Non Linear Free-Surface Flow Over a Submerged Obstacle

Free-surface flow over a triangular obstacle is considered. The fluid is assumed to be inviscid, incompressible and the flow is assumed to be steady and irrotational. Both gravity and surface tension are included in the dynamic boundary condition. Far upstream, the flow is assumed to be uniform. The problem is solved numerically using a boundary integral equation method. The problem is solved by first deriving integro-differential equations. These equations are discretized and the resulting nonlinear algebraic equations are solved by Newton method. When surface tension and gravity are included, there are two additional parameters in the problem known as the Weber number and Froude number. Indexing terms/


Introduction
Research has been done in this area for more than 150 years and it still continues today. Over these years problems have been solved including only gravity g, or surface tension T and also problems that consider both.
The reason for neglecting or including such properties is to isolate what effect surface tension or gravity has on a free surface. This paper considers the problem of steady free-surface of a two-dimensional, irrotational, inviscid and incompressible flow. The problem is first formulated as an integral equation for the unknown shapes of the free surfaces. This equation is then discretized and the resulting algebraic equations are solved by Newton's method. Later on, we found numerical solutions of free-surface flows over a triangular obstacle with the effects of surface tension and the gravity. The problem is formulated in section 2, the numerical procedure is described in section 4 and the results are discussed in section 5.

Mathematical Formulation
We consider the fluid flow over a triangular obstacle. The fluid is inviscid and incompressible and the flow is irrotational. The flow domain is bounded above by a free surface EGF and below by a horizontal bottom and the triangle BCD with angle γ where 2 0     . We introduce Cartesian coordinates   y x, (see Figure 1).
Here T is the surface tension g the gravity and  is the fluid density. We introduce the complex potential function and the complex velocity Using the capillary Laplace's equation defined by: is the curvature, p* is the fluid pressure, p 0 is the atmospheric pressure, * is the fluid density and g is the gravitational constant. Substituting (2.4) into (2.3) and in terms of the dimensionless variables, Bernoulli's equation on the free surface becomes: The kinematic boundary conditions are Now we reformulate the problem as an integral equation. We define the function We map the flow domain onto the upper half of the  -plane by the transformation  The velocity terms, first, become: Secondly, the curvature K of a streamline, in terms of , is

Boundary integral techniques
By using (3.4), we rewrite (3.8) as: By substituting (3.5) and (3.9) into (3.6), an integro-differential equation is created and this is solved numerically in the following section.

Numerical Procedure
In Section 3, a nonlinear integro-differential equation was derived. In this Section, the numerical procedure used to calculate solutions to this equation will be discussed. The expression (3.5) is used to calculate  along the free surface. It is required to have points, 0  , along the free surface at which  values can be evaluated. This is done by creating an equally spaced mesh, in the potential function, over the region that relates to the free surface. Let has been introduced to ease notation. Then, using the trapezoidal rule gives:

Discussion of the Result
We used the numerical scheme described in section 4 to compute solutions for various values of F and  .
Most of the calculations were performed with N = 300,