inﬂuence of rapid changes in a channel bottom on free-surface … · 2008. 2. 4. · free-surface...

IMA Journal of Applied Mathematics (2008) 73, 254−273doi:10.1093/imamat/hxm049Advance Access publication on October 24, 2007

Influence of rapid changes in a channel bottom on free-surface flows

B. J. BINDER

School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia

F. DIAS

Centre de Mathematiques et de Leurs Applications, Ecole Normale Superieure de Cachan,UniverSud, 61 Avenue President Wilson, F-94230 Cachan, France

AND

J.-M. VANDEN-BROECK†

School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK

[Received on 26 January 2007; accepted on 1 June 2007]

Two-dimensional non-linear free-surface flows in a channel bounded below by an uneven bottom withrapid changes are considered. Numerical solutions are computed by a boundary integral equation methodsimilar to that first introduced by King & Bloor (1987, J. Fluid Mech., 182, 193–208). Free-surface flowspast localized disturbances, steps and sluice gates are calculated. In addition, weakly non-linear solutionsare discussed.

Keywords: free-surface flow; boundary integral equation method; potential flow.

1. Introduction

Many problems in fluid mechanics involve free-surface flows past disturbances. Here, a free surfacerefers to the interface between a fluid (e.g. water) and the atmosphere (assumed to be characterizedby a constant atmospheric pressure). Examples include free-surface flows generated by moving surface-piercing objects (e.g. ships) or moving submerged objects. These flows reduce to flows past disturbanceswhen viewed in a frame of reference moving with the objects. Other examples involve free-surfaceflows past an uneven channel bottom. These problems are often modelled within the framework ofpotential theory. Efficient numerical methods based on boundary integral equation formulations havebeen developed to solve these non-linear problems both in 2D (see, e.g. Forbes & Schwartz, 1982;King & Bloor, 1987; Vanden-Broeck, 1987; Forbes, 1988; Dias & Vanden-Broeck, 1989; Asavanant &Vanden-Broeck, 1994; Binder & Vanden-Broeck, 2007; Binder et al., 2006) and in 3D (see, e.g. Parauet al., 2005a,b).

In this paper, we consider mainly two flow configurations. Gravity is included in the dynamic bound-ary condition but surface tension is neglected. The first configuration is the free-surface flow past a stepin a channel (see Fig. 1a). The second one is obtained by replacing the step of Fig. 1(a) by a rectangularobstacle of finite length b (see Fig. 2a). The flow of Fig. 1(a) was first considered by King & Bloor(1987), who developed a boundary integral equation method to compute non-linear solutions. Flowspast submerged disturbances such as the one of Fig. 2(a) were studied by many previous investigators

†Email: [email protected]. Present address: Department of Mathematics, University College London, Gover Street,London WC1E 6BT, UK

c© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 255

FIG. 1. (a) Sketch of flow past a step in physical coordinates (x∗, y∗). (b) Sketch of flow in the plane of the complex potential( f -plane). (c) Sketch of flow in the lower half-plane (ζ -plane).

including Forbes (1981), Forbes & Schwartz (1982), Vanden-Broeck (1987), Forbes (1988), Dias &Vanden-Broeck (1989), Dias & Vanden-Broeck (2002), Dias & Vanden-Broeck (2004) and Binderet al. (2005). The results presented in this paper supplement these previous investigations. Our approachfollows that of Dias & Vanden-Broeck (2002), Binder et al. (2005) and Binder et al. (2006). In partic-ular, we present new results when the height of the step is large. We also show that the structure of thevarious families of solutions for the flow of Fig. 2(a) is different when s∗ > 0 or s∗ < 0. Finally, wepresent some new hybrid solutions involving multiple disturbances.

Cartesian coordinates (x∗, y∗) are introduced in Figs 1(a) and 2(a) and the flows are assumed toapproach uniform streams with constant velocity U and constant depth H as x∗ → ∞. (All the flowsconsidered in the present paper are reversible from a mathematical point of view. From a physical pointof view, some of the flows shown in the various figures ought to be reversed to be more realistic.) Wedefine the downstream Froude number

F = U

(gH)1/2, (1.1)

where g is the acceleration due to gravity. The flow as x∗ → −∞ can either be a uniform stream withconstant velocity V and constant depth D − h∗ or be characterized by a train of waves. Here, h∗ �= 0in Fig. 1(a) and h∗ = 0 in Fig. 2(a). When the flow is uniform as x∗ → −∞, we define the upstream

256 B. J. BINDER ET AL.

FIG. 2. (a) Sketch of flow past a rectangular obstacle in physical coordinates (x∗, y∗). (b) Sketch of flow in the plane of thecomplex potential ( f -plane). (c) Sketch of flow in the lower half-plane (ζ -plane).

Froude number

F∗ = V

[g(D − h∗)]1/2. (1.2)

By using a weakly non-linear theory, Shen (1995), Dias & Vanden-Broeck (2002), Binder et al.(2005), Binder & Vanden-Broeck (2005), Binder et al. (2006), Binder & Vanden-Broeck (2007) andothers identified four types of solutions. They are illustrated in Fig. 3 for the flow past a rectangularobstacle sketched in Fig. 2(a). The first type is a waveless supercritical flow with F = F∗ > 1 (seeFig. 3a). The second type is a subcritical flow (F < 1) with a train of waves as x → −∞ (see Fig. 3b).The third type has uniform streams both far upstream and far downstream with F > 1 and F∗ < 1 (seesolid line in Fig. 3c). These three types of solutions have been investigated in many previous studiesby Forbes (1981), Forbes & Schwartz (1982), Vanden-Broeck (1987), Forbes (1988) and Dias &Vanden-Broeck (1989). The fourth type was discovered by Dias & Vanden-Broeck (2002). It is a flowwith a supercritical uniform stream (F > 1) as x∗ → ∞ and a train of waves as x∗ → −∞ (see dashedline in Fig. 3c).

The flow past a step sketched in Fig. 1(a) has been studied by King & Bloor (1987), Binder et al.(2006) and Chapman & Vanden-Broeck (2006). The results of Binder et al. (2006) show that the fourtypes of flows found in Fig. 3 also exist for the flow past a step.


FIG. 3. The well-known basic flow types for a submerged obstacle. (a) Supercritical flows with given values of F , s and b(F = 1.10, s = 0.03 and b = 1.20). (b) Subcritical flow with given values of F , s and b (F = 0.76, s = 0.01 and b = 1.90).(c) The solid curve corresponds to a hydraulic fall with given values of s and b (s = 0.02 and b = 1.60). The Froude number,F = 1.12, comes as part of the solution. The broken curve corresponds to a generalized critical flow with given values of F , s andb (F = 1.12, s = 0.02 and b = 1.60) and a fourth parameter taken here as the free-surface elevation far away from the obstacleon the wavy side: η(−∞) = 0.16.

An interesting related problem is that of the free-surface flow under a sluice gate (see Fig. 4a).This problem has been intensively studied in the past by Benjamin (1956), Frangmeier & Strelkoff(1968), Larock (1969), Chung (1972), Vanden-Broeck & Keller (1989), Vanden-Broeck (1996) andBinder & Vanden-Broeck (2005). Vanden-Broeck (1996) and Binder & Vanden-Broeck (2005) showedthat the flow of Fig. 4(a) differs from that of Fig. 2(a) in the sense that only solutions of the first andfourth types exist. In other words, there are no subcritical flow with waves only on one side of the gate orflows characterized by uniform streams far upstream and far downstream with F > 1 and F∗ < 1. Suchflows can, however, exist if a second disturbance is introduced in the flow (see Binder & Vanden-Broeck,2007).

The paper is organized as follows. The problems are formulated in Section 2 and the boundaryintegral equation methods used to compute non-linear solutions are described in Section 3. Analyticalmethods based on a weakly non-linear theory are summarized in Section 4. The numerical results aredescribed in Section 5.


FIG. 4. (a) Sketch of flow past a sluice gate in physical coordinates (x∗, y∗). (b) Sketch of flow in the plane of the complexpotential ( f -plane). (c) Sketch of flow in the lower half-plane (ζ -plane).

2. Governing equations

We consider the steady 2D irrotational flows of an incompressible inviscid fluid shown in Figs 1(a), 2(a)and 4(a).

We first formulate the free-surface flow past a step (Fig. 1a). The flow domain is bounded below bythe bottom of the channel A′B ′C ′D′ and above by the free surface AB. The equations of the bottom ofthe channel and of the free surface are denoted by y∗ = σ ∗(x∗) and y∗ = H + η∗(x∗), respectively.The function η∗(x∗) is assumed to vanish as x∗ → ∞.

On the free surface AB, the dynamic boundary condition gives

1

2(u∗2 + v∗2) + gy∗ = 1

2U 2 + gH, on y∗ = H + η∗(x∗), (2.1)

where u∗ and v∗ denote the horizontal and vertical components of the velocity. Here, we have used theconditions

u∗ → U, v∗ → 0, η∗(x∗) → 0, as x∗ → ∞, (2.2)

to evaluate the Bernoulli constant on the right-hand side of (2.1).


The mathematical problem is then to seek the complex velocity u∗ − iv∗ as an analytic function ofz∗ = x∗ + iy∗ in the fluid domain, satisfying (2.1), (2.2) and the kinematic boundary conditions

v∗ = u∗ dη∗

dx∗ , on AB, (2.3)

v∗ = 0, on A′B ′ and C ′D′, (2.4)

u∗ = 0, on B ′C ′. (2.5)

The formulation for the flow of Fig. 2(a) is identical except that the kinematic boundary conditions(2.4) and (2.5) are now replaced by

v∗ = 0, on A′ B ′, C ′E ′ and F ′G ′, (2.6)

u∗ = 0, on B ′C ′ and E ′F ′. (2.7)

The flow of Fig. 4(a) can also be formulated in a similar way. The differences are now that thedynamic boundary condition (2.1) only holds on AB and C D and that the kinematic boundary conditionon BC gives

v∗ = −u∗ tan σc. (2.8)

3. Boundary integral equation

King & Bloor (1987) derived a boundary integral equation method to solve the flow configuration ofFig. 1(a). Here, we present a similar method which applies to arbitrary bottom shapes consisting ofstraight segments. As we shall see, the method also works if a portion of the free surface is replaced bya flat plate (sluice gate or surfboard) (see Fig. 4a). For the sake of clarity, we first present the method forthe particular flow configurations sketched in Figs 1(a) and 2(a). The numerical procedure is derived asa combination of the methods used by Vanden-Broeck (1996), Dias & Vanden-Broeck (2002), Dias &Vanden-Broeck (2004), Binder et al. (2005), Binder & Vanden-Broeck (2005), Binder et al. (2006) andBinder & Vanden-Broeck (2007) to compute flows past submerged obstacles, steps and flat plates. Someof the details are repeated for completeness and further details can be found in these papers.

We define dimensionless variables by taking H as the reference length and U as the reference ve-locity. The dimensionless quantities are denoted by letters without a star.

The dynamic boundary condition (2.1) on the free surface AB then takes the form

1

2(u2 + v2) + 1

F2y = 1

2+ 1

F2, on y = 1 + η. (3.1)

We introduce the complex potential function, f = φ + iψ , and the complex velocity, w = d f/dz =u − iv . Without loss of generality, we choose ψ = 0 on the streamline AB. It follows that ψ = −1 onthe channel bottom streamline. We also choose φ = 0 at C ′ in Fig. 1(a) and D′ in Fig. 2(a). We denoteby φ = φb the value of φ at the point B ′ in Fig. 1(a). Similarly, we denote the values of φ at the cornersB ′, C ′, E ′ and F ′ in Fig. 2(a) by φ1, φ2, φ3 and φ4, respectively. In the complex potential plane, the fluidis in the strips −1 < ψ < 0 and −∞ < φ < ∞, see Figs 1(b) and 2(b).

We then map the strips of Figs 1(b) and 2(b) onto the lower half of the ζ -plane by the transformation

ζ = α + iβ = eπ f . (3.2)


The flows in the ζ -plane are shown in Figs 1(c) and 2(c). The value of α at the point B ′ in Fig. 1(c) isαb = −eπφb . The values of α at the points B ′, C ′, E ′ and F ′ of Fig. 2(c) are α1 = −eπφ1 , α2 = −eπφ2 ,α3 = −eπφ3 and α4 = −eπφ4 , respectively. The aim is now to derive integral equation relations thatonly involve unknown quantities on the free surfaces, subject to the kinematic boundary conditions (2.4)and (2.5) for the flow of Fig. 1(a) and (2.6) and (2.7) for the flow of Fig. 2(a).

We define the function τ − iθ by

w = u − iv = eτ−iθ (3.3)

and we apply Cauchy’s integral formula to the function τ − iθ in the ζ -plane with a contour consistingof the α-axis and a semicircle of arbitrary large radius in the lower half-plane. Since (2.2) implies thatτ − iθ → 0 as |ζ | → ∞, there is no contribution from the semicircle and we obtain after taking the realpart

τ (α) = 1

π

∫ ∞

−∞θ (α0)

α0 − αdα0. (3.4)

Here, τ (α) and θ (α) denote the values of τ and θ on the α-axis. The integral in (3.4) is a Cauchyprincipal value.

Next we note that the kinematic boundary conditions (2.4–2.7) imply

θ (α) = 0, on αb < α < 0 and α < −1, (3.5)

θ (φ) = π

2, for − 1 < α < αb, (3.6)

θ (α) = 0, for α < α4, α3 < α < α2 and α1 < α < 0, (3.7)

θ (α) = π

2, for α2 < α < α1, (3.8)

θ (α) = −π

2, for α4 < α < α3. (3.9)

Substituting (3.5) and (3.6) into (3.4) we obtain the following relation between τ and θ on the freesurface AB of Fig. 1

τ (α) = 1

2ln

|αb − α||1 + α| + 1

π

∫ ∞

0

θ (α0)

α0 − αdα0. (3.10)

Similarly, substituting (3.7–3.9) into (3.4) we obtain on the free surface AB of Fig. 2 the relation

τ (α) = 1

2ln

|α1 − α||α − α2| − 1

2ln

|α3 − α||α − α4| + 1

π

∫ ∞

0

θ (α0)

α0 − αdα0. (3.11)

Since α > 0 on the free surface AB, we can rewrite (3.10) and (3.11) in terms of φ by using the changeof variables

α = eπφ, α0 = eπφ0 . (3.12)

This yields

τ(φ) = 1

2ln

|αb − eπφ ||1 + eπφ | +

∫ ∞

−∞θ(φ0)eπφ0

eπφ0 − eπφdφ0 (3.13)


for the flow of Fig. 1 and

τ(φ) = 1

2ln

|α1 − eπφ ||eπφ − α2| − 1

2ln

|α3 − eπφ ||eπφ − α4| +

∫ ∞

−∞θ(φ0)eπφ0

eπφ0 − eπφdφ0 (3.14)

for the flow of Fig. 2. In (3.13) and (3.14), τ(φ) = τ (eπφ) and θ(φ) = θ (eπφ).Integrating the identity

x(φ) + iy(φ) = 1

u − iv= e−τ+iθ , (3.15)

we obtain the following parametric representation of the free surface AB

x(φ) = x(∞) +∫ φ

∞e−τ(φ0) cos θ(φ0)dφ0, (3.16)

y(φ) = 1 +∫ φ

∞e−τ(φ0) sin θ(φ0)dφ0. (3.17)

The dynamic boundary condition (3.1) and (3.3) give

e2τ(φ) + 2

F2y(φ) = 1 + 2

F2. (3.18)

Equations (3.13), (3.17) and (3.18) define an integro-differential equation for the unknown functionθ(φ) on the free surface AB of Fig. 1. Similarly, (3.14), (3.17) and (3.18) define an integro-differentialon the free surface AB of Fig. 2. These equations can be solved by using the numerical proceduredescribed in Binder et al. (2005). Once they have been solved, (3.16) and (3.17) give the shape of thefree surface.

We conclude this section by showing that the numerical method also applies if a portion of the freesurface is replaced by a flat plate (see Fig. 4a). Since the plate is inclined at the angle σc, one has

θ = −σc, for φb < φ < φc, (3.19)

where φb and φc are the values of φ at the end points B and C of the plate. Assuming a flat bottom, itcan easily be shown that (3.13) or (3.14) is replaced by

τ(φ) = −σc

πln

|eπφc − eπφ ||eπφb − eπφ | +

∫ φb

−∞θ(φ0)eπφ0

eπφ0 − eπφdφ0 +

∫ ∞

φc

θ(φ0)eπφ0

eπφ0 − eπφdφ0. (3.20)

The integro-differential equation on the free surface is then defined by (3.17), (3.18) and (3.20).Further details on the numerical methods for free-surface flows with gates and surfboards can be found inBinder & Vanden-Broeck (2005).

4. Weakly non-linear theory

The number of independent parameters needed to obtain a unique solution to a free-surface problem isoften not obvious. There are two natural ways to find it. The first one is by careful numerical experimen-tation (fixing too many or too few parameters fails to yield convergence). The second one is to perform


a weakly non-linear analysis in the phase plane. This second approach has the advantage of allowing asystematic determination of all the possible solutions (within the range of validity of the weakly non-linear analysis). In all the examples presented in this paper, we checked that both approaches lead to thesame number of independent parameters.

Several investigators (see, e.g. Shen, 1995; Dias & Vanden-Broeck, 2002; Binder et al., 2005) havederived a forced Korteweg–de Vries equation to model the flow past an obstacle at the bottom of achannel. They showed that the forcing can be approximated by a jump in ηx . Therefore, one writes

ηxx + 9

2η2 − 6(F − 1)η = 0, for x �= xt ,

with the vertical jump condition

ηx (x+t ) − ηx (x−

t ) = −δ. (4.1)

Here, xt denotes the position of the obstacle and δ is related to the size of the disturbance. For therectangular obstacle of Fig. 2(a), δ = 3sb where s and b are the height and the length of the rectangle.We note that δ > 0 when s > 0 and δ < 0 when s < 0.

Binder & Vanden-Broeck (2005) derived the corresponding weakly non-linear solutions for flowspast a sluice gate or surfboard (the numerical procedure to compute the corresponding fully non-linearsolutions was outlined at the end of Section 3). They showed that the flow is described on the portionsof free surface on the right and left of the gate by the Korteweg–de Vries equation (KdV equation)

ηxx + 9

2η2 − 6(F − 1)η = 0, (4.2)

with the conditions

ηr − ηl = L sin σc, (4.3)

ηrx = ηl

x = − tan σc. (4.4)

Here, the superscripts r and l refer to the end points on the right and left of the gate, L is the length ofthe plate and σc is the inclination (see Fig. 4a). Since ηr

x = ηlx , (4.3) and (4.4) imply that the gate is

represented in the phase plane by a horizontal segment of length L sin σc. We refer to that segment as ahorizontal jump.

In the absence of disturbances, the KdV equation (4.2) holds for all x . The corresponding solutionsare shown in the phase plane ηx versus η in Fig. 5. To construct weakly non-linear solutions for flowspast disturbances, we need to combine the trajectories of the phase plane of Fig. 5 with both verticaljumps (submerged objects) and horizontal jumps (sluice gates or plates).

Binder et al. (2006) showed that the weakly non-linear theory can also be used for the flow past thestep of Fig. 1. The main difference is that the depths are now different far upstream and far downstream.There are then no vertical or horizontal jumps but instead a superposition of two phase planes. Solutionsare obtained by moving continuously from the orbits of one phase plane to those of the other.

5. Discussion of the results

In Section 5.1, we first present results for the flow configuration of Fig. 2. Here, we contrast the proper-ties of solutions with s > 0 and s < 0. We then describe solutions for the free-surface flow past a step(see Fig. 1) and concentrate our attention on solutions for large steps (Section 5.2 ). In Section 5.3, wesummarize some of our findings for free-surface flows under a sluice gate. More results for submergedobstacles are shown in Section 5.4.


FIG. 5. Weakly non-linear phase portraits, dη/dx versus η, h = 0. (a) Supercritical flow, F > 1. There is a saddle point atη = 0, ηx = 0 and a centre at η = 4/3(F − 1), ηx = 0. The inner closed trajectories are periodic solutions. The closed trajectoryis the solitary wave. The maximum value of ηx at η = 4/3(F − 1) on the solitary wave trajectory is η′

m = 4/3√

2(F − 1)3/2.(b) Subcritical flow, F < 1. There is a saddle point at η = 4/3(F − 1), ηx = 0 and a centre at η = 0, ηx = 0. The inner closedtrajectories are periodic solutions. The closed trajectory is the solitary wave. The maximum value of ηx at η = 0 on the solitarywave trajectory is η′

m = 4/3√

2(1 − F)3/2.

5.1 Free-surface flows past a rectangular submerged disturbance

As mentioned in Section 1, there are four types of solutions when s > 0 (see Fig. 3). Their existencecan be predicted by using the weakly non-linear theory.

Let us first consider the case F > 1. We then need to combine the phase portraits of Fig. 5(a) withthe downward vertical jump (4.1) modelling the obstacle. We start at the saddle point η = ηx = 0


and move clockwise on the solitary wave trajectory. We then have a vertical downward jump. There areseveral possibilities. If the obstacle is too large (i.e. 3δ > 2η′

m), there are no solutions. If 3δ = 2η′m ,

the vertical jump brings back the solution to the solitary wave trajectory along η = 4/3(F − 1) andwe then return to the saddle point η = 0 along the solitary wave trajectory. If 3δ < 2η′

m , the solutioncan either jump back on the solitary wave trajectory or jump on a periodic wave solution. In the formercase, there are two possibilities: one to the left and one to the right of η = 4/3(F − 1). They are illus-trated in Fig. 3(a). In the latter case, there are infinitely many solutions. They were first discovered byDias & Vanden-Broeck (2002), who called them generalized critical solutions. An example is shown inFig. 3(c) (broken curve). If 3δ = η′

m , the wavelength of the periodic wave vanishes and we obtain ahydraulic fall. A profile is shown in Fig. 3(c) (solid curve).

We now consider the case F < 1. Since η = ηx = 0 as x → ∞, the solution has to end at the originin Fig. 5(b). There are three possibilities. The first is to start at the saddle point, move clockwise on thesolitary wave and jump vertically to the origin along η = 0. This is the hydraulic fall solution that wehave already considered. The second possibility is to start on a periodic solution and then to jump back

FIG. 6. The basic flow types for a dip in the bottom of a channel. (a) Supercritical flow for given values of F = 1.10, s = −0.20and b = 0.65. (b) Subcritical flow for given values of F = 0.85, s = −0.05 and b = 0.33. (c) The solid curve is for a hydraulicfall for given values of F = 1.10 and b = 0.49. The step height, s = −0.11, came as part of the solution. The broken curve is fora generalized hydraulic fall for given values of F = 1.10, s = −0.11, b = 0.49 and elevation η(−∞) = 0.10.


FIG. 7. Supercritical flow. (a) Profiles for a step height of h = 0.02. The values of the Froude number for the curves, from topto bottom, are F = 1.175, F = 1.20, F = 1.50 and F = 3.00. The corresponding values of the uniform depth far upstreamare d = 1.13, d = 1.08, d = 1.04 and d = 1.02. The corresponding values of the upstream Froude number are F∗ = 1.04,F∗ = 1.08, F∗ = 1.46 and F∗ = 2.99. (b) Profile for a step height of h = 2.35 and value of the Froude number F = 5.00. Theuniform depth upstream is d = 3.48 and the upstream Froude number is F∗ = 4.16. (c) Profile for a step height of h = 5.00 andvalue of the Froude number F = 5.00. The uniform depth upstream is d = 6.34 and the upstream Froude number is F∗ = 3.22.(d) Graph of d − h versus F . Top to bottom curves are for values of h = 3.00, 2.35, 1.00, 0.50, 0.10, 0.02. (e) Profile for a stepheight of h = −3.85 and value of the Froude number F = 2.00. The uniform depth upstream is d = 0.56 and the upstreamFroude number is F∗ = 4.77. *b, c and e have the same vertical and horizontal scale.


FIG. 8. Subcritical flow for h > 0. (a) Profiles for a step height of h = 0.01. The value of the Froude number for the solid curveis F = 0.50. The value of the Froude number for the dash curve is F = 0.76. The value of the Froude number for the dot–dashcurve is F = 0.85. (b) Profiles for a step height of h = 0.10. The value of the Froude number for the solid curve is F = 0.30. Thevalue of the Froude number for the dash curve is F = 0.40. The value of the Froude number for the dot–dash curve is F = 0.50.(c) Profile for a step height of h = 0.15. The value of the Froude number is F = 0.30. (d) Profile for a step height of h = 0.25.The value of the Froude number is F = 0.30.

FIG. 9. Subcritical flow for h < 0. (a) Free-surface profiles for a step height of h = −0.01. The value of the Froude number forthe solid curve is F = 0.50. The value of the Froude number for the dash curve is F = 0.76. The value of the Froude number forthe dot–dash curve is F = 0.90. (b) Free-surface profiles for a step height of h = −0.25. The values of the Froude number fromthe bottom to the top curves are F = 0.10, F = 0.30 and F = 0.50.


FIG. 10. Subcritical hydraulic falls for F < 1 and h > 0. (a) Free-surface profile for a value of the Froude number F = 0.50.The value of the upstream Froude number is F∗ = 1.16. The step height is h = 0.17 and d = 0.74. (b) Free-surface profile for avalue of the Froude number F = 0.10. The value of the upstream Froude number is F∗ = 1.12. The step height is h = 0.67 andd = 0.87. (c) The solid curve is a plot of h versus F . The dash curve is a plot of d versus F . The dot–dash curve is a plot of d − hversus F . Note, the horizontal and vertical scales are the same in a and b.

to the origin along η = 0. This case is illustrated in Fig. 3(b). The third possibility is that 3δ > η′m , there

are then no solutions.We can at this stage summarize our approach as follows. We have used the weakly non-linear anal-

ysis to identify all the possible solutions. The weakly non-linear analysis gave us also the number ofparameters needed to determine uniquely a solution. For example, when F > 1, we first fix F (thisdefines the phase portrait of Fig. 5a). In Fig. 3(a), we need to fix a value of δ (i.e. the size of the object)for each of the two solutions. Therefore, the solutions of Fig. 3(a) depend on two parameters. For thesolid curve in Fig. 3(c), δ is given, so the size of the obstacle cannot be assigned. Therefore, this solutiondepends on one parameter. For the broken curve in Fig. 3(c), we start on the solitary wave trajectory andjump on a periodic solution. We can then choose δ and the particular periodic trajectory we want to


FIG. 11. Subcritical flow past a two-pressure distribution and an inclined plate for a given value of F = 0.76. (a) Fully non-linearfree-surface profile for L = 1.92, σc = 5.4◦. (b) Values of dy/dx = tan(θ) versus y − 1 = η, showing the fully non-linear phasetrajectories for a. (c) Weakly non-linear phase portrait for a, dη/dx versus η.

jump to. Therefore, this solution depends on three parameters. Similarly, we see that the solution of Fig.3(b) depends on two parameters.

Binder et al. (2005) showed on similar problems that the weakly non-linear free-surface profilesare in good agreement with fully non-linear computations when F is close to 1 and when the size ofthe obstacle is small. In Fig. 3, we presented only fully non-linear solutions computed by the boundaryintegral equation of Section 3. The reader interested in explicit weakly non-linear free-surface profilesis refered to Binder et al. (2005).

When s < 0, the obstacle is replaced by a depression in the bottom (see Fig. 6). The weakly non-linear analysis proceeds as in the case s > 0, except that the vertical jumps are now upwards. Proceedingas before, we find that the four types of solutions of Fig. 3 also exist for s < 0 (see Fig. 6). However,Fig. 6(a) shows that there are now three possible supercritical solutions (instead of two when s > 0).The first two (the two top curves) are similar to the ones found in Fig. 3(a). The third one (lower curve)does not have an equivalent in Fig. 3(a). Here, we start at the saddle point at η = ηx = 0 in Fig.5(a) and move downwards along the trajectory in the third quadrant of Fig. 5(a). We then jump ver-tically upwards to the corresponding trajectory in the second quadrant and come back to the originalong it.


5.2 Free-surface flows past a step

Binder et al. (2006) showed that the four types of solutions discussed in Section 5.1 also occur for theflow past a step (Fig. 1). Here, we supplement their findings by presenting fully non-linear solutionsfor large steps. We concentrate on three of the four types, namely the supercritical flows, the subcriticalflows and the hydraulic falls.

Figure 7 shows fully non-linear supercritical solutions. The supercritical solutions are characterizedby uniform streams at infinity with F > 1 and F∗ > 1. Figure 7(a) shows the effect of increasing F fora given value h > 0. As F → ∞, F∗ → ∞ and the solution approaches a free streamline solution witha constant velocity 1 on the free surface. Figure 7(b and c) shows similar profiles for larger steps. SinceF is large, these two solutions are close to free streamline solutions.

Different behaviours were observed for h > 0 and h < 0. For a given value of h > 0, solutionsexist only for F greater than a critical value F which depends on h. This is illustrated in Fig. 7(d) (thetop points on each of the curves correspond to values of F close to F). As F → F, F∗ → 1. The topprofile in Fig. 7(a) is a solution for F close to F . We note that Binder et al. (2006) used conservation of

FIG. 12. (a) Supercritical flow for given values of F = 1.20, s = 0.02 and b = 10.00. (b) Supercritical flow for given values ofF = 1.20, s = −0.03 and b = 8.41. (c) Supercritical flow for given values of F = 1.10 and b = 5.13. The free surface wasforced flat at x = 0 (θ(0) = 0). The step height, s = −0.0356, came as part of the solution. (d) Supercritical flow for given valuesof F = 1.10, s = −0.07 and b = 5.48.


FIG. 13. (a) Supercritical flow for given values of F = 1.10 and b = 20.24. The free surface was forced flat at x = 0 (θ(0) = 0).The step height, s = −0.0356, came as part of the solution. (b) Supercritical flow for given values of F = 1.10, b = 19.93 ands = −0.03. (c) Supercritical flow for given values of F = 1.10, s = −0.01 and b = 13.20. (d) Supercritical flow for given valuesof F = 1.10, s = −0.05 and b = 20.81.

mass and Bernoulli equation to derive the exact non-linear relations

2d

F2(d − h)2 −

(1 + 2

F2

)(d − h)2 + 1 = 0, (5.1)

F∗ = F

(1

d − h

)3/2

. (5.2)

The numerical results of Fig. 7(d) were found to be in close agreement with the exact results predictedby (5.1) and (5.2). In particular

F = (d − h)3/2. (5.3)

On the other hand, when h < 0 it can be shown that solutions exist for arbitrary large steps. A typicalsolution is shown in Fig. 7(e). As h → −∞, the free surface near the bottom of the step approaches aninfinitely thin jet.

Figures 8 and 9 illustrate subcritical free-surface flows. Figures 8(a,b) and 9(a,b) show that for agiven value of h, the amplitude of the waves increases as F increases. Similarly for a fixed value of F ,the amplitude of the waves increases as |h| increases (see Fig. 8c,d).


FIG. 14. (a) Subcritical flow for given values of F = 0.50, s = 0.10 and b = 8.38. (b) Subcritical flow for given values ofF = 0.50 and s = 0.10. The length of step, b = 8.81, came as part of the solution. (c) Subcritical flow for given values ofF = 0.50, s = −0.30 and b = 8.30. (d) Subcritical flow for given values of F = 0.50 and s = −0.30. The length of step,b = 8.85, came as part of the solution. (e) Subcritical flow for a given value of F = 0.50. The step height, s = 0.08, and lengthof step, b = 1.08, came as part of the solution. (f) Subcritical flow for a given value of F = 0.50. The step height, s = −0.29,and length of step, b = 1.35, came as part of the solution.


Finally, Fig. 10 shows hydraulic falls for h > 0. Two typical free-surface profiles for small andlarge steps are presented in Fig 10(a,b). Figure 10(c) shows values of d, d − h and h versus F .For F = 1, h = 0 and the flow reduces to a uniform stream. For F = 0, the free surface is flatand h = 1.

5.3 Free-surface flows past a flat plate

Free-surface flows past a flat plate (surfboard or sluice gate) were studied by Lamb (1945),Benjamin (1956), Frangmeier & Strelkoff (1968), Chung (1972), Vanden-Broeck & Keller (1989),Binder & Vanden-Broeck (2005), Binder & Vanden-Broeck (2007) and others. Vanden-Broeck (1996)and Binder & Vanden-Broeck (2005) showed that there are no subcritical flow with waves only onone side of the plate or flows characterized by uniform streams in the far field with F > 1 and F∗ < 1.Binder & Vanden-Broeck (2007) then showed that such solutions can be constructed by introducing an-other disturbance. In particular, they obtained subcritical waves with waves only on one side of the plate.

Here, we show in Fig. 11 that subcritical solutions with no waves in the far field can be obtained byintroducing two additional disturbances. Here, we choose for simplicity two pressure distributions cen-tered at x = −9.32 and x = 3.60. It can be shown that such distributions of pressure are modelled in theweakly non-linear theory by vertical jumps. Therefore, we need, in the weakly non-linear approxima-tion, to combine two vertical jumps and a horizontal jump (modelling the plate) with the phase portraitof Fig. 5(b). This is shown in Fig. 11(c). Here, the parameters were adjusted to eliminate the waves bothfar upstream and far downstream. Non-linear values of ηx versus η are shown in Fig. 11(b). The resultsare in qualitative agreement with those of Fig. 11(c) (the agreement is not very close because F is notclose to 1).

5.4 Free-surface flows past long submerged disturbances

We conclude the paper by showing in Figs 12–14 the effect on the solutions of increasing the length bof the submerged obstacle. Figures 12 and 13 show supercritical solutions. As can be expected some ofthe solutions for large b are essentially the superposition of two solutions past a step (one step up andone step down). For example, Fig. 13(b) and (d) are close to the superposition of two generalized criticalflows. Similarly, Fig. 13(a) is close to the superposition of two hydraulic falls. The superposition is ofcourse approximative but becomes better and better as b → ∞.

Figure 14 shows subcritical flows. Figure 14(b and d) shows that it is possible to trap the waves ontop of the disturbance. A similar property was found in Dias & Vanden-Broeck (2004) and Binder et al.(2005) for flows past two submerged disturbances.

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