Torrential forced KdV equation: soliton solutions over a hole

Torrential flow of an incompressible and a perfect fluid over a hole in infinite channel is investigated. The flow is supposed to be two-dimensional, steady and irrotational. The surface tension is neglected but the gravity is included in the dynamic boundary condition. The issue has been programed and run on a computer to solve a nonlinear ordinary differential equation of the third order, called the forced Korteweg de Vries equation by using the Newton’s method. The obtained results in the torrential flow have been represented graphically by varying the shape of the hole and its depth, also by giving a different values of the Froude number λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}.


Introduction
Free streamline flows over several obstacle shapes have been studied in a considerable amount of work. Solutions to the linearized problem of flow past topography on the stream bed were developed by (Lamb [19]; Wehausen and Laitone [23]), with related problems examined by Havelock [15,16]. Some interesting concerned free surface flow problems have been investigated, for instance Long [20] and Baines [2]. Mina et al. [22] studied a problem of flow generated by a ramp placed on the bottom of a running stream, using the linearized theory. Results depending on the solutions to the full nonlinear problem were obtained numerically by Forbes [11,12], who considered a semi-elliptical obstruction on the stream bed, Schwartz et al. [13] and Vanden-Broeck [6], computed nonlinear solutions for flow over a semi-circular obstruction. An alternative was given by Boutros et al. [5] using the Hilbert method for the nonlinear solution of a two dimensional flow past a triangular obstacle in a stream bed, with large depth Froude numbers. S. N. Hanna et al. [14] suggested a numerical method for the solution of the fully nonlinear problem based on series truncation, where the flow is over a two-dimensional trapezoidal obstacle.
King and Bloor [18] used conformal mapping to calculate the flow over an arbitrary topography, with Gaussian, triangular and semicircular obstructions. Zhang and Zhu [25,26] computed the flow over a semicircular obstruction using a hodograph method and also computed the second order perturbation solution for flow over a semicircular trench which they then compared with solutions to the full nonlinear problem. Weakly nonlinear solutions have been found using long wave theory like the Kortewegde Vries (KdV) equation. Trains of solitary waves have been computed in similar problems. Chu et al. [7] computed solitary waves by solving the KdV equation without a forcing term. Akylas [1] derived the KdV equation with a forcing term to examine the waves that generated by a pressure distribution moving on the surface. Dias and Vanden-Broeck [8,9] used KdV theory to find critical solutions over a single obstruction called 'generalized hydraulic falls'. These solutions are characterized by a supercritical flow on one side of the obstacle and a train of waves on the other, also they have found that if they introduced a second obstruction, the waves would be trapped between the two obstructions. In [17], Keeler et al. examined a critical free-surface flow over a Gaussian dip topography by using the forced Korteweg-de Vries (fKdV) equation, the weakly nonlinear steady solutions space is examined in detail for the particular case of a Gaussian dip by utilizing a combination of asymptotic analysis and numerical computations. Beyoud et al. [3], presented a numerical solution of a free boundary problem over a bump. In the present work, our study is based on the determination of the equilibrium free surface of a bidimensional, stationary and irrotational flow of a perfect and incompressible fluid, which is perturbed by a hole placed on the bed of an infinite channel. We take into account of the gravity g and we neglect the effects of the superficial tension.
The remainder of this paper is organized as follows: In Sect. 2, we introduce the mathematical foundations, where we put the unsteady fully nonlinear equations and we do their nondimensionalization to obtain the forced Korteweg-de Vries equation. In Sect. 3 we represent the numerical method used to solve the problem. The graphs that show the results reached in the unsteady case are illustrated in Sect. 4, also a discussion about the obtained solutions were given. Section 5 summarize the results of this work and draws conclusion.

The governing equations
This section is divoted to the formulation of the problem. We consider an unsteady, twodimensional, irrotational flow of a perfect, incompressible fluid over a hole localised on the horizontal stream bed.  The fluid domain is given by: characterize the bed topography and the free surface profile of the channel flow, respectively; u(x) is the disturbance of the horizontal flow cause of the hole located on the bottom of the channel. "σ " and "u" are regular functions with compact support included in R. Remind that our main unknown is the function "u".
The quantities H , C and g define a nondimension parameter which describe the flow, called the Froude number given by λ = C √ g H , it is the ratio of the upstream uniform velocity to the critical speed of shallow fluid waves. The flow is said to be torrential when λ > 1, else (λ < 1) the flow it is called fluvial.
• Condition on ω Under the assumption of irrotationality, the flow of an incompressible fluid is governed by the following Laplace's equation for the velocity potential φ, • Boundary conditions In the boundary value problems under consideration the flow boundaries will be free, in which case their shape is unknown beforehand, so two conditions must be satisfied which are sufficient, in conjunction with the other datas to balance the incomplete knowledge of the boundary and also to determine all unknowns of the flow problem. The first condition, kinematic-in nature, states that on both the free boundary is a material surface; particles initially on the surface remain thereon, this later also valid for the stream bed. Mathematically are expressed as follow: or and or such that D Dt is the material derivative operator, u t is the derivative of u with respect to t and φ x , φ y are the derivatives with respect to x, y respectively of φ. The second boundary condition, dynamic one, is due to the principle of energy conservation. Thus, in unsteady motion under the influence of gravity, Bernoulli's equation states that ρ is the fluid density and C is the speed of the upstream flow.

Non-dimensional system
Recall that the Eqs. 1,3,5 and 6 are in terms of dimensional quantities, to nondimensionlize it, we pressent the following dimensionless variables: l is a typical wave lengh and ε = H l 2 . As consequence, we obtain the following system:

The perturbed potential wave motion
The solution of the system (P) is trivial if σ (x) = 0, so our problem is when σ (x) = 0, for this, we define the perturbed potential wave motion γ where: φ = γ + λx, then applying the Taylor expansion on the boundary equations, results in: Due to the alternatives are numerical and approximated solutions, we assume the following asymptotic expansions: where λ 0 and λ 1 are the critical and the perturbed speed of the upstream uniform flow, respectively. Combining the Eqs. 11-13 and 7-10, we can reach to a sequence of boundary value problems through rearranging the terms according to the power of ε, as follow Lowest order: First order: Second order : We can deduce from integrating the Eqs. 14-17 that: • The function γ 1 is dependent only on x and t.
• The function γ 1x is: Using the Eqs. 18-21 and 25, results in thus the equation 19 yields to By integrating the last two equations, we can obtain The value of the critical upstream speed λ 0 can be deduced from the Eqs. 20 and 26 which equal to one. Integrating the Eq. 22 then using both the Eqs. 23 and 24, we get In the end, we reach to The equation above is said to be the forced Korteweg de Vries equation, where the free surface is dependent of time. The Eq. 30 has been derived in the first time by Akylas [1], to model the free surface of a flow subjected to a localised pressure distribution. Here, the effects of surface tension have been neglected, and the parameter λ 1 is proportional to λ − 1, where λ is the dimensionless Froude number defining the ratio of flow speed to the linear shallow fluid wave speed V = √ g H. It is given by where g is the acceleration due to the gravity, C is the upstream velocity of the fluid, and H is the constant upstream fluid depth. The fKdV equation includes the balancing effects of weak nonlinearity and dispersion, as well as the effects of the forcing (for example, by some underlying topography σ (x) or some applied pressure distribution p(x)). Further derivation of the fKdV equation have been given for several underlying topographic forcings, see for instance the work of Beyoud and Boukari-Hernane [3].
In the absence of the forcing, i.e. with σ (x) = 0, the equation reduces to the classical KdV equation.

Numerical method
The forced Korteweg-de Vries Eq. 30 provides an approximated solutions to the problem. In this paper, we consider the stationary case through omitting the first term in Eq. (30), to have: The unknown function u 1 (x) represents the amplitude of the relevant linear long wave mode. σ (x) is the forcing term due to the hole on the bottom. Therefore, for a given: y = σ (x) and a given upstream near critical flow speed λ = 1 + ελ 1 , we can find an approximated shape of the free surface u = εu 1 + O(ε), by solving the differential Eq. 31. Given a stationary flow, the integrated form of the Eq. 31 is A finite difference method is commonly used to solve the Eq. 32 through approximating its second derivative with a centred-difference leads to a system of nonlinear algebraic equations at each grid point, which is solved using Newton's method. The numerical computations were performed by assuming that the obstacles have a cavity shape and the flow is torrential (λ > 1). The idea behind such a method is to calculate the unknowns at just the boundaries of the problem, in order to determine the behaviour of the fluid as a whole. This reduces the dimension of the problem by one, and so the two-dimensional problem becomes one-dimensional; it is only necessary to consider what happens at the free surface. As we solve the Eq. 32 numerically, we choose x in the truncated domain [−L, L], (L > 0) and we write the boundary conditions at ±L. According to the experimental results we can write u 1 (−L) = u 1 (L) = 0 for λ 1 > 0. So, our whole problem is expressed as follows: As consequence, the resolution of the problem 33, is equivalent to solve the second order nonlinear ordinary differential equation. For a given hole σ (x) and a given upstream near critical flow speed F 1 , we can find an asymptotically approximate shape of the free surface u 1 (x).

Results and discussion
In the system 33, for a certain values λ and the function σ (x) we obtain a system of N + 1 equations with N + 1 unknowns which solved by Newton's method. The curves shown in the Figs. 2,3,4,5,6,7,8, 9are examples of free surface profiles calculated using the numerical scheme discussed in Sect. 3. To validate our results we have taken a several shape holes with Froude number λ = 1.1, we mention for instance: • Semi elliptical hole defined by the equation: 0 else.
• A semi circular hole of the radius h = 0.1.
• An arbitrary triangular pit.

Remark 1
The plotting program Gnuplot is used to draw the tabulated data files which contain three columns, the first one for the x-axis, the second and the third one are saved for the free surface and the bed streamlines, respectively. In the following figures, the free surface profile, is scaled to y = 0.5 + u(x) instead of y = 1 + u(x) for having an obvious and precise curve.     Fig. 9 illustrates the effect of the Froude number on the shape of the free surface when the depth of the hole is fixed (h = 0.1), for a bed stream given by:  The Fig. 10 shows the behaviour of the free surface flow profile when the depth of the hole defined by 34, is varied and the Froude number λ is fixed.
It is noted in the Fig. 9 that whenever the Froude number decreases, the cavity over the hole increases. But in the Fig. 10, both the depression shown on the free surface and the hole are proportional.

Conclusion
In this work, a torrential-steady flow problem over a hole has been considered with the presence of the gravity. The flow is supposed to be stationary, two dimensional and irrotational; the fluid is perfect and incompressible. The effects of the superficial tension have been neglected. To solve our problem, we have considered the governing equations with the velocity potential φ, in this way, the problem is reduced to the resolution of a nonlinear ordinary differential equation of the third order. The finite difference method is directly applied to resolve it for the case λ > 1. The obtained results show that the free surface profile takes a form as the shape of the bottom when the flow is torrential (λ > 1). We have remarked that in the torrential regime, the function representing the cavity on the free surface decreases when the Froude number λ increases. Also for λ fixed, the depth of the free surface depression and the cavity of the hole are proportional.