study of breakingwaves, accurate nonlinear solutions to the ‘almost-highest’ waves, those just shorterthan the tallest wave, were found (Longuet-Higgins & Fenton 1974; Byatt-Smith &Longuet-Higgins 1976; Longuet-Higgins & Fox 1977; Williams 1981; Schwartz &Fenton 1982; Hunter & Vanden-Broeck 1983; Vanden-Broeck 1986; Longuet-Higgins& Fox 1996; Maklakov 2002). In addition, an understanding of the instabilities ofthese almost-highest waves has been established (Tanaka 1986; Longuet-Higgins &Tanaka 1997).

In the case of steady unforced flow, there are some highly nonlinear phenomenaobserved as the waves approach the tallest configuration. Many properties of the wave,for example its total energy, pass through local maxima and minima as the waveheight increases (Longuet-Higgins & Fox 1996; Maklakov 2002). This phenomena hasbeen interpreted in terms of a non-uniqueness of solutions with respect to certainflow parameters. Tanaka (1986) performed a stability analysis of the almost-highestsolutions, and showed that the first local maxima of the total energy of the wave

† Email address for correspondence: [email protected]

mailto:[email protected]

2 S. L. Wade, B. J. Binder, T. W. Mattner & J. P. Denier

corresponds to the onset of a superharmonic (short-wavelength) instability. Longuet-Higgins & Cleaver (1994) demonstrated that this instability is caused by the flow nearthe crest.

These results for the unforced large-amplitude waves can naturally be extended toconsider forced waves in which a localized forcing is prescribed in the flow (forexample Dias & Vanden-Broeck 1989). For a finite-depth fluid with solid bottomtopography, the localized forcing can be due to either a bump in the otherwisehorizontally flat topography (Vanden-Broeck 1987; Dias & Vanden-Broeck 1989), ora pressure distribution on the free surface (Binder & Vanden-Broeck 2007). In thispaper we examine the effect on the almost-highest wave of the latter of these twotypes of disturbances. Our work draws some parallels with earlier work on unforcedlarge-amplitude waves as well as extending those related to topographic forcing to thecase of surface forcing, such as may occur in marine propulsion applications.

For the case of steady forced flow, Dias & Vanden-Broeck (1989) calculatedboth the limiting configurations and almost-highest waves for flow past a localizedtopographical disturbance. In figure 6 of their paper, although they did not commenton this, it appears there is non-uniqueness of solutions as the wave approaches thehighest wave courtesy of turning points in the wave height as a function of a non-dimensional Froude number, which we define in the following paragraph. We explorethis issue here, in the context of a pressure disturbance, demonstrating non-uniquenessin the Froude number–wave amplitude space.

In this study a frame of reference moving with the pressure disturbance is chosen,and we will focus on the flows that are steady in this reference frame. It is assumedthat the finite-depth fluid is inviscid and incompressible and the flow is irrotational.The large-amplitude solitary wave propagates under the influence of gravity g, andthe effect of surface tension on the free surface is neglected. Both far upstream anddownstream the flow approaches a uniform stream with constant speed U and depthH. The flow can then be conveniently characterized by the dimensionless depth-basedFroude number

F = U

(gH)1/2. (1.1)

For locally forced solitary waves the flow is assumed to be supercritical with F > 1,and only symmetric free-surface profiles about the central position of the pressuredisturbance are considered. A sketch of the flow problem is illustrated in figure 1, andwe take two different approaches to study the effect the localized forcing on steepsolitary waves. The first approach is numerical, requiring the solution of the fullynonlinear (i.e. finite-amplitude) governing equations. The second approach exploits aweakly nonlinear analysis, allowing considerable analytic progress to be made. Thelatter also provides the stepping off point from which the fully nonlinear calculationscan be started.

Our numerical approach is based upon a boundary integral equation technique todetermine the free-surface elevation; Levi-Civita (1925), Lim & Smith (1978) andothers established the fundamental ideas upon which our the derivation is based. Weuse a concentrated clustering of mesh points to compute the rapid change in slope ofthe free surface near the crest of a steep wave, similar to the method used by Hunter &Vanden-Broeck (1983). It is well known that the highest unforced solitary wave has a

On the free-surface flow of very steep forced solitary waves 3

H

U

0

FIGURE 1. A schematic of the steady flow problem. The free surface is denoted byy∗ = H + η∗ and the pressure disturbance p∗ is travelling with the crest of the wave.

corresponding value of Froude number (in agreement up to four decimal places)

Fu = 1.2909, (1.2)

as calculated by Williams (1981), Hunter & Vanden-Broeck (1983), Longuet-Higgins& Fox (1996) and others. We compare this value of the Froude number with ourresults for unforced flow to validate the computational method.

Our second approach is an analytical study of a weakly nonlinear model of theproblem. Akylas (1984) was the first to derive the (unsteady) forced Korteweg–deVries (fKdV) equation (Korteweg & de Vries 1895) for resonant forcing given by apressure disturbance, which has also been used in many later studies (e.g. Miles 1986;Shen, Shen & Sun 1989; Shen 1993; Binder, Dias & Vanden-Broeck 2005; Chardardet al. 2011 and others). The same equation can be derived for the waves due to aflow perturbed by a topographical disturbance (Cole 1985; Grimshaw & Smyth 1986).There are two regimes where this asymptotic theory may not be applicable, the firstbeing when the waves are steep and highly nonlinear, the second regime being a bandof F around unity for which the flow is considered resonant and steady solutionswill not exist (Grimshaw & Smyth 1986). Nevertheless, an analysis of the weaklynonlinear phase portraits in the phase plane of the problem provides a systematic andintuitive way to classify the various nonlinear waves that can exist within the flow. Ourclassification of the solution types is similar to that presented by Shen (1993), whoexamined analytic solutions of the forced Korteweg–de Vries (fKdV) equation.

Using the weakly nonlinear theory, we identify and classify five forced solutions fora given value of the Froude number 1 < F < Fu. When the amplitude of the pressuredisturbance A is positive and less than some critical value depending on F, as shownby Miles (1986) and others, there are two flow types, which we denote as type I andII, these correspond to perturbations of a uniform stream (I) and a solitary wave (II).Vanden-Broeck (1987) and later Dias & Vanden-Broeck (1989) used this classificationfor the nonlinear solutions to flow over a localized bump in the channel bottomtopography, but without reference to a weakly nonlinear analysis of the problem.

When the amplitude of forcing A is negative, there are three flow types, which wewill denote by III, IV and V, which are classified as perturbations of a single solitarywave (III), two solitary waves (IV) and a uniform stream (V). The existence of wavetypes III and IV depends on A being greater than some critical value depending onF much like the positively forced solutions (Miles 1986). In contrast, solution type Vis found for all A < 0 and F > 1. Nonlinear solutions for these three flow types havebeen shown by Binder, Vanden-Broeck & Dias (2008), who considered flow over alocalized dip in the bottom of a channel. Following the work of Grimshaw, Zhang &


Chow (2007) and Chardard et al. (2011) we find that only weakly nonlinear solutionsfor flow types I and V are (numerically) stable but that their strongly nonlinearcounterparts II–IV are unstable.

Here we use a continuation method to compute the five types of forced solutions,as identified by the weakly nonlinear theory, using the nonlinear boundary integralequation technique to be described in § 2.1. We found that the flow types II, III andIV all appear to approach the Stokes-limiting configuration with stagnation points atthe wave crests. This was also found for an additional qualitatively different solutiontype not predicted by the weakly nonlinear theory. The height of the almost highestwave, and the corresponding value of the Froude number, in the forced solutions is notnecessarily the same as occurs in the unforced case where F = Fu. At wave heightsclose to the limiting height a rich solution structure is observed, similar to that foundin the unforced case (see Byatt-Smith & Longuet-Higgins 1976; Longuet-Higgins &Fox 1977, 1996; Hunter & Vanden-Broeck 1983).

The details on the formulation of the problem are given in the next section. Wepresent our results and discuss their significance in subsequent sections.

2. FormulationThe steady two-dimensional irrotational flow of an incompressible inviscid fluid (of

uniform constant density ρ) is considered. The flow domain is bounded above bya free surface y∗ = H + η∗ and below by the channel bottom y∗ = 0. A Cartesiancoordinate system (x∗, y∗) is chosen with the y∗-axis intersecting the central positionof a symmetric distribution of pressure p∗ on the free surface (see figure 1). Gravityacts with force g in the negative y∗ direction. Only symmetric (about the y-axis)supercritical flows are considered here, and both far upstream and downstream the flowis taken to be uniform with

u∗→ U, v∗→ 0, η∗→ 0, as x∗→±∞. (2.1)

Here u∗ and v∗ are the horizontal and vertical components of the velocity.Non-dimensional quantities

(x, y, η, p, u, v)= (x∗/H, y∗/H, η∗/H, p∗/(ρgH), u∗/U, v∗/U) (2.2)

are defined by taking H as the reference length and U as the reference velocity. Interms of these dimensionless quantities, the dynamic boundary condition on the freesurface is given by

12(u2 + v2)+ 1

F2(y+ p)= 1

2+ 1

F2on y= 1+ η, (2.3)

and the condition of no normal flow on the channel bottom implies that

v = 0 on y= 0. (2.4)

Following Binder & Vanden-Broeck (2007), the equation for the pressuredistribution on the free-surface is taken to be

p= AB√π

exp[−(Bx)2], (2.5)

where A and B are both constants. It can be shown that

p→ Aδ(x) as B→∞, (2.6)


where δ is Dirac delta function and A is the amplitude of forcing. This relationshipwill prove convenient when comparing our fully nonlinear solutions with predictionsfrom weakly nonlinear theory. In the nonlinear computations a finite value of B needsto be prescribed. Equation (2.5) then represents a symmetric localized forcing on thefree surface, centred on x= 0. For the results presented in this paper we chose B= 2.8,but found qualitatively similar results for other values of B.

2.1. Boundary-integral formulationTo determine the shape of the free surface we must solve Laplace’s equations for thevelocity potential φ subject to the boundary conditions (2.1), (2.3) and (2.4). We dothis by exploiting a boundary-integral formulation. To do this we first introduce thecomplex potential f = φ + iψ and complex velocity w = df /dz = u − iv, where φ isthe velocity potential and ψ the associated streamfunction for the flow. Without loss ofgenerality, we set ψ = 1 on the free surface, and it follows that ψ = 0 on the bottomof the channel.

Following Vanden-Broeck (1997), Lustri, McCue & Binder (2012) and others (seealso the references therein), the transformation

ζ = α + iβ = eπf (2.7)

is chosen to map the flow domain in the f plane to the upper half of the ζ plane.Applying the Cauchy integral formula to (u − 1) − iv in the ζ plane with the contourof integration consisting of the α-axis and a semicircle of arbitrary large radius in theupper half-plane gives

u(α, 0)= 1− 1π−∫ ∞−∞

v(α, 0)α − α dα, −∞< α <∞, (2.8)

where −∫

denotes the Cauchy principal value. Note that there is no contribution fromthe integral over the large semicircle as the non-dimensional form of the far-fieldcondition (2.1) implies that (u − 1) − iv→ 0 as |ζ | →∞. Using the no penetrationcondition (2.4) on the channel bottom streamline (α > 0 with v(α, 0) = 0), and thesubstitutions u(φ)= u(−eπφ, 0) and v(φ)= v(−eπφ, 0), equation (2.8) becomes

u(φ)= 1+−∫ ∞−∞

v(φ)

eπφ − eπφeπφ dφ, −∞< φ <∞. (2.9)

By exploiting the symmetry in the problem, namely that,

v(−φ)=−v(φ) and u(φ)= u(−φ) (2.10)

the integral (2.9) can be expressed as

u(φ)= 1+ 12−∫ ∞

0G(φ, φ) dφ, −∞< φ <∞, (2.11)

where

G(φ, φ)= v(φ)

eπ(φ+φ) − 1+ v(φ)

1− eπ(φ−φ)+ v(φ)

eπ(φ−φ) − 1+ v(φ)

1− e−π(φ+φ), (2.12)

and we have dispensed with the tilde notation. From a computational standpoint wemust necessarily truncate the semi-infinite domain appearing in the integral (2.11). To


facilitate this we rewrite (2.11) as

u(φ)= 1+ 12−∫ φm

0G(φ, φ) dφ + 1

2

∫ ∞φm

G(φ, φ) dφ, −φm < φ < φm, (2.13)

with φm � 1. The solution for the velocity in the far-field and the second integralin (2.13) are approximated using Stokes’s result (see, for example, Byatt-Smith &Longuet-Higgins 1976; Hunter & Vanden-Broeck 1983) on the asymptotic behaviourof w as |φ| →∞. This yields

u(φ)≈ 1− D cos(λ) exp(−λ|φ|) as |φ| →∞, (2.14)

and

v(φ)≈±D sin(λ) exp(−λ|φ|) as φ→∓∞. (2.15)

Here D is an unknown constant and λ is the smallest positive solution to

tan λ= F2λ. (2.16)

The second integral of (2.13) can be evaluated in terms of hypergeometric functions(denoted as 2F1), using (2.15) and formula (3.194) of Gradshtein, Ryzhik & Jeffrey(2000). This, together with the removal of the singularity via the substitutionv(φ)= v(φ)− v(φ)+ v(φ) for the first integral in (2.13), gives

u(φ)≈ 1+ 12

∫ φm

0coth(π(φ + φ)/2)(v(φ)+ v(φ)) dφ

+ 12

∫ φm

0coth(π(φ − φ)/2)(v(φ)− v(φ)) dφ + v(φ)

πln

∣∣∣∣sinh(π(φ − φm)/2)sinh(π(φ + φm)/2)

∣∣∣∣− D sin λ

2

{e−π(φm+φ)e−λφm

π+ λ 2F1(1, 1+ λ/π; 2+ λ/π; e−π(φm+φ))

+ e−π(φm−φ)e−λφm

π+ λ 2F1(1, 1+ λ/π; 2+ λ/π; e−π(φm−φ))

+ e−λφm

λ2F1(1, λ/π; 1+ λ/π; e−π(φm−φ))

+ e−λφm

λ2F1(1, λ/π; 1+ λ/π; e−π(φm+φ))

}, −φm < φ < φm. (2.17)

Finally, integrating the identity

dz

df= 1

u− iv(2.18)

provides the following parametric representation for the shape of the free surface,

x(φ)= x(φm)+∫ φ

φm

u(φ)

u2(φ)+ v2(φ)dφ, −φm < φ < φm (2.19)

and

y(φ)= y(φm)+∫ φ

φm

v(φ)

u2(φ)+ v2(φ)dφ, −φm < φ < φm. (2.20)


2.2. Numerical methodAs the flows considered are symmetric, only half of the finite domain needs to bediscretized (0 < φ < φm) to obtain a numerical solution. An irregular set of meshpoints in the φ domain is used in the computations, as it is anticipated that thesolutions will have a rapidly increasing change in the slope of the free surface near thecrest of a wave.

Hunter & Vanden-Broeck (1983) introduced a new variable ξ with φ ∼ ξ 3 for theunforced solitary wave. This ensures a clustering of mesh points near the wave crestif the grid is regular in ξ . We extended this idea to a piecewise cubic relationshipbetween ξ and φ, as we required a clustering of mesh points at more than onewave crest (for the multiple crest waves we identify here). The piecewise cubicpolynomials were constructed by dividing the truncated φ domain into M subintervalswith endpoints φ[0], φ[1], . . . , φ[M]. The polynomial in each subinterval ξ ∈ [j, j+ 1] forj= 0, . . . ,M − 1 is given by a linear combination of four basis functions,

φ(ξ)= φ[j]a[j](ξ)+ φ[j+1]b[j](ξ)+ s[j]c[j](ξ)+ s[j+1]d[j](ξ) (2.21)

where

a[j](ξ)= 2(ξ − (j+ 1))2(ξ − (j− 1/2)), (2.22a)

b[j](ξ)=−2(ξ − j)2(ξ − (j+ 3/2)), (2.22b)

c[j](ξ)= (ξ − (j+ 1))2(ξ − j), (2.22c)

d[j](ξ)= (ξ − j)2(ξ − (j+ 1)). (2.22d)

With this choice of basis functions φ(ξ) = φ[j] at ξ = j and s[j] is exactly the slope ofthe tangent of φ(ξ) at the point ξ = j, for all j. Note that φ[0] = 0 and φ[M] = φm.

In our computations there are only two cases to consider in terms of the numberof subintervals M. The first case is when there is one wave crest and one subintervalwith M = 1. If we were to set s[0] = 0 and s[1] = 3φm, then φ = φmξ

3 for ξ ∈ [0, 1],thus recovering the relation of Hunter & Vanden-Broeck (1983). The second case iswhen there are two wave crests and two subintervals, so that M = 2. The location φ[1]

of the downstream crest is unknown and must be determined as part of the solution.Typically, the values of s[0] and s[1] are taken to be small positive numbers to ensure aclustering of mesh points near the crest and trough, with the value of s[2] taken to bemuch larger.

The regular grid of N mesh points for ξ is taken to be

ξi = (i− 1)M/(N − 1) for i= 1, 2, . . . ,N (2.23)

with the corresponding irregular grid in φ given by

φi = φ(ξ = ξi), (2.24)

with midpoints

φi+1/2 = φ(ξ = (ξi + ξi+1)/2) for i= 1, 2, . . . ,N − 1. (2.25)

A numerical solution can now be computed by discretizing and solving the integro-differential equation defined by (2.3) and (2.5), (2.17) and (2.19) and (2.20) on thefinite domain, which is matched with the far-field solution (2.14) and (2.15).

On the finite domain, the dynamic boundary (2.3) is satisfied at the midpoints (2.25)by approximating the integral (2.17) using the trapezoidal rule and the parametricrelations (2.19) and (2.20) by the midpoint rule. This yields N − 2 algebraic equations,


because the last midpoint is used to match the finite domain solution with the far-fieldsolution (2.14) and (2.15), thus giving another equation. The last equation is then(2.16), and this gives a total of N + 1 equations.

We explore the space of permissible solutions by using the predictor–correctormethod of Allgower & Georg (1990) to find one-parameter families of solutions (or‘branches’) in an automated fashion, with the parameter being similar to the arclengthof the branch in the space of the unknowns. The predictor–corrector method requiresan underdetermined system which has one more unknown than equations; this differsfrom the usual approach of using Newton’s method on an exactly determined system.The uniform stream solution for a fixed Froude number F > 1 is the starting point ofour calculations. Given that solutions with one crest are found in the neighbourhoodof the uniform stream, there are initially N + 2 unknowns A, D, λ and v(φi) fori= 2, . . . ,N (as v(φ1 = 0)= 0, due to symmetry), with N + 1 equations as above. Foreach solution computed we search for any i such that v(φi+1)v(φi) < 0 with v(φi) > 0,which corresponds to the two crest solutions being detected. When this occurs, themost recent solution is interpolated onto a new grid with an extra unknown φ[1] andan extra equation v(φ[1])= 0, and the continuation algorithm then resumes on the newsystem. Numerical solutions were checked to be independent of the choice of N andφm (the results presented here are converged to within graphical accuracy). TypicallyN > 1000 and φm > 10 in our calculations.

3. ResultsBefore turning to a discussion of the results for the fully nonlinear problem we

briefly describe a weakly nonlinear phase plane analysis of the fKdV model of thisflow, which provides a systematic way to identify and classify the nonlinear solutions.Further detailed discussion of the fKdV model and its solutions may be found inBaines (1995), Ee et al. (2010, 2011).

Following Akylas (1984), Cole (1985) and Grimshaw & Smyth (1986), the fKdVequation takes the form

6ηt + ηxxx + 9ηηx − 6(F − 1)ηx =−3px, (3.1)

when written in terms of the variables used in the nonlinear computations. Equation(3.1) is strictly only valid for small disturbances and when the Froude number F isclose to unity. We explore the solutions of this equation as they provide a usefulclassification scheme for our fully nonlinear boundary-integral calculations. Assumingthat the forcing is given by (2.6) (in the limit B→∞), i.e. p(x) = Aδ(x), the steady,integrated form of (3.1) is

ηxx + 92η

2 − 6(F − 1)η = 0 for x 6= 0, (3.2)

with

ηx(0+)− ηx(0−)=−3A. (3.3)

For a given value of the Froude number F > 1, there are two fixed points located at(0, 0) and (4/3(F − 1), 0) in the (η, ηx) phase plane. The fixed point at the origin is asaddle node whilst the other fixed point is a centre. Equation (3.3) introduces verticaljump in the phase plane, with magnitude equal to three times the amplitude of forcingA. When A > 0 the vertical jump is downwards and when A < 0 the vertical jump isupwards. The jump condition (3.3) provides a way of jumping between fixed points,periodic orbits, unbounded trajectories and the homoclinic orbit in the phase plane.


Integrating (3.2) yields

η2x = 6(F − 1)η2 − 3η3 + C for x 6= 0, (3.4)

for the trajectories in the phase plane, where C is a constant of integration. The valueof C = 0 corresponds to trajectories that intersect with the origin in the phase plane.

For the flow problem that we are interested in the solution’s journey through thephase plane must begin and end with the saddle point as a consequence of the factthat the supercritical flow with F > 1 approaches a uniform stream with unit depthas x→±∞. If the (non-trivial) solution is unforced (with A = 0) there is only oneway of achieving this, namely by traversing the homoclinic (or solitary wave) orbit ina clockwise direction in the phase plane. This case is shown in figure 2(b) with thecorresponding surface displacement shown in figure 2(a). However, for given values of|A|> 0 and F > 1, there is more than one type of solution (see figure 2d,f ).

When the amplitude of forcing A > 0 there are two types of solutions, as foundby Grimshaw & Smyth (1986) and Miles (1986), here denoted as types I and II,which are illustrated in the phase-plane diagram of figure 2(d). The journey in thephase plane for these two types of solution starts with the saddle point, moving in aclockwise direction along the homoclinic orbit in the upper half of the phase plane.For the same value of A> 0 there are two opportunities to jump vertically downwardsonto the homoclinic orbit in the lower half of the phase plane. The journey through thephase plane for both solution types I and II then continues along the homoclinic orbit(in the lower half plane) back to the saddle point. The weakly nonlinear solution type I(see figure 2d) has a vertical jump to the left of the centre, and as A→ 0 the solutionapproaches a uniform stream. On the other hand, the weakly nonlinear solution type IIhas a vertical jump to the right of the centre, and as A→ 0 the solution approachesthe unforced solitary wave solution. Hence, the solution types I and II are classified asperturbations of a uniform stream and (single) solitary wave, respectively.

The weakly nonlinear analysis in the phase plane of the problem predicts threetypes of solution III, IV and V when A < 0 (see figure 2f ). Solution types III andIV are found as solutions to narrow disturbances in Miles (1986), amongst others,while type V was found as a solution to the fKdV equation for a narrow disturbanceby Malomed (1988). Starting with the saddle point, the solution types III and IVboth traverse the homoclinic orbit from the upper half to the lower half of the phaseplane, before jumping vertically upwards back onto the homoclinic orbit in the upperhalf of the phase plane. The journey in the phase plane for these two solution typesthen continues around the homoclinic orbit ending with the saddle point. The typeIII solution exhibits a vertical jump to the right of the centre, and as A→ 0 thesolution approaches the unforced solitary wave solution. The type IV solution has avertical jump to the left of the centre, and as A→ 0 the solution approaches twounforced solitary waves that are infinitely far apart. The type III and IV solutionsare therefore classified as perturbations of a single solitary wave and two solitarywaves, respectively. The last type V solution, which exhibits a jump between twounbounded phase plane trajectories that intersect with the saddle point, is classified asa perturbation of a uniform stream.

For F = 1.26 and the values of A given in the caption, the fully nonlinear solutionsplotted in figure 2(a), (c) and (e) can be compared with the weakly nonlinear phaseplane diagrams of figure 2(b), (d) and (f ), respectively. The qualitative agreementbetween the weakly and fully nonlinear calculations is excellent. This is furtherillustrated in figure 3(a), where we plot the free-surface height, at x = 0, obtainedfrom the nonlinear (solid curve) and weakly nonlinear (broken curve) calculations,


II

I

III

I II

IV

V

V IV III

y

y

y

x

1.0

1.3

1.6

–3 0 3

–3 0 3

1.0

1.3

1.6

–3 0 3

1.0

1.3

1.6

0

0

0

0 0.35

0 0.35

0 0.35

(c)

(e)

(a)

(d )

( f )

(b)

FIGURE 2. Unforced and forced solutions for a value of the Froude number F = 1.26.Shown are: (a) the nonlinear free-surface profile for an unforced solitary wave with A = 0and η(0) = 0.63; (b) the weakly nonlinear phase portrait for (a) with η(0) = 0.52; (c)the nonlinear profiles for A = 0.09, where the flow type II is a forced solitary wave withη(0) = 0.62 and the flow type I is a perturbation of a uniform stream with η(0) = 0.093; (d)the weakly nonlinear phase portrait for (a) with η(0) = 0.12 and η(0) = 0.50, for the flowtypes I and II, respectively; (e) the nonlinear profiles for A = −0.09, where the flow typesIII and IV are forced solitary waves with η(0) = 0.55 and η(0) = 0.091, respectively, and theflow type V is a perturbation of a uniform stream with η(0) = −0.063; and (f ) the weaklynonlinear phase portraits for (c) with η(0) = 0.50, η(0) = 0.12 and η(0) = −0.099, for theflow types III, IV and V, respectively.

as a function of the forcing amplitude A. For 1 < F < 1.26 the qualitative natureof the solution space in the (A, η(0)) plane was found to be the same, with the‘loop’ contracting towards the origin as F→ 1, with a tail in the lower half planeas A→ −∞. The weakly nonlinear results, whilst still providing good qualitative


Unforced solitary wave

V

IV

IIIII

I

II

III IIIIUnforced solitary wave

IV

(a) (b)

(c) (d)

0

0.5

1.0

–0.2 0.20

0

0.5

1.0

–0.2 0.20

0

0.5

1.0

0

0

0.5

1.0

0–0.2 0.2

A–0.2 0.2

A

FIGURE 3. Plots of the variation of η(0) versus forcing amplitude for the four values ofFroude number F = {1.26, 1.262, 1.2909, 1.32}. The solid curves are derived from the fullynonlinear calculations. In (a) the dotted curved indicates values derived from the weaklynonlinear analysis. The dashed curves in (b)–(d) are the locus of the ‘near-limiting’ values,with u = 0.05 at the wave crests. The markers in (a) indicate the location in parameterspace of the unforced solution and the five types of forced solution shown in figure 2(b)–(d),whereas in (b) they identify the forced almost-highest solitary waves, in (c) the upper andlower markers denote the unforced and forced almost-highest solitary waves, and finally in (d)the upper and middle markers denote the forced almost-highest solitary waves.

agreement with the fully nonlinear calculations, begin to diverge from the fullynonlinear results as F increases away from 1. For F > 1.26 the curve for the nonlinearvalues η(0) becomes disconnected (see figure 3b–d), which (unsurprisingly) is notpredicted by the weakly nonlinear analysis. The broken curves in figure 3(b–d) arenot related to the weakly nonlinear fKdV analysis, but are instead for ‘near limiting’values of wave amplitude, where the flow is approaching stagnation points at the wavecrests.

Before continuing with the discussion of the plots shown in figure 3(b–d), weturn our attention to the general layout of the results shown in figures 4, 5, 6, 7and 8. The results presented in figures 4, 5, 6, 7 and 8 are for the type II, IIIand IV fully nonlinear forced and unforced types of solution (classified accordingto their corresponding weakly nonlinear counterparts) and a new fully nonlinearsolution (figure 7). These results were computed using solutions found in figure 3as starting points for the same continuation algorithm as described at the end of § 2,except now the amplitude of forcing is used as the continuation parameter whilst theFroude number is determined as part of the solution. All types of solution whichare perturbations to solitary waves approach the Stokes-limiting configuration of an


1.0

1.2

1.4

1.6

–5 50

0

0–5 5

1.262

1.263

0.94 0.96 0.98 1.00 0.92 0.96 1.00

0.58

0.59

y

F E

(a) (b)

(c) (d )

x x

FIGURE 4. Forced solitary waves of type II, with amplitude of forcing A = 0.07.(a) Nonlinear profile for a value of the Froude number F = 1.262. The solid and dashedcurves are for values of ω = 0.999 and ω = 0.933, respectively. (b) Plot of θ versus x for (a).(c,d) Plots of the Froude number F and wave energy E versus ω, respectively.

included angle of 120◦ and a stagnation point at their crests. The layout of the panels(a–d) in figures 4, 5, 6, 7 and 8 is described next, with reference to the results foundin the unforced case (figure 5), and then we will discuss the relationship betweenfigure 3(b–d) and figures figure 4–8 in detail.

Shown in figure 5(a) are surface elevations for two unforced nonlinear waves(broken and solid curves) that are both close to the Stokes-limiting configuration,for the same value of the Froude number (F = Fu in this case). The broken andsolid curves plotted in figure 5(b) are for the angle θ the free surface makes withthe horizontal, for each of the two profiles shown in figure 5(a). Following Longuet-Higgins & Fox (1996) and others, we introduce the parameter

ω = 1− (u(xc)F)2, (3.5)

where xc is the location of the wave crest. This parameter represents a rescaled waveheight, with a maximum value ω = 1. Figure 5(c) is a plot of the Froude numberagainst the wave height parameter ω, close to the Stokes-limiting configuration (whenω = 1). A plot of the total energy E = T + V versus ω is shown in figure 5(d). Thetotal energy is calculated using the formulation of Longuet-Higgins & Fenton (1974),where T is the wave kinetic energy and V is the wave potential energy.

The wave height and the Froude number of the almost-highest waves, where ω

is near to 1, differs between the forced solution types. In the unforced case analmost-highest solitary wave, where ω = 0.993 (see figure 5a) was found with a valueof F = Fu, which is in agreement with the work of Hunter & Vanden-Broeck (1983)


0

E

1.0

1.4

1.8

–5 0 5

1.292

1.294

0.90 1.000.85 0.95

0.99

1.01

–5 50

0.8 1.00.9

y

F

(a) (b)

(c) (d )

x x

FIGURE 5. Results for unforced solitary waves. Shown are: (a) the free-surface profiles fora value of the Froude number F = 1.2909, where the solid and dashed curves are for valuesof ω = 0.995 and ω = 0.849, respectively; (b) a plot of θ versus x for (a); (c,d) plots of theFroude number F and wave energy E versus ω defined in (3.5), respectively. The crosses in (c)indicate the asymptotic predictions of Longuet-Higgins & Fox (1996) for the extrema.

and Longuet-Higgins & Fox (1996) (amongst others). However, only type IV wavesexhibit the property of F being near to Fu (see figure 6a) when ω is close to 1.This is explained by the position of the two wave crests for wave form IV being aconsiderable distance away from the centre of the free-surface pressure distribution(2.5). Thus, the wave crests are not affected by the localized forcing of the pressureand are therefore effectively unforced, and retain the limiting value of F being veryclose to Fu. For the other types, wave forms II and III, the support of the pressuredistribution overlaps the region of the crest(s), and influences the almost-highest waveheight and the corresponding limiting value of the Froude number.

An additional effect on the crest due to the nearby pressure disturbance was foundfor the type III solutions. The angle of the computed free surfaces (figures 4b–8b)illustrate the near-Stokes-limiting configuration by way of a rapid change from π/6 to−π/6 near the crest, except in the case of figure 8(b). At the crests of the solutionshown in figure 8(b), the graph of the rapid change in θ is of similar magnitude, butshifted vertically. In other words, the corner flow has the same included angle but theleft and right crests are perturbed by a small rotation: counter-clockwise for the leftcrest and clockwise for the right. This is consistent with a modified Stokes’s analysisincluding the presence of a pressure disturbance detailed in the Appendix.

A striking feature common to both the unforced and forced solutions isthe oscillatory way in which they all approach the limiting configuration; seefigures 4(c)–8(c). Our computations show the first two turning points of the oscillation,and it is conjectured that there are infinitely many oscillations (decreasing in


E

y

F

0

1.0

1.4

1.8

0x x

–5 5 –5 50

1.292

1.294

0.85 0.90 1.000.95

1.94

1.97

2.00

0.8 0.9 1.0

(a) (b)

(c) (d)

FIGURE 6. Forced solitary waves of type IV, with amplitude of forcing A = −0.10. (a) Free-surface profile for a value of the Froude number F = 1.2909. The solid and dashed curves arefor values of ω = 0.993 and ω = 0.850, respectively. (b) Plot of θ versus x for (a). (c,d) Plotsof the Froude number F and wave energy E versus ω, respectively.

amplitude) as ω→ 1, as in the unforced case (Longuet-Higgins & Fox 1996). Furthercalculations, with better grid refinement, were able to determine the next turning pointon the F−ω curve; this agreed with the result of Longuet-Higgins & Tanaka (1997) tofive significant figures. Figures 4(c)–8(c) also demonstrate the resulting non-uniquenessof solutions for given values of the Froude number F and amplitude of forcing A; thisis also clearly seen in figure 3. Longuet-Higgins & Fox (1996) formulated predictionsfor the extrema of the unforced waves, which are in agreement with our computations;see figure 5(c). This nonlinear oscillatory behaviour is also seen in the energy plots offigures 4(d)–8(d). In all types of forced solutions the almost-highest wave is not themost energetic nor is it the fastest, which is similar to what is found in the unforcedcase.

We now return to our discussion to the results of figure 3(b–d), representing thechanges for given (increasing) values of the Froude number. For Froude numberF = 1.262, figure 3(b) demonstrates that the branch of nonlinear solutions becomesdisconnected in the upper right-hand side of the (A, η(0)) plane. Here the circularmarkers represent two solutions close to the Stokes-limiting configuration, both ofwhich are type II waves. The left-most marker in figure 3(b) corresponds to theprofile (solid curve) in figure 4(a). As the value of the Froude number increases,to F = 1.2909 in figure 3(c) and F = 1.32 in figure 3(d), the distance between thetwo markers in the (A, η(0)) plane increases. The markers move apart in opposite


y 0

F E

(a) (b)

(c) (d)

1.0

1.4

1.8

0x

0x

5–5

1.320

1.324

1.328

0.8 0.9 1.0 0.8 0.9 1.0

1.30

1.34

1.38

–5 5

FIGURE 7. Forced solitary wave of type II with amplitude of forcing A = −0.044. Thistype of solution is not predicted by the weakly nonlinear analysis. (a) Nonlinear profiles fora value of the Froude number F = 1.320. The solid and dashed curves are for values ofω = 0.991 and ω = 0.750, respectively. (b) Plot of θ versus x for (a). (c,d) Plots of the Froudenumber F and wave energy E versus ω, respectively.

directions, along the upper broken curve which defines the locus of the ‘near-limiting’values in the (A, η(0)) plane.

When the value of the Froude number equals Fu the upper (originally left-most)marker coincides with the almost-highest unforced solitary wave, as is shown infigure 3(c). The upper marker in figure 3(c) corresponds to the profile (solid curve)in figure 5(a). The entire branch of forced type IV waves, emanating from the origininto the left-hand side of the upper half of the (A, η(0)) plane of figure 3(c), are closeto the Stokes-limiting configuration. The lower marker on this branch of solutions infigure 3(c) corresponds to the profile (solid curve) in figure 6(a).

The almost-highest unforced and forced type IV solitary waves do not exist forvalues of the Froude number F > Fu. For a value of the Froude number F = 1.32,the upper marker that was for the unforced solution in figure 3(c) has now movedfurther to the left along the upper broken curve of ‘near-limiting’ values in figure 3(d),and is now a forced (with A < 0) type solution approaching the Stokes limitingconfiguration. This solution is not predicted by the weakly nonlinear analysis and isclassified as a perturbation of a single solitary wave (i.e. type II). The upper markerin figure 3(d) corresponds to the free-surface profile in figure 7(a) (solid curve). Theleft-most marker in figure 3(d) indicates a type III solution with the Stokes-limitingconfiguration at the wave crests, which is shown in figure 8(a). It is found thatthe left-most upper branch of solutions connecting the two curves of ‘near-limiting’values in the (A, η(0)) plane decreases in length as the Froude number increases (seefigure 3(d)). These ‘near-limiting’ type II and III solutions remain qualitatively distinct


y

E

0

F

(a) (b)

(a) (b)

1.0

1.2

1.4

1.6

1.8

2.0

0x x

1.320

1.322

1.324

0.90 0.95 1.00

1.80

1.82

1.84

0.90 0.95 1.00

–5 50–5 5

FIGURE 8. Forced solitary waves of type III, with amplitude of forcing A = −0.158. (a)Nonlinear profile for a value of the Froude number F = 1.320. The solid and dashed curvesare for values of ω = 0.997 and ω = 0.888, respectively. (b) Plot of θ versus x for (a). (c) and(d) Plots of the Froude number F and wave energy E versus ω, respectively.

as the amplitude of forcing increases up to the limit of our computational schemeused; this was around A≈−0.21 for these types of wave.

4. Concluding remarksWe have considered the free-surface flow of almost-highest unforced and forced

solitary waves. The forced solutions that approach the Stokes-limiting configurationconsist of perturbations of either one or two solitary waves. For all values of theFroude number F > 1.262 there are type II waves with amplitude of forcing A > 0(e.g. figure 3(b–d), right-most markers). The almost-highest unforced and forced typeIV waves (with A < 0) only exist for values of the Froude number F = Fu (seefigure 3c). When the Froude number Fu < F there are almost-highest type II and IIIwaves, with A < 0 (see figure 3(d), left-most and upper markers). For each wave typethe almost-highest unforced and forced solitary waves are not the fastest nor the mostenergetic.

Although we have not examined the stability of the almost-highest forced solitarywaves, it is reasonable to expect that since the energy of both the unforced andforced waves (see figures 4(d)–8(d)) exhibit the same oscillatory behaviour, thestability characteristics will be qualitatively similar. In the unforced case, Tanaka(1986), Longuet-Higgins & Fox (1996) and Kataoka (2006) (amongst others) provide adiscussion on the modes of instability occurring at the stationary values of the energyin a solitary wave.


0

200

400

x

t

–50 0 50 100 150 200 250

0

1

600

FIGURE 9. Evolution of perturbed stationary weakly nonlinear solutions for a value of theFroude number F = 1.26. Flow type IV with A = −0.13, initial profile is 0.98 times thesteady-state profile.

0

200

400

x

t

–50 0 50 100 150 200 250

0

1

600

FIGURE 10. Evolution of perturbed stationary weakly nonlinear solutions for a value of theFroude number F = 1.26. Flow type IV with A = −0.13, initial profile is 1.02 times thesteady-state profile.

To determine the stability of our solutions before they approach the Stokes-limitingconfiguration, the time-dependent form of the fKdV (3.1) was used with the pressureforcing given by (2.5) (as has been demonstrated earlier, the fKdV equation providesgood qualitative agreement with our nonlinear calculations at larger Froude numbers.Thus, although the fKdV is only formally valid for Froude number near unity,we believe its stability characteristics will match well those obtained from a fullunsteady boundary-element approach, the latter being considerably more computationalexpensive than the weakly nonlinear unsteady fKdV approach). Stationary solutionswere found numerically, perturbed by multiplying the steady-state profile by a constant,and substituted as the initial condition for (3.1). We used both the spectral and finite-difference approaches described by Grimshaw et al. (2007) and Chardard et al. (2011);the evolution with time was observed and typical results are presented in figures 9 and10. Only the forced solution types I and V that are perturbations of a uniform streamwere found to be (numerically) stable.

We considered two different classes of perturbation, where the perturbation constantis slightly larger than one, or slightly less than one. We focus on the value F = 1.26as it is near to the almost-highest solutions, and to ensure that we are not nearthe resonance region around F = 1. For positive amplitude of forcing, the unsteady


response of solution type II has been considered with similar initial conditions byCamassa & Wu (1991), where the profile near the disturbance will decay to the profileof solution type I. In the case of the larger initial state an upstream solitary wave isemitted together with downstream oscillatory wave train. In the smaller initial state,there is no solitary wave emitted upstream, and there is one small localized wavetravelling downstream.

For the negative amplitude types III and IV waves the response is composedof similar behaviour. We focus on solution type IV, where there initial state hasa considerable influence on behaviour. Given the smaller initial state, shown infigure 9, the upstream and downstream humps are initially drawn towards theforcing, coalescing, and eventually emitted as a solitary wave upstream. For the largerinitial state (figure 10), the upstream hump is very quickly emitted upstream. Thedownstream hump is initially drawn towards the forcing, then briefly downstream, thendrawn back upstream where it is emitted as a solitary wave. The downstream responsein either case contains small localized waves, as well as a decaying oscillatory wavetrains, emitted near to the time that the upstream solitary waves are emitted.

Finally, we mention that it would interesting to study the effect of a different type oflocalized forcing on very steep waves, rather than the pressure distribution consideredin this paper. For example, one could readily consider a localized depression in thebottom topography of the channel. In terms of the weakly nonlinear analysis, themodel (3.2)–(3.4) are the same, but with the amplitude of forcing given by the area ofdepression (see, for example, Binder et al. 2005). The same qualitative picture as thepressure distribution will therefore be seen when the Froude number is close to unity.However, there are likely to be significant differences in the solution behaviour whenapproaching the Stokes-limiting configuration between the pressure disturbance resultspresented here and those that arise for a bottom depression.

AcknowledgementsS.L.W. was the recipient of a University of Adelaide Divisional Scholarship. We

gratefully acknowledge the helpful and insightful comments of the three referees.

Appendix. The Stokes corner flow at the crest in the presence of a nearbypressure disturbance

To proceed we use the reference frame given by the complex coordinate z = x + iywhere x = x − xc and y = y − (1 + ηc), where xc is the horizontal displacement ofthe crest relative to the pressure disturbance and ηc is the displacement of the crestfrom uniform stream. Thus, the crest is at the origin in the z plane. Dropping thetilde notation from here on (but maintaining the new reference frame), the complexpotential near the crest is assumed to be of the form f (z) = φ(z) + iψ(z) = Reiτ zn forreal numbers R and τ with the index n> 1 to be determined.

In these coordinates, equation (2.5) becomes

p(x;A,B, xc)= AB√π

e−(B[x+xc])2 . (A 1)

Expanding the pressure disturbance in x about x= 0 using Taylor series,

p(x;A,B, xc)= A(1− xB)+ · · · (A 2)

for A= ABπ−1/2e−(Bxc)2 and B= 2Bxc. Here · · · denotes higher-order terms.


The assumption that for small x the streamline of the free surface is asymptoticallyy = m+x downstream and y = m−x upstream, combined with the expansion of thepressure term, allows us to rewrite Bernouilli’s equation (2.3) as

R2n2((1+ m±2)x2)n−1 + 2

F2(m±x+ ηc + A(1− xB+ · · · ))= 1. (A 3)

As x→ 0, the leading-order component of (A 3) yields 2(ηc + A)/F2 = 1. To determinethe index n, we proceed to higher order and balance the O(x2n−2) terms with the O(x)term in (A 3) which yields a value n= 3/2. It then follows from (A 3) that

R= 23F

√±2(AB− m±)√

1+ m±2, (A 4)

where ± is taken on the downstream and upstream sides, respectively, and AB > m+for x > 0 and AB < m− for x < 0. As R is a constant, the following equation needs tobe satisfied:

(AB− m+)√

1+ m−2 + (AB− m−)√

1+ m+2 = 0. (A 5)

For the assumed form of the potential and the flow domain Im (f ) ∈ (−1, 0) (acondition imposed by our non-dimensionalization), it follows that if the downstreamstreamline of the free surface is given by z+(r) = reiθ+ and upstream is given byz−(r)= reiθ− , then

θ+ =−23τ, θ− =−2

3(τ + π). (A 6)

As fz = φx − iφy, and φy/φx is the slope of the streamline, then by

df

dz= 3R

2eiτ z1/2, (A 7)

a relationship between τ and m± is found by substituting z− and z+ into the aboveexpression and using the previous two equations relating τ and θ±;

tan(

23τ

)=−m+, tan

(π

3− 2

3τ

)= m−. (A 8)

Solving (A 5) and (A 8) determines the slope m± of the downstream and upstreamsurfaces, which must have an included angle of 120◦. These provide the same resultas that of Stokes when xc = 0 (and when A = 0), i.e. when the pressure disturbance isat the location of the crest (and when there is no pressure disturbance). In this casethe flow has a crest with a 120◦ included angle and it is symmetric in the sense that−m+ = m− = tan(π/6).

When xc 6= 0 and A 6= 0 then the solutions to (A 5) and (A 8) suggest a crest thatwill be re-orientated, but still having a 120◦ included angle. When xc is positive, thecrest is rotated slightly clockwise and when xc is negative the crest will be rotatedcounter-clockwise, similar to figure 8. As |xc| gets large, the solutions will limit to−m+→ m− and m−→ tan(π/6), so the symmetric result will be observed for creststhat are far from the pressure disturbance, as in figure 6.


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J. Fluid Mech. (2014), . 739, pp. doi:10.1017/jfm.2013.590 ... · 2 S. L. Wade, B. J. Binder, T. W. Mattner & J. P. Denier corresponds to the onset of a superharmonic (short-wavelength)