Mathematical Modeling of Pressure on Cylindrical Ellipse using Side-by-Side Configuration

The application of the concept of fluid is often used to solve problems in the daily life. One of them is the problem of fluid around an elliptical cylinder. This study aims to solve the problems of the fluid around two elliptical cylinder configuration with side-by-side using the Navier-Stokes eq uations. Navier-Stokes equations–incompressible, viscous and uns teady– are solved using finite difference method staggered grid and SIMPLE (Semi Implicit Method for Pressure-Linked Equation ) algorithms. Finite difference method is used to complete th e grid arrangement, whereas the SIMPLE algorithm is used to obtain components of velocity and pressure value. Results o f this study are the pressure value based on fluid flow profile and a mathematical model which received an elliptical cylinder pressure. Profile of fluid flow is simulated by varying the Reynolds number of 100, 1000, 7000, and 10000 as well as variations in t he distance between the cylinder with a ratio of2 ≤ S/a ≤ 6 where L is the distance between the cylinder anda is the minor axis of the cylinder ellipse. Then the pressure is calculated bas ed on the value of the received cylinder pressure components. Aft er obtaining the pressure value, then we create a mathematical model of the stresses imposed on the elliptical cylinder.


I. INTRODUCTION
M INIMIZING the drag accepted by body of the car is needed to obtain a suitable car body design, especially for racing cars.This technique is also used in catamaran boat design.The basic concept for analyzing the drag is also used in a variety of shapes ranging from boxes, cylinders, circular, elliptical and other shapes modifications.Then it is developed with the object of more than one configuration with tandem or side-by-side.
Singha and Sinhamahapatra conduct research fluid flow passing through the cylindrical single circular object with Reynolds numbers at 45 ≤ R e ≤ 250 and perform experiments on the distance between the dividing wall which is 2 ≤ H/D ≤ 8 [1].Sarvghad-Moghaddam and Nooreddin conducted a study of fluid flow around two circular cylinders arranged side-by-side with the Reynolds number 100 and 200 for a laminar flow, 10 4 to turbulent flow.Then the position within the cylinder using a ratio that varies at 2 ≤ L/D ≤ 4 is 1.5, 3, 4 [2].
This paper discusses a two-dimensional fluid flow around two elliptical cylinders arranged side-by-side to determine the pressure received at four different points with attention to fluid flow and pressure profiles received by both cylinder ellipses.

II. NUMERICAL METHOD
The equations used for unsteady, viscous and incompressible fluid is the Navier-Stokes equations where u is the velocity, R e is the Reynolds number, and P is the pressure.Navier-Stokes equations above can be solved numerically using the SIMPLE algorithm.The algorithm consists of several steps.The first step is to provide an initial value for each variable, and then completed the momentum equations to obtain the velocity components ignoring initial pressure components, so that (1) becomes the right-hand side is solved using finite difference, and the second step is carried out We now put divergence on both sides, and the result is by considering (2), we obtain ∇u * * = 0.It follows that the equation turns into Equation ( 6) is called Poisson's equation.Completion of the Poisson equation solution will more quickly to achieve convergence, if SOR (Successive Over Relaxation) is used, as follows This equation will yield a value of P .The final step is the correction of velocity equation

III. MAIN RESULTS
We simulate the Navier-Stokes equations by using a numerical method.The space of interest is partitioned into 25a×50a grids, where a is length of minor axis of ellipse cylinder and S is the distance between the cylinder.Two cylindrical ellipses using side-by-side configuration are placed at distance 20a from the front side.The value of S/2a is varied between 1.0, 2.0 and 3.0, as shown in Fig. 1.
The results of a simulation program is accurate, if the simulation results are closer to the experimental value.Drag coefficient of the experimental results compared with simulation results can be seen in Table I.The result of the simulation program with R e = 100, 1000, 7000, and 10000 and S/2a = 1.0, 2.0, and 3.0, can be seen in Table II, and the graphs can be seen in Fig. 3.

TABLE I COMPARISON
OF DRAG COEFFICIENT OF A CYLINDER FOR Re = 100

TABLE II DRAG
COEFFICIENT C D ON Re WITH S/D = 1.0, 2.0 AND 3.0