Toward wave-body interaction roblems using CIP method: A demonstrating 2 phase problem

CIP (constrained interpolation profile) is one of the CFD (computational fluid dynamics) methods developed by Japanese professor Takashi Yabe. It is used to simulate 3 phase problems including air on the surface, liquid and structure in solid form. To check the validity of CIP theory, experiments with different problems have been implemented and obtained very positive results. This proves the correctness of the CIP method. Based on the need of simulation of wave structure interaction (water wave with float of seaplanes, wing in ground effect crafts, piers, drilling, casing ships...), this paper applies the theory of CIP method to find the answer to the problem of 2D simulation via a obstacle. Objectives to do are understanding the physics, finding out the differential equations describing the phenomenon, then proceeding discrete, setting up algorithms and finding out solution of the equations. This paper uses Matlab software to write programs and displays the results.


1.1.Objectives
It is very important to know interaction of water waves on structures (body and float of seaplanes, flying boats, piers, drilling, casing ships...). The main objective of this paper is to establish a numerical prediction way for how water waves impact to a solid body.
Purpose of this paper includes constructing algorithms and computational simulation modules, calculating the fluid forces acting on the structure (lift, drag, torque) and processing and displaying calculated results.

Missions
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K7-2015 Trang 190 CIP is a CFD method developed by a Japanese professor [1]. CIP is used to simulate 3phase environments consisting of air over the surface, liquid and a structure. The problem can be understood simply as follows: -Using equations to describe the movement of water waves.
-Discretizing mathematical equations to establish algorithms programmed on the computer to find the answer.
-Using the programming language to calculate an explanation of the equations.
-Using graphics software to display the results of the problem found in graphs image. Software used in this paper is Matlab.

GOVERNING EQUATIONS [1]
From the basic conservation equations: Where t is the time variable; xi (i =1,2) are the coordinates of a Cartesian coordinate system; ρ is the mass density; ui (i=1,2) are the velocity components; fi (i=1,2) are due to the gravityorce. where: σij is the total stress p is the pressure; μ is the dynamic viscosity coefficient; δij is the Kronecker delta function; Kronecker delta function: In order to identify which part is the air, the water or the solid body, density functions φm (m=1, 2, 3) is introduced: where Ωm : domain occupied by the liquid, gas and solid phase, respectively. These functions satisfy: . Density function ϕm (m=1,2,3) for multiphase problems with 0≤ ϕm ≤ 1 and ϕ1 + ϕ2 + ϕ3 = 1 in the computational cells.

Principle of CIP Method [2]
CIP method has some advantages over other methods with respect to the treatment of advection terms. In this section, the principle of CIP method is explained. Figure 3 shows the principle of CIP method. Here, a onedimensional advection equation is used to simplify the explanation of CIP method. As mentioned in the previous section, a onedimensional advection equation is described as below, The approximate solution of the above equation is given as: Where xi is the coordinates of calculation grid. The above equation indicates that a specific profile of f at time t + t is obtained by shifting the profile at time t with a distance u∆t as shown in Figure 3(a). In the numerical simulation, however, only the values at grid points can be obtained, as shown in Figure 3(b). If we eliminate the dashed line shown in Figure 3 (a), it is difficult to imagine the original profile and is naturally to retrieve the original profile depicted by solid line in (c). This process is called as the first order upwind scheme [3]. On the other hand, the use of quadratic interpolation, which is called as Lax-Wendroff scheme [4] or Leith scheme [5], suffers from overshooting. In CIP method, a spatial profile within each cell is interpolated by a cubic polynomial.
Differentiating equation (5) with a spatial variable x gives: given at two grid points, the profile between the points can be described by a cubic polynomial: The profile at n+1 step can be obtained by shifting the profile with u∆t,

Separation of Equations
The governing equations of the fluid and the density function is: Non-advection phase 1: Non-advection phase 2: Instead of calculating 1 n n f f   (n is time step) directly from Equation (7)

Problem Statement
Two-dimensional water interacting with a solid body is considered in this section. The fluid is assumed to be incompressible and inviscid. Temperature variations are neglected. The problem is described in Figure 6.
2-phase problem is the first step, the base premise to write programs for 3-phase problem and absolutely no experimental verification` [5]. Computational domain is presented by two points P1 and P2.
Obstacle is presented by two points P3 and P4, as shown in Figure 6.

Boundary Grid Structure
Boundary grid structure is shown in Figure  7, 8 and 9.

Boundary Conditions
Inlet boundary condition:
Number of mesh in two axis x, y are: nx and ny respectively.
The size of a small grid is : h (h = x=y) . The velocity vector fields, u-velocity contour, v-velocity contour, pressure contour are presented in Fig. 10

CONCLUSIONS
This paper presented an applicable method for simulating the wave body interaction problems. This method is cip (constrained interpolation profile). Throughout the research, we obtained some results as follows: from the physical phenomena, in particular here is the flow through an object in three phase environments (solid, liquid, gas). Then, we proceed to discretize these mathematical equations to create an algorithm and used computer to find the solution. This study uses matlab software as a tool for programming and presenting the results as graphs.
This paper has built a solver for two dimensional flows in a two phase (liquid, solid) environment. These results can be used to develop a three phase flow (liquid, air, and solid) [5].
Due to limited on the basis of information technology, mathematical knowledge, and fluid dynamic, this paper stops at the simulation of two phases flow problems and much remains unresolved, specifically error analysis and validation by experimental results.
In order to develop this work, it is necessary to analyze more simulations cases and invest more time. That is the future work. This method can be developed successfully to find the answer of three phase flow problem [6].