Stability of gravity-capillary waves generated by a moving pressure disturbance in water of finite depth

Transcription

1 Stability of gravity-capillary waves generated by a moving disturbance in water of finite depth Roger Grimshaw 1, Montri Maleewong 2, and Jack Asavanant 3 1 Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK 2 Department of Mathematics, Kasetsart University, Bangkok 19, Thailand 3 Department of Mathematics, Chulalongkorn University, Bangkok 13, Thailand Abstract In previous work, we investigated two-dimensional steady gravity-capillary waves generated by a localized distribution moving with constant speed U in water of finite depth h. Localized solitary waves can only exist in subcritical flows where the Froude number F = U/(gh) 1/2 < 1, and were found using a combination of numerical simulations of the fully nonlinear inviscid, irrotational equations, and analytically from a weakly nonlinear long-wave model, the steady forced Korteweg-de Vries equation. The solution branches depended on three parameters, the Froude number, F < 1, the Bond number, τ > 1/3, and the magnitude and sign of the distribution, ǫ. In this paper, we examine the two-dimensional stability of these waves using numerical simulations of the fully nonlinear unsteady equations. The results are favourably compared with analogous numerical solutions of the unsteady forced Korteweg-de Vries equation. We find that, for ǫ >, the small-amplitude steady depression wave is stable whereas the large-amplitude steady depression wave is unstable. The depression wave with a dimple at its crest, which occurs only when ǫ < is unstable, but the small-amplitude elevation wave with ǫ < is stable. Key words: gravity-capillary, depression wave, elevation wave, fkdv, stability 1 Introduction Two-dimensional surface waves generated by an applied localized distribution in the presence of a uniform stream have received much attention over many decades, especially for pure gravity waves when the effects of surface tension are ignored. The key parameters describing the generated waves are then the Froude number F = U/(gh) 1/2 and the magnitude of applied ǫ. Here U is the velocity of uniform stream, h is the undisturbed water depth and g is the acceleration of gravity. When F is not close to unity, linearized theory gives a good prediction for small amplitude wave, but the linear model breaks down as F 1. In this limit, Akylas [1] developed a weakly nonlinear theory to study the resonant generation of surface gravity waves, analogous to related models of transcritical flow over a localized obstacle (see [9] [13] [21], for instance). This weakly nonlinear theory is based on the forced Korteweg-de Vries equation (fkdv), and complements numerical solutions of the fully nonlinear equations (see [6] [18], for instance). 1

2 When surface tension is included, there is an additional parameter, the Bond number τ = T/ρgh 2, where T is the coefficient of surface tension and ρ is the water density. The critical value of τ determined from linear theory is τ = 1/3, and solitary waves of the KdV-type can only arise when either τ =, or τ > 1/3, which is the case of interest in this paper. The appropriate weakly nonlinear model is then the fkdv equation, and several branches of steady solutions have been found, see our previous work Maleewong, Asavanant and Grimshaw [15] (denoted MAG hereafter) and the references therein. We will present a brief summary of our findings in section 4.1 below. Although not the subject of this paper, we note that when τ < 1/3, the situation is more complicated. Steady depression and elevation solitary wave solutions are then found in envelope form,and the forced nonlinear Schrödinger equation (fnls) is a suitable weakly nonlinear model. Many steady solutions have been found, see [4] for water of infinite depth, and [16] for water of finite depth. There have been several studies of the stability of free steady gravity-capillary solitary waves. For the present case when τ > 1/3, it is known that the waves are stable to two-dimensional long-wave perturbations, see Haragus and Scheel [11] for a rigorous linear stability analysis, but it is unstable to three-dimensional perturbations, see Groves et al [1]. On the other hand, for the case τ < 1/3, it would seem that only asymptotic and numerical results are available which suggest that elevation waves are unstable and depression waves are stable in a two-dimensional configuration, see Calvo and Akylas [4], but both are unstable at small amplitudes in a three-dimensional setting, see Akylas and Cho [2] and the references therein. The purpose of this present work is to investigate the stability of steady gravitycapillary waves in water of finite depth when τ > 1/3. We study the stability of steady depression and elevation waves found in MAG by considering the unsteady initial-value problem in two scenarios. First, we solve numerically the unsteady fkdv equation using a pseudo-spectral method, to investigate the stability of the steady solutions of that equation. This approach is similar to the studies of the fkdv equation by Akylas [1] in the absence of surface tension ( τ = ), or by Zhu [22] when τ > 1/3, but here we use a different initial condition; specifically we use a small perturbation of the steady wave as the initial condition, instead of the zero initial condition used in most previous studies. Second, we solve numerically the fully nonlinear unsteady two-dimensional Euler equations, using an accurate boundary integral equation method. Again, the simulations are initialized with the a small perturbation imposed on the known steady solution. Overall, we find good agreement between these two approaches, except that wave breaking may occur in the fully nonlinear model for the unstable cases. Note that our use of the term stability in this paper refers always to the (numerical) solution of an appropriate initial-value problem, whereas a more common and rigorous use would be to solve the eigenvalue problem for the growth rate, formed by linearization of the unsteady equations about the known steady solution. This is the approach used, for instance, by Camassa and Wu [5] in a study of the stability of some steady solutions of the fkdv equation for the case of τ =. In section 2, we describe the formulation of the fully nonlinear two-dimensional unsteady problem on the one hand, and the weakly nonlinear fkdv model on the other hand. In section 3, we provide some details of the numerical methods to be used, 2

3 namely a boundary integral method for the fully nonlinear unsteady equations, and a pseudo-spectral method for the unsteady fkdv equation. Our numerical results and are described in section 4. We conclude with a discussion in section 5. 2 Formulation 2.1 Fully nonlinear unsteady equations For inviscid, incompressible two-dimensional flow above a rigid boundary, the governing equations in the fluid domain D are φ y 2 φ =, in D, (1) =, on y = h. (2) Here φ is a velocity potential such that the velocity field is (u, v) = (φ x, φ y ). The free surface conditions are then expressed in a semi-lagrangian form. Thus, we let X = (x(α, t), y(α, t)) represent a point on the free surface, where α is a label for a fluid particle. This is a parametric representation of the free surface, where elimination of α yields the equation of the free surface as y = η(x, t). Then the kinematic and dynamic boundary conditions on the free surface can be written as ( Dx Dt, Dy ) Dt = DX Dt = φ, (3) Dφ Dt = 1 2 (φ2 x + φ 2 y) gy p ρ + TK ρ, (4) where D/Dt = / t + φ. (5) Here D/Dt is the material time derivative, T is the surface tension coefficient and K is the curvature of the free surface, while p(x, t) is an applied localized distribution moving to the left with a constant speed U. The fluid flow is at rest with an undisturbed depth h at the initial time t =. Then the is suddenly applied to the free surface for all time t >. Note that this problem can also be constructed as a fixed applied distribution with a uniform flow of speed U maintained at infinity. The sketch of the physical plane is shown in Fig. 1. The governing equations (1)-(4) can be written in nondimensional form by introducing h as the length scale, the speed U of the applied as a velocity scale, and the time scale is then h/u. We obtain 2 φ = in D, (6) φ y =, on y = 1, (7) 3

4 U y x h D h Figure 1: Physical plane ( Dx Dt, Dy ) Dt = DX Dt = φ, (8) F 2Dφ Dt = F 2 2 (φ2 x + φ2 y ) y F 2 p + τk. (9) Here F = U/ gh is the Froude number, τ = T/ρgh 2 is the Bond number, and p = p/ρu 2 is the moving distribution. In the present work, we will consider the applied distribution in the form of a localized function. Thus, for all the results shown here, we set p(x + t) = ǫ sech 2 (x + t) (1) where ǫ is the magnitude of the applied. Our simulations were repeated for the case when p(x + t) = ǫ sech 4 (x + t), with completely analogous results. As noted in MAG, the forcing (1) is analytically exceptional in that it allows steady solutions for F > 1, as well as the expected steady solutions for F < 1, which for any compact and fully localized forcing expressions is clearly forbidden. The reason for this anomaly is that a sech 2 -profile (unlike the sech 4 -profile) retains the KdV balance between nonlinearity and dispersion into the far-field, at least analytically. However, in our numerical solutions, the forcing expression (1) becomes effectively zero in the far-field, and hence numerically no steady solutions with F > 1 can then be found. 2.2 Forced Korteweg-de Vries equation In our previous paper MAG [15], we considered the steady problem, and as discussed in the Introduction, the main issue in this present paper is whether those steady solutions are stable, and can be reached in the long-time limit. Note that with respect to the 4

5 present formulation, the steady problem of MAG is obtained by putting ˆx = x + t, ŷ = y, ˆt = t, and ˆφ = x + φ, (11) and then requiring that ˆφ/ ˆt =. In MAG, our main interest was in the transcritical regime F 1, when the waves produced have long wavelengths and in the weakly nonlinear regime could be described by the steady forced Korteweg-de Vries equation. The derivation in MAG is readily extended to the unsteady forced KdV equation (fkdv), given in the moving-frame coordinates by 2ηˆt + (F 2 1)ηˆx 3 2 (η2 )ˆx + (τ 1 3 )ηˆxˆxˆx = F 2 pˆx (ˆx). (12) 3 Numerical methods 3.1 Fully nonlinear unsteady equations In the past three decades, boundary integral methods have been developed in order to study two-dimensional unsteady free surface flows, espectially the formation of steep waves. The pioneering work by Louguet-Higgins and Cokelet [14] was followed by the works of Vinje and Brevig [19], New, McIver and Peregrine [17], Cooker [7], and Dold [8] amongst many others. In this work, we apply the numerical method used by Cooker [7] to solve numerically the system of nonlinear equations (6)-(9). Here, gravity, surface tension and the applied distribution effects are all included in the dynamic boundary condition. Γ 1 h Γ 2 Γ 4 h Γ 3 Figure 2: Reflected physical domain Let P(s, t) = x(s, t)+iy(s, t) be the position of a point on the free surface. Note that t represents time and s is a parameter indicating the location of a grid point. To satisfy the bottom boundary condition, we reflect the flow domain D with respect 5

6 to the bottom line (see Fig. 2 ). The complex velocity is defined by w(z) = φ x iφ y where z = x + iy. We can then express w on the free surface in terms of tangential ( s ) and normal ( n ) components to the free surface as w = (φ s iφ n ) P s P s 2, (13) where represents the complex conjugate. Applying the Cauchy integral formula to w(z) where z is a point on the boundary, yields w(z) = 1 U(z ), (14) πi z z dz where the contour C is composed of Γ 1 Γ 2 Γ 3 Γ 4 (see Fig. 2), and t he integral (14) is in principle value form. Substituting (13) into (14) at the point z = P, we get (φ s iφ n ) P s = 1 P s 2 πi C C (φ s iφ n ) z P P P s 2 dz. (15) Then, on changing the variable z to s, so that z = P (s ), dz = P s )ds, (s Ps P s/ P s 2 = 1, and multiplying (15) by ip s, we obtain iφ s + φ n = 1 π b a (φ s iφ n)p s ds, (16) P P where a and b are the limits on the range of s describing the contour C. The flow is assumed to be at rest as x ±, so w(p) is zero when the point P is on the vertical boundaries Γ 2 and Γ 4. Hence the main contribution of the contour integral (16) comes from the line integrals Γ 1 and Γ 3, and taking the real part, we obtain φ n = 1 π + 1 π [ { φ s Re Ps } { + φ P n P Im Ps P P { [φ s Re P } { s φ n P 2ih P Im }] ds P s P 2ih P }] ds. (17) For a given initial shape of the free surface, we know φ, P, P at the discretization points s i, i = 1...N. The derivatives of P and φ along the free surface P s and φ s can then be calculated. Thus the integral equation (17) involves the unknown φ n at each point on the free suface. The first term on the right-hand side of (17) has a singular integrand, which is treated by subtracting the quantity, φ s s s ds =. The resulting integrals are then discretized using a trapezoidal rule with the outcome that the integral equation (17) is converted to the matrix system 6

7 πφ n (s) = φ ss (s) + N A(s, s )φ s (s ) + s =1 { A(s, s Ps ) = Re P P + { A(s, s Pss ) = Re + 2P s { B(s, s Ps ) = Im P P { B(s, s Pss ) = Im 2P s N B(s, s )φ n (s ), s = 1, 2,..., N, (18) s =1 P } s, if s s, P 2ih P P } s, if s = s, P 2ih P P } s, if s s, P 2ih P P } s, if s = s. P 2ih P For a given φ, we approximate φ s and φ ss using a six-point Lagrange polynomial. The linear matrix system (18) involves only the unknown φ n at each point along the free surface, and is solved iteratively. This completes the step of solving the integral equation. Next, we march the numerical solutions forwards in time by solving (8) and (9). These equations can be regarded as first-order ordinary differential equations in t of the form dw = f(t, w), dt which is solved numerically by a standard fourth-order predictor-corrector method, Applying the predictor-corrector method to equations (8) and (9) means that we have to solve the integral equation for φ/ n twice for each time step. Because this method requires information from the three previous time steps, we used a single step method such as a fourth order Runge-Kutta method to make the first three time steps from the initial conditions. The overall steps can be summarized as follows. First, given the initial velocity and initial profile of the free surface at time t, we find φ(s, t) and P(s, t), and then use them to approximate φ s and φ ss from a five-point interpolation formula. Then we solve iteratively the linear system (18) to find φ n (s, t). Next, calculate w(p) from (13). Then, calculate DP/Dt from (8) and calculate Dφ/Dt from (9). Finally, go forward in time by applying the predictor-corrector method to obtain P(s, t+ t) and φ(s, t + t),and then repeat the process until reaching the final time. Note that in this present problem we restrict attentions to the Bond number dominated case, τ =.4. For this parameter value, our presented numerical method can give accurate results. For a case of smaller τ, however, this numerical method still can be used to obtain the evolution of the free surface, see for instance [12]. But the numerical solutions will diverge rapidly for the case of very small τ due to capillary effects in the dynamic boundary condition. In this case a more efficient method presented by [3] can be applied to solve the problem. It is based on a Fourier boundary integral 7

8 method that approximates space derivatives by using a Fourier spectral method. In general, this method is stable espectially for the case of very small τ. 3.2 Forced Korteweg-de Vries equation We solve (12) numerically with the applied forcing (1) by using a combination of a finite difference scheme and the Fourier pseudo-spectal method. The time derivative is discretized by using a fourth-order Runge-Kutta routine. The spatial terms are then evaluated by a pseudo-spectral method, that is we introduce the approximate solution η(ˆx, ˆt) = N k= N a k (ˆt)Φ k (ˆx), (19) where Φ k (ˆx) = exp (ikˆx) are the Fourier exponentials, and a k (ˆt) are the coefficients to be determined; note that a k = ā k on order to keep η real-valued. Then, let ηj n denotes the value of η at grid point ˆx j and time ˆt n. To find ηj n, we summarize the main steps as follows. First, given ηj n = η(ˆx j, ˆt n ), we evaluate a n k = a k(ˆt n ) from (19). using the Fast Fourier Transform (FFT) algorithm. The derivatives are easily obtained in spectral space, for instance ηˆx becomes ika k. The nonlinear term is found similarly. Finally, given F, τ and p(ˆx), we finally calculate η n+1 j at ˆx = ˆx j and ˆt = ˆt n Resolution We solved the unsteady fkdv equation using the Fourier spectral method described in section 3.2, with a time step of t =.4. The x domain is 25 x 25 where the distribution is applied at the origin. We checked the resolution by comparing two cases, N = 128, N = 256. The corresponding numerical solutions for F =.9 and ǫ =.1 at t = 4 are shown in Fig. 3, and exhibit mesh independence. Thus we set N = 128 is in all the simulations reported here. As described in section 3.1, we solved the unsteady fully nonlinear equations using a mixed Eulerian-Lagrangian formulation, based on an accurate boundary integral method, using a time-step t =.2. Initially, the distribution is applied at the origin, and then moves to the left with a speed F. Thus, where the x domain is truncated depends on the simulatation time, and is determined so that the free surface is zero far upstream and downstream, as required by the boundary conditions in the formulation. Fig. 4 shows the fully nonlinear results when F =.9 and ǫ =.1 for three cases of mesh spacing E. These are E =.2,.3,.4, corresponding to the number of mesh points N which are 6, 8, 12 respectively. We see that the results are identical, and so we set E =.3 in all the simulations reported here. 8

9 F =.9, τ =.4, ε =.1, t = η N = 128 N = x Figure 3: fkdv solutions in case of F =.9 and ǫ =.1.2 F =.9, ε =.1, t = E =.2 E =.3 E = Figure 4: Fully nonlinear solutions in case of F =.9 and ǫ =.1 4 Results 4.1 Steady forced Korteweg-de Vries equation We recall that the steady solutions were obtained in MAGby solving the steady forced KdV equation, which is (see MAG or (12) with the time derivative term omitted). (F 2 1)η 3 2 η2 + (τ 1 3 )η xx = F 2 p. (2) We consider the distribution in the form (1). For τ =.4, the steady depression wave branches obtained by solving (2) are shown in Fig. 5, and the elevation wave branches are shown in Fig. 6. 9

10 τ = ε =.1 ε =..1 4 Amplitude ε = F Figure 5: Branches of steady depression wave solutions obtained when τ = τ = ε =.1.5 Amplitude F Figure 6: Branches of steady elevation wave solutions obtained when τ = Unsteady forced Korteweg-de Vries equation We next describe some simulations using the unsteady fkdv equation (12), obtained with the numerical scheme described in section 3.2. Two cases of initial data are considered. These are (i) Uniform flow, that is η(x, t = ) =, and (ii) The steady solution found previously in subsection 4.1, perturbed by random noise. In case (1), there will be no steady solution if F 2 1, as discussed in Grimshaw and Smyth [9]. The precise criterion depends on the forcing, but for a sufficiently wide forcing function Grimshaw and Smyth [9] find that there are no steady solutions emerging from a zero initial condition when (F 2 1) 2 < 3 F 2 ǫ, 1

11 t t On the other hand, it can be shown that in the same limit (wide forcing), the condition for a localized steady solution to exist is that since τ > 1/3, F < 1 and (F 2 1) 2 > 3 F 2 ǫ. The time evolution of the free surface profiles corresponding to the steady cases labeled 1-4 in Fig. 5 are shown in Figs 7-1 respectively, while that for the case labeled 5 in Fig. 6 is shown in Fig. 11. Here we have used the steady solutions as the initial condition, allowing the numerical discretization error to provide a small perturbation. Essentially identical results are obtained when the perturbation is imposed by random noise F =.9, τ =.4, ε =.1.63 η x Figure 7: Time evolution of free surface by unsteady fkdv, label F =.9, τ =.4, ε =.1.26 η x Figure 8: Time evolution of free surface by unsteady fkdv, label 2 From these simulations, we infer that in the framework of the unsteady fkdv equation (12) that the steady localized solutions which can be characterized as perturbations of a uniform flow are stable for both depression (label 1, Fig. 7) and elevation (label 5, 11

12 t t F =.9, τ =.4, ε =.1.27 η x Figure 9: Time evolution of free surface by unsteady fkdv, label 3 F =.947, τ =.4, ε = η x Figure 1: Time evolution of free surface by unsteady fkdv, label 4 Fig. 11) waves. On the other hand, the steady solutions which for both ǫ > (label 2, Fig. 8) and when ǫ < (label 3, Fig. 9), can be characterized as perturbations of a free solitary wave are unstable. In the case of the solution with label 2 (Fig. 8), we see that a large solitary wave is generated and propagates downstream, leaving behind a very small steady solitary wave, which is just the solution with label 1. This an illustration of the well-known result that there is an exchange of stability at the turning point on this branch with ǫ > in Fig. 5. For ǫ <, the solitary wave without a dimple (label 3, Fig. 9) oscillates in the zone of applied for a finite time, emitting a small amount of radiation, and with a slightly increasing oscillation amplitude; eventually a large amplitude solitary wave separates and moves downstream. The solitary wave with a dimple (label 4, Fig. 1) is initially stable, but after a finite time again a large amplitude solitary wave separates and moves downstream leaving behind a solitary wave similar to that with label 3; that is, its amplitude oscillates and a small amount of radiation is emitted; presumably eventually it will collapse and generate another large amplitude solitary 12

13 t F =.9, τ =.4, ε = x.27 η.23 Figure 11: Time evolution of free surface by unsteady fkdv, label 5 wave propagating downstream. We characterize both these cases, labels 3 and 4, as unstable, but with initially a rather slow oscillatory instability, which is associated with the weak emitted radiation. These results are in agreement with the analogous results of Camassa and Wu [5] who studied the stability of forced steady solitary waves when τ = in the framework of a forced KdV equation, using a combination of linear and nonlinear stability theory and numerical simulations. They considered two special types of forcing function, namely a sech 2 -profile as here, and also a sech 4 -profile. Also the situation is analogous to that described by Wright and Scheel [2] in a stability analysis of solitary waves in a coupled KdV system..2 F =.9, τ =.4, ε = η at t = η at t = 1 η at t = Figure 12: Evolution of free surface at time t = 1, 2, and 3 when F =.9, τ =.4 and ǫ =.1 (label 1) 13

14 .5 F =.9, τ =.4, ε = steady fkdv nonlinear, t = Figure 13: Comparison between steady fkdv and unsteady fully nonlinear solutions at t = 3 for F =.9, τ =.4 and ǫ =.1 (label 1) 4.3 Unsteady fully nonlinear solutions We now turn to simulations of the fully nonlinear Euler equations, using the numerical method described in section 3.1. Although in MAG we obtained a set of fully nonlinear steady solutions, these were found numerically in a hodograph plane, and cannot be immediately used as initial conditions here. Since it was found in MAG that at least for small forcing amplitudes, the fully nonlinear steady solutions were quite well approximated by the corresponding steady solutions of the steady fkdv equation, we used the latter as initial conditions here. Specifically the initial surface profile is obtained from the steady fkdv model (2), and the corresponding initial velocity field is then reconstructed using the fkdv scalings for consistency. Note that unlike the fkdv results above, the results shown in this section are in a fixed frame of reference in which the applied moves steadily to the left. In Fig. 12, we show the case for a small-amplitude depression wave with ǫ >, label 1 in Fig. 5. Because the initial condition is now only approximate, there is some initial adjustment, after which a stable solitary wave emerges, attached to the applied forcing. It is close to the corresponding fkdv solution, see Fig. 13. Then in Fig. 14, we show the corresponding case of a large-amplitude depression wave with ǫ >, label 2 in Fig. 5. Again, there is an initial adjustment, after which at an early time, see Fig. 15, the solution becomes a steady solitary wave close to the corresponding fkdv solution. But at later time it begins to separate from the forcing, and eventually breaks, see Fig. 16. We infer that this case is unstable. This pair of results agrees with those from the fkdv model, except of course that the fkdv model cannot predict breaking. Next, in Fig. 17, we show a simulation of a depression wave with ǫ <, label 3 in Fig. 5. It is immediately unstable, and soon breaks. Then in Fig. 18 we show the simulation of the dimpled solitary wave on this same branch, namely label 4 in Fig. 5. We see that it develops into a similar solution but with a much larger dimple. Thus it is unstable, but it is not clear whether or not the emerging solitary wave is then stable. 14

15 F =.9, τ =.4, ε = η at t = η at t = η at t = Figure 14: Evolution of free surface at time t = 1, 1, and 13.5 when F =.9, τ =.4 and ǫ =.1 (label 2) Although instability is again predicted, these results differ from the fkdv model which predicted an oscillatory instabilty, whereas, especially in Fig. 18 this is not evident. In these two cases, finite amplitude effects are more important. Finally, we show a simulation of an elevation wave with ǫ < in Fig. 19. After an initial adjustment, the solution settles down to a stable solitary wave attached to the applied forcing, in agreement with the fkdv model. A similar simulation of an elevation wave is shown in Fig. 2. The initial condition in this case is flow at rest. the upstream flow. It is found that the elevation wave is stable. The agreement with the steady fkdv model is shown in Fig Discussion In our previous paper MAG [15] we obtained a family of steady solitary waves forced by an applied localized distribution in water of finite depth, for the case when the Bond number τ > 1/3. In this paper, we have investigated the stability of these steady waves, using two models, namely an unsteady forced KdV model (12) appropriate for the weakly nonlinear regime when the Froude number F 1, and the fully nonlinear Euler equations for unsteady inviscid incompressible flow. Our approach to stability has been through numerical simulations, where the initial condition is the steady solution perturbed either by the numerical discretization error, or through small random noise, or in the case of the fully nonlinear equations by using the steady fkdv approximation to the initial conditions. A pseudo-spectral numerical method was used for the fkdv model, while a boundary integral method was used to obtain the fully nonlinear numerical solutions. For depression waves, steady solutions exist for both ǫ > and ǫ <, where ǫ measures the amplitude of the applied forcing. Along the solution branch 15

16 .5 steady fkdv nonlinear, t = Figure 15: Comparison between steady fkdv and unsteady fully nonlinear solutions at t = 1 for F =.9, τ =.4 and ǫ =.1 (label 2) ǫ >, there are two depression waves for the same value of the Froude number, F < F c < 1, where F c denotes a turning point on this branch. The wave with the smaller amplitude is a perturbation from the uniform flow, and the wave with the larger amplitude is a perturbation from a free depression solitary wave. Our simulations from both the fkdv model and fully nonlinear equations agree and show that the smallamplitude solution is stable. But the larger-amplitude wave is unstable and breaks down into a stable small-amplitude wave and a larger wave which propagates away, and in the fully nonlinear model may eventually break. For the case of ǫ <, there are two types of steady solutions. These are large-amplitude depression waves, and when the Froude number is close to unity, a depression wave with a dimple. We found that both these two types of waves are unstable. For the large-amplitude wave, the fkdv model predicts an oscillatory instability whereas the fully nonlinear shows that wave breaking eventually occurs. For the dimple wave, the fkdv shows the shedding of a wave and the development of the larger-amplitude wave, which still has an oscillatory instability. On the other hand, the fully nonlinear model shows the development of a wave with a larger dimple still trapped in the area of the applied. For the elevation wave, steady solutions exist only for a negative forcing,and are a perturbation from a uniform flow. The simulations from both the fkdv and fully nonlinear model agree and show that the solution is stable. The case of τ < 1/3 which was not considered here, is more complicated. There are many braches of steady depression and elevation waves [16]. The appropriate model from the weakly nonlinear theory is now the nonlinear Schrödinger equation. In the future work we propose a numerical investigation of the stability of the steady solutions by solving the unsteady nonlinear Schrödinger equation, together with the same fully nonlinear model as that used here. Acknowledgements This research was financially supported by the Thailand Research Fund (Grant No. MRG58231) to the second author. 16

17 .1.5 steady fkdv nonlinear, t = Figure 16: Comparison between steady fkdv and unsteady fully nonlinear solutions at t = 13.5 for F =.9, τ =.4 and ǫ =.1 (label 2). The simulation breaks down when t > 13.5 References [1] Akylas, T.R.: On the excitation of long nonlinear water waves by a moving distribution. J. Fluid Mech. 141, (1984) [2] Akylas, T.R., Cho, Y.: On the stability of lumps and wave collapse in water waves, Proc. Roy. Soc. A, (28) [3] Beale, J.T., Hou T.Y., Lowengrub J.: Convergence of a boundary integral method for water waves, SIAM J. Numer Anal., 33, (1996) [4] Calvo, D.C, Akylas, T.R.: Stability of steep gravity-capillary solitary waves in deep water. J. Fluid Mech. 452, (22) [5] Camassa R., Wu, T.Y.: Stability of forced steady solitary waves, Phil. Trans. R. Soc. Lond. A, 337, (1991) [6] Casciola, C.M., Landrini, M.: Nonlinear long waves generated by a moving disturbance. J. Fluid Mech. 325, (1996) [7] Cooker, M.J.: A boundary-integral method for water wave motion over irregular beds. Eng. Anal. Boundary Elements. 7, (199) [8] Dold, J.W.: An efficient surface-integral algorithm applied to unsteady gravity waves. J. Comp. Phys. 13, (1992) [9] Grimshaw, R., Smyth, N.: Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, (1986) [1] Groves, M. D., Haragus, M. and Sun, S-M.: Transverse instability of gravitycapillary line solitary water waves, C. R. Acad. Sci. Paris, 333, (21) 17

18 F =.9, τ =.4, ε = η at t = η at t = η at t = Figure 17: Evolution of free surface at time t =.1, 2, and 2.5 when F =.9, τ =.4 and ǫ =.1 (label 3). The simulation breaks down when t > 2.5 [11] Haragus, M., Scheel, A.: Finite-wavelength stability of capillary-gravity solitary waves, Comm. Math. Phys., 225, (22) [12] Henderson, K.L., Peregrine, D.H., Dold, J.W.: Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation, Wave motion, 29, (1999) [13] Lee, S., Yates, G.T., Wu, T.: Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mech. 199, (1989) [14] Longuet-Higgins, M.S., Cokelet, E.D.: The deformation of steep surface waves on water I. A numerical method of computation. Proc. R. Soc. Lond. A. 35, 1 26 (1976) [15] Maleewong, M., Asavanant, J., Grimshaw, R.: Free surface flow under gravity and surface tension due to an applied distribution: I Bond number greater than one-third. Theor. Comput. Fluid Dyn. 19, (25) [16] Maleewong, M., Grimshaw, R., Asavanant, J.: Free surface flow under gravity and surface tension due to an applied distribution: II Bond number less than one-third. Eur. J. Mech. B Fluids 24, (25) [17] New, A.L., McIver, P., Peregrine, D.H.: Computations of overturning waves. J. Fluid Mech. 15, (1985) [18] Schwartz L.W.: Nonlinear solutions for an applied over on a moving stream. J. Eng. Math., 12, (1981). 18

19 η at t =.1 F =.947, τ =.4, ε = η at t = 2 η at t = Figure 18: Evolution of free surface (dimple solution) at time t =.1, 2, and 4 when F =.947, τ =.4 and ǫ =.1 (label 4) [19] Vinje, T., Brevig, P.: Numerical simulation of breaking waves. Adv. Water Resources. 4, (1981) [2] Wright, J.D., Scheel, A.: Solitary waves and their linear stability in weakly coupled KdV equations, Z. angew. Math. Phys. 58, (27) [21] Wu, T.Y.: Generation of upstream solitons by moving disturbances. J. Fluid Mech. 184, (1987) [22] Zhu, Y.: Rosonant generation of nonlinear capillary-gravity waves. Phys. Fluids. 7, (1995) 19

20 F =.9, τ =.4, ε = η at t = 2 η at t = 1 η at t = Figure 19: Evolution of free surface at time t = 2, 1, and 3 when F =.9, τ =.4 and ǫ =.1 (label 5).4.2 η at t = 1 F =.9, τ =.4, ε = η at t = η at t = Figure 2: Evolution of free surface at time t = 1, 5, and 1 when F =.9, τ =.4 and ǫ =.1 (label 5). Flow at rest is set as initial condition 2

21 .3.25 F =.9, τ =.4, ε =.1 Nonlinear, t = 135 steady fkdv Figure 21: Comparison between steady fkdv and unsteady fully nonlinear solutions at t = 135 for F =.9, τ =.4 and ǫ =.1 (label 5) 21