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

Transcription

1 IMA Journal of Applied Mathematics (2008) 73, doi: /imamat/hxm049 Advance 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 Mathématiques et de Leurs Applications, Ecole Normale Supérieure de Cachan, UniverSud, 61 Avenue President Wilson, F 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 with rapid changes are considered. Numerical solutions are computed by a boundary integral equation method similar to that first introduced by King & Bloor (1987, J. Fluid Mech., 182, ). Free-surface flows past localized disturbances, steps and sluice gates are calculated. In addition, weakly non-linear solutions are 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 surface refers to the interface between a fluid (e.g. water) and the atmosphere (assumed to be characterized by a constant atmospheric pressure). Examples include free-surface flows generated by moving surfacepiercing objects (e.g. ships) or moving submerged objects. These flows reduce to flows past disturbances when viewed in a frame of reference moving with the objects. Other examples involve free-surface flows past an uneven channel bottom. These problems are often modelled within the framework of potential theory. Efficient numerical methods based on boundary integral equation formulations have been 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. Parau et al., 2005a,b). In this paper, we consider mainly two flow configurations. Gravity is included in the dynamic boundary condition but surface tension is neglected. The first configuration is the free-surface flow past a step in a channel (see Fig. 1a). The second one is obtained by replacing the step of Fig. 1(a) by a rectangular obstacle 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. Flows past submerged disturbances such as the one of Fig. 2(a) were studied by many previous investigators j.vanden-broeck@uea.ac.uk. Present address: Department of Mathematics, University College London, Gover Street, London WC1E 6BT, UK c The Author Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 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 Binder et al. (2005). The results presented in this paper supplement these previous investigations. Our approach follows that of Dias & Vanden-Broeck (2002), Binder et al. (2005) and Binder et al. (2006). In particular, we present new results when the height of the step is large. We also show that the structure of the various families of solutions for the flow of Fig. 2(a) is different when s > 0ors < 0. Finally, we present 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 to approach uniform streams with constant velocity U and constant depth H as x. (All the flows considered in the present paper are reversible from a mathematical point of view. From a physical point of view, some of the flows shown in the various figures ought to be reversed to be more realistic.) We define the downstream Froude number U F =, (1.1) (gh) 1/2 where g is the acceleration due to gravity. The flow as x can either be a uniform stream with constant velocity V and constant depth D h or be characterized by a train of waves. Here, h 0 in Fig. 1(a) and h = 0 in Fig. 2(a). When the flow is uniform as x, we define the upstream

3 256 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 the complex 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) and others identified four types of solutions. They are illustrated in Fig. 3 for the flow past a rectangular obstacle sketched in Fig. 2(a). The first type is a waveless supercritical flow with F = F > 1 (see Fig. 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 (see solid line in Fig. 3c). These three types of solutions have been investigated in many previous studies by 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 flow with a supercritical uniform stream (F > 1) as x and a train of waves as x (see dashed line 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 four types of flows found in Fig. 3 also exist for the flow past a step.

4 INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 257 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 and b (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 obstacle on the wavy side: η( ) = 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) and Binder & Vanden-Broeck (2005). Vanden-Broeck (1996) and Binder & Vanden-Broeck (2005) showed that the flow of Fig. 4(a) differs from that of Fig. 2(a) in the sense that only solutions of the first and fourth types exist. In other words, there are no subcritical flow with waves only on one side of the gate or flows characterized by uniform streams far upstream and far downstream with F > 1 and F < 1. Such flows 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 boundary integral equation methods used to compute non-linear solutions are described in Section 3. Analytical methods based on a weakly non-linear theory are summarized in Section 4. The numerical results are described in Section 5.

5 258 B. J. BINDER ET AL. FIG. 4. (a) Sketch of flow past a sluice gate 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). 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 by the bottom of the channel A B C D and above by the free surface AB. The equations of the bottom of the 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 2 U 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 the conditions u U, v 0, η (x ) 0, as x, (2.2) to evaluate the Bernoulli constant on the right-hand side of (2.1).

6 INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 259 The mathematical problem is then to seek the complex velocity u iv as an analytic function of z = x + iy in the fluid domain, satisfying (2.1), (2.2) and the kinematic boundary conditions v = u dη, on AB, (2.3) dx 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 the dynamic boundary condition (2.1) only holds on AB and CDand that the kinematic boundary condition on BC gives v = u tan σ c. (2.8) 3. Boundary integral equation King & Bloor (1987) derived a boundary integral equation method to solve the flow configuration of Fig. 1(a). Here, we present a similar method which applies to arbitrary bottom shapes consisting of straight segments. As we shall see, the method also works if a portion of the free surface is replaced by a flat plate (sluice gate or surfboard) (see Fig. 4a). For the sake of clarity, we first present the method for the particular flow configurations sketched in Figs 1(a) and 2(a). The numerical procedure is derived as a 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) and Binder & Vanden-Broeck (2007) to compute flows past submerged obstacles, steps and flat plates. Some of 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 velocity. 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 + v 2 ) + 1 F 2 y = , on y = 1 + η. (3.1) F2 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 = 1on the channel bottom streamline. We also choose φ = 0atC in Fig. 1(a) and D in Fig. 2(a). We denote by φ = φ b the value of φ at the point B in Fig. 1(a). Similarly, we denote the values of φ at the corners B, C, E and F in Fig. 2(a) by φ 1,φ 2,φ 3 and φ 4, respectively. In the complex potential plane, the fluid is in the strips 1 <ψ<0and <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 = e π f. (3.2)

7 260 B. J. BINDER ET AL. 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) is α b = e πφ b. The values of α at the points B, C, E and F of Fig. 2(c) are α 1 = e πφ 1,α 2 = e πφ 2, α 3 = e πφ 3 and α 4 = e πφ 4, respectively. The aim is now to derive integral equation relations that only 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 = e τ iθ (3.3) and we apply Cauchy s integral formula to the function τ iθ in the ζ -plane with a contour consisting of the α-axis and a semicircle of arbitrary large radius in the lower half-plane. Since (2.2) implies that τ iθ 0as ζ, there is no contribution from the semicircle and we obtain after taking the real part τ(α) = 1 θ(α 0 ) π α 0 α dα 0. (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 Cauchy principal value. Next we note that the kinematic boundary conditions ( ) imply θ(α) = 0, on α b <α<0 and α< 1, (3.5) θ(φ) = π 2, for 1 <α<α b, (3.6) θ(α) = 0, for α<α 4,α 3 <α<α 2 and α 1 <α<0, (3.7) θ(α) = π 2, for α 2 <α<α 1, (3.8) θ(α) = π 2, for α 4 <α<α 3. (3.9) Substituting (3.5) and (3.6) into (3.4) we obtain the following relation between τ and θ on the free surface AB of Fig. 1 τ(α) = 1 2 ln α b α + 1 θ(α 0 ) 1 + α π 0 α 0 α dα 0. (3.10) Similarly, substituting ( ) into (3.4) we obtain on the free surface AB of Fig. 2 the relation τ(α) = 1 2 ln α 1 α α α ln α 3 α α α θ(α 0 ) π 0 α 0 α dα 0. (3.11) Since α>0on the free surface AB, we can rewrite (3.10) and (3.11) in terms of φ by using the change of variables α = e πφ, α 0 = e πφ 0. (3.12) This yields τ(φ) = 1 2 ln α b e πφ 1 + e πφ θ(φ 0 )e πφ 0 + e πφ 0 e πφ dφ 0 (3.13)

8 INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 261 for the flow of Fig. 1 and τ(φ) = 1 2 ln α 1 e πφ e πφ α ln α 3 e πφ e πφ α 4 + θ(φ 0 )e πφ 0 e πφ 0 e πφ dφ 0 (3.14) for the flow of Fig. 2. In (3.13) and (3.14), τ(φ) = τ(e πφ ) and θ(φ) = θ(e πφ ). 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 φ x(φ) = x( ) + e τ(φ0) cos θ(φ 0 )dφ 0, (3.16) φ y(φ) = 1 + e τ(φ0) sin θ(φ 0 )dφ 0. (3.17) The dynamic boundary condition (3.1) and (3.3) give e 2τ(φ) + 2 F 2 y(φ) = F 2. (3.18) Equations (3.13), (3.17) and (3.18) define an integro-differential equation for the unknown function θ(φ) on the free surface AB of Fig. 1. Similarly, (3.14), (3.17) and (3.18) define an integro-differential on the free surface AB of Fig. 2. These equations can be solved by using the numerical procedure described in Binder et al. (2005). Once they have been solved, (3.16) and (3.17) give the shape of the free surface. We conclude this section by showing that the numerical method also applies if a portion of the free surface is replaced by a flat plate (see Fig. 4a). Since the plate is inclined at the angle σ c, one has θ = σ c, for φ b <φ<φ c, (3.19) where φ b and φ c are the values of φ at the end points B and C of the plate. Assuming a flat bottom, it can easily be shown that (3.13) or (3.14) is replaced by τ(φ) = σ c π ln eπφc e πφ φb e πφ b e πφ + θ(φ 0 )e πφ 0 e πφ 0 e πφ dφ 0 + φ c θ(φ 0 )e πφ0 e πφ 0 e πφ dφ 0. (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 in Binder & Vanden-Broeck (2005). 4. Weakly non-linear theory The number of independent parameters needed to obtain a unique solution to a free-surface problem is often not obvious. There are two natural ways to find it. The first one is by careful numerical experimentation (fixing too many or too few parameters fails to yield convergence). The second one is to perform

9 262 B. J. BINDER ET AL. a weakly non-linear analysis in the phase plane. This second approach has the advantage of allowing a systematic determination of all the possible solutions (within the range of validity of the weakly nonlinear analysis). In all the examples presented in this paper, we checked that both approaches lead to the same number of independent parameters. Several investigators (see, e.g. Shen, 1995; Dias & Vanden-Broeck, 2002; Binder et al., 2005) have derived a forced Korteweg de Vries equation to model the flow past an obstacle at the bottom of a channel. They showed that the forcing can be approximated by a jump in η x. Therefore, one writes η xx η2 6(F 1)η = 0, for x x t, with the vertical jump condition η x (x t + ) η x (xt ) = δ. (4.1) Here, x t denotes the position of the obstacle and δ is related to the size of the disturbance. For the rectangular obstacle of Fig. 2(a), δ = 3sb where s and b are the height and the length of the rectangle. We note that δ>0when s > 0 and δ<0when s < 0. Binder & Vanden-Broeck (2005) derived the corresponding weakly non-linear solutions for flows past a sluice gate or surfboard (the numerical procedure to compute the corresponding fully non-linear solutions was outlined at the end of Section 3). They showed that the flow is described on the portions of free surface on the right and left of the gate by the Korteweg de Vries equation (KdV equation) η xx η2 6(F 1)η = 0, (4.2) with the conditions η r η l = L sin σ c, (4.3) η r x = ηl x = tan σ c. (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 of the plate and σ c is the inclination (see Fig. 4a). Since η r x = ηl x, (4.3) and (4.4) imply that the gate is represented in the phase plane by a horizontal segment of length L sin σ c. We refer to that segment as a horizontal jump. In the absence of disturbances, the KdV equation (4.2) holds for all x. The corresponding solutions are shown in the phase plane η x versus η in Fig. 5. To construct weakly non-linear solutions for flows past disturbances, we need to combine the trajectories of the phase plane of Fig. 5 with both vertical jumps (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 the step 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. Solutions are 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 properties 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, we summarize some of our findings for free-surface flows under a sluice gate. More results for submerged obstacles are shown in Section 5.4.

10 INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 263 FIG. 5. Weakly non-linear phase portraits, dη/dx versus η, h = 0. (a) Supercritical flow, F > 1. There is a saddle point at η = 0,η x = 0 and a centre at η = 4/3(F 1), η x = 0. The inner closed trajectories are periodic solutions. The closed trajectory is the solitary wave. The maximum value of η x 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), η x = 0 and a centre at η = 0,η x = 0. The inner closed trajectories are periodic solutions. The closed trajectory is the solitary wave. The maximum value of η x at η = 0 on the solitary wave trajectory is η m = 4/3 2(1 F) 3/ 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 existence can 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) with the downward vertical jump (4.1) modelling the obstacle. We start at the saddle point η = η x = 0

11 264 B. J. BINDER ET AL. and move clockwise on the solitary wave trajectory. We then have a vertical downward jump. There are several 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) and we then return to the saddle point η = 0 along the solitary wave trajectory. If 3δ <2η m, the solution can either jump back on the solitary wave trajectory or jump on a periodic wave solution. In the former case, there are two possibilities: one to the left and one to the right of η = 4/3(F 1). They are illustrated in Fig. 3(a). In the latter case, there are infinitely many solutions. They were first discovered by Dias & Vanden-Broeck (2002), who called them generalized critical solutions. An example is shown in Fig. 3(c) (broken curve). If 3δ = η m, the wavelength of the periodic wave vanishes and we obtain a hydraulic fall. A profile is shown in Fig. 3(c) (solid curve). We now consider the case F < 1. Since η = η x = 0asx, the solution has to end at the origin in Fig. 5(b). There are three possibilities. The first is to start at the saddle point, move clockwise on the solitary wave and jump vertically to the origin along η = 0. This is the hydraulic fall solution that we have 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 = 0.20 and b = (b) Subcritical flow for given values of F = 0.85, s = 0.05 and b = (c) The solid curve is for a hydraulic fall for given values of F = 1.10 and b = The step height, s = 0.11, came as part of the solution. The broken curve is for a generalized hydraulic fall for given values of F = 1.10, s = 0.11, b = 0.49 and elevation η( ) = 0.10.

12 INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 265 FIG. 7. Supercritical flow. (a) Profiles for a step height of h = The values of the Froude number for the curves, from top to bottom, are F = 1.175, F = 1.20, F = 1.50 and F = The corresponding values of the uniform depth far upstream are d = 1.13, d = 1.08, d = 1.04 and d = The corresponding values of the upstream Froude number are F = 1.04, F = 1.08, F = 1.46 and F = (b) Profile for a step height of h = 2.35 and value of the Froude number F = The uniform depth upstream is d = 3.48 and the upstream Froude number is F = (c) Profile for a step height of h = 5.00 and value of the Froude number F = The uniform depth upstream is d = 6.34 and the upstream Froude number is F = (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, (e) Profile for a step height of h = 3.85 and value of the Froude number F = The uniform depth upstream is d = 0.56 and the upstream Froude number is F = *b, c and e have the same vertical and horizontal scale.

13 266 B. J. BINDER ET AL. FIG. 8. Subcritical flow for h > 0. (a) Profiles for a step height of h = The value of the Froude number for the solid curve is F = The value of the Froude number for the dash curve is F = The value of the Froude number for the dot dash curve is F = (b) Profiles for a step height of h = The value of the Froude number for the solid curve is F = The value of the Froude number for the dash curve is F = The value of the Froude number for the dot dash curve is F = (c) Profile for a step height of h = The value of the Froude number is F = (d) Profile for a step height of h = The value of the Froude number is F = FIG. 9. Subcritical flow for h < 0. (a) Free-surface profiles for a step height of h = The value of the Froude number for the solid curve is F = The value of the Froude number for the dash curve is F = The value of the Froude number for the dot dash curve is F = (b) Free-surface profiles for a step height of h = The values of the Froude number from the bottom to the top curves are F = 0.10, F = 0.30 and F = 0.50.

14 INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 267 FIG. 10. Subcritical hydraulic falls for F < 1 and h > 0. (a) Free-surface profile for a value of the Froude number F = The value of the upstream Froude number is F = The step height is h = 0.17 and d = (b) Free-surface profile for a value of the Froude number F = The value of the upstream Froude number is F = The step height is h = 0.67 and d = (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 h versus 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 analysis to identify all the possible solutions. The weakly non-linear analysis gave us also the number of parameters needed to determine uniquely a solution. For example, when F > 1, we first fix F (this defines 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 the solid curve in Fig. 3(c), δ is given, so the size of the obstacle cannot be assigned. Therefore, this solution depends on one parameter. For the broken curve in Fig. 3(c), we start on the solitary wave trajectory and jump on a periodic solution. We can then choose δ and the particular periodic trajectory we want to

15 268 B. J. BINDER ET AL. FIG. 11. Subcritical flow past a two-pressure distribution and an inclined plate for a given value of F = (a) Fully non-linear free-surface profile for L = 1.92, σ c = 5.4. (b) Values of dy/dx = tan(θ) versus y 1 = η, showing the fully non-linear phase trajectories 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 profiles are in good agreement with fully non-linear computations when F is close to 1 and when the size of the obstacle is small. In Fig. 3, we presented only fully non-linear solutions computed by the boundary integral equation of Section 3. The reader interested in explicit weakly non-linear free-surface profiles is refered to Binder et al. (2005). When s < 0, the obstacle is replaced by a depression in the bottom (see Fig. 6). The weakly nonlinear analysis proceeds as in the case s > 0, except that the vertical jumps are now upwards. Proceeding as 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 = η x = 0 in Fig. 5(a) and move downwards along the trajectory in the third quadrant of Fig. 5(a). We then jump vertically upwards to the corresponding trajectory in the second quadrant and come back to the origin along it.

16 INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 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 the flow past a step (Fig. 1). Here, we supplement their findings by presenting fully non-linear solutions for large steps. We concentrate on three of the four types, namely the supercritical flows, the subcritical flows and the hydraulic falls. Figure 7 shows fully non-linear supercritical solutions. The supercritical solutions are characterized by uniform streams at infinity with F > 1 and F > 1. Figure 7(a) shows the effect of increasing F for a given value h > 0. As F, F and the solution approaches a free streamline solution with a constant velocity 1 on the free surface. Figure 7(b and c) shows similar profiles for larger steps. Since F 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, solutions exist only for F greater than a critical value F which depends on h. This is illustrated in Fig. 7(d) (the top points on each of the curves correspond to values of F close to F). As F F, F 1. The top profile 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 = (b) Supercritical flow for given values of F = 1.20, s = 0.03 and b = (c) Supercritical flow for given values of F = 1.10 and b = The free surface was forced flat at x = 0(θ(0) = 0). The step height, s = , came as part of the solution. (d) Supercritical flow for given values of F = 1.10, s = 0.07 and b = 5.48.

17 270 B. J. BINDER ET AL. FIG. 13. (a) Supercritical flow for given values of F = 1.10 and b = The free surface was forced flat at x = 0(θ(0) = 0). The step height, s = , came as part of the solution. (b) Supercritical flow for given values of F = 1.10, b = and s = (c) Supercritical flow for given values of F = 1.10, s = 0.01 and b = (d) Supercritical flow for given values of F = 1.10, s = 0.05 and b = mass and Bernoulli equation to derive the exact non-linear relations 2d F 2 (d h)2 (1 + 2F 2 ) (d h) = 0, (5.1) ( ) 1 3/2 F = F. (5.2) d h The numerical results of Fig. 7(d) were found to be in close agreement with the exact results predicted by (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 typical solution is shown in Fig. 7(e). As h, the free surface near the bottom of the step approaches an infinitely thin jet. Figures 8 and 9 illustrate subcritical free-surface flows. Figures 8(a,b) and 9(a,b) show that for a given 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).

18 INFLUENCE OF RAPID CHANGES ON FREE-SURFACE FLOWS 271 FIG. 14. (a) Subcritical flow for given values of F = 0.50, s = 0.10 and b = (b) Subcritical flow for given values of F = 0.50 and s = The length of step, b = 8.81, came as part of the solution. (c) Subcritical flow for given values of F = 0.50, s = 0.30 and b = (d) Subcritical flow for given values of F = 0.50 and s = The length of step, b = 8.85, came as part of the solution. (e) Subcritical flow for a given value of F = The step height, s = 0.08, and length of step, b = 1.08, came as part of the solution. (f) Subcritical flow for a given value of F = The step height, s = 0.29, and length of step, b = 1.35, came as part of the solution.

19 272 B. J. BINDER ET AL. Finally, Fig. 10 shows hydraulic falls for h > 0. Two typical free-surface profiles for small and large 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 flat and h = 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 on one 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 another 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 by introducing two additional disturbances. Here, we choose for simplicity two pressure distributions centered at x = 9.32 and x = It can be shown that such distributions of pressure are modelled in the weakly non-linear theory by vertical jumps. Therefore, we need, in the weakly non-linear approximation, to combine two vertical jumps and a horizontal jump (modelling the plate) with the phase portrait of Fig. 5(b). This is shown in Fig. 11(c). Here, the parameters were adjusted to eliminate the waves both far upstream and far downstream. Non-linear values of η x versus η are shown in Fig. 11(b). The results are in qualitative agreement with those of Fig. 11(c) (the agreement is not very close because F is not close to 1). 5.4 Free-surface flows past long submerged disturbances We conclude the paper by showing in Figs the effect on the solutions of increasing the length b of the submerged obstacle. Figures 12 and 13 show supercritical solutions. As can be expected some of the solutions for large b are essentially the superposition of two solutions past a step (one step up and one step down). For example, Fig. 13(b) and (d) are close to the superposition of two generalized critical flows. Similarly, Fig. 13(a) is close to the superposition of two hydraulic falls. The superposition is of course 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 on top of the disturbance. A similar property was found in Dias & Vanden-Broeck (2004) and Binder et al. (2005) for flows past two submerged disturbances. REFERENCES ASAVANANT, J.& VANDEN-BROECK, J.-M. (1994) Free-surface flows past a surface-piercing object of finite length. J. Fluid Mech., 273, BENJAMIN, B. (1956) On the flow in channels when rigid obstacles are placed in the stream. J. Fluid Mech., 1, BINDER, B. J., DIAS, F.& VANDEN-BROECK, J.-M. (2005) Forced solitary waves and fronts past submerged obstacles. Chaos, 15, BINDER, B. J., DIAS, F.& VANDEN-BROECK, J.-M. (2006) Steady free-surface flow past an uneven channel bottom. Theor. Comp. Fluid Dyn., 20, BINDER, B. J.& VANDEN-BROECK, J.-M. (2005) Free surface flows past surfboards and sluice gates. Eur. J. Appl. Math., 16,

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