**1. Introduction**

60 Computational Simulations and Applications

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The quest for understanding the mechanisms responsible for the vortex shedding process comes from past decades, but it is still challenging. The turbulent wake downstream of bluff bodies induces alternating and dynamic loads in the bodies like (antennas, chimneys, suspended bridges, a set of risers and structures in general). These structures can produce disastrous results. Extrapolating the scale of the phenomena, one can refer to petroleum exploration, which starting from the platforms to the seabed, there are risers that are cylindrical tubes of great length. They are subjected to ocean currents and suffer fluiddynamic effort. The consequence is that the phenomenon as fatigue and wear are accelerated, reducing the useful life of equipment and could lead them to collapse.

Thus, the study of problems involving immersed bodies is motivated by great technological challenges, both within the academic and industrial environment. Fluid-structure interaction is today one of the crucial problems in various areas of mechanical and civil engineering, because of the necessity of extensive structures subjected to fluid-dynamic random efforts. Therefore, it is important to appreciate the great importance of the study of flow around stationary circular cylinder in order to extrapolate to moving bodies or to set of moving bodies. This is a model, used to simulate, for example, a set of risers. We also emphasize the importance of such studies, including transition and turbulent flows, in order to better approximate the real conditions.

In the context of fluid mechanics, the study of fluid flows and how they interact with solid materials has been of great interest in various fields such as civil and mechanical engineering, meteorology and environment. In recent decades, great efforts have been made for the development of new numerical methods to analyze the wide range of problems in fluid mechanics, as well as improving existing ones. The Computational Fluid Dynamics has been considered an interesting tool to simulate various problems of practical interest in engineering. The literature shows different computational methods with several techniques to solve differential equations aiming to accuracy of results. Different numerical methods developed for the study of flow in the presence of immersed bodies are basically divided into techniques based on the immersed boundary method, and those based on meshes that are able to adapt to the immersed body inside the flow. However, there is no method that can be considered absolutely superior to others. The choice of the most appropriate method should be made case by case, taking into account the specific characteristics of the focused problem.

An Introduction of Central Difference Scheme Stability for High Reynolds Number 63

points coincident or close to the Lagrangean mesh, being zero for the remaining points of the calculation domain. This term is calculated by distributing the components of the Lagrangean interfacial force vector, using a distribution function (Peskin & McQueen, 1994):

> <sup>2</sup> *ij kk k*

where *f x* [N] is the Eulerian force vector, *x* [m] and *<sup>k</sup> x* [m] are respectively the position vectors of Eulerian and Lagrangean points, *S* [m] is the arc length centered on each Lagrangean points, *F x <sup>k</sup>* [N] is the interfacial force calculated by the IBM and *Dij* [m-2] is a interpolation/distribution function, which properties are the same of the Gaussian

**2.1.2 Mathematical formulation for the immersed interface – Virtual Physical Model** 

The VPM allows the calculation of Lagrangean force based on physical interaction of the fluid and immersed solid surface in the flow. This model is based on applying the balance of momentum quantity over the fluid particles located at the Lagrangean points. The equation

, , ,, , , ,

where *Fa* [N] is the acceleration force, *Fi* [N] is the inertial force, *Fv* [N] is the viscous force

Turbulence is one of the most challenging problems of modern physics and is among the most complex and beautiful phenomena in nature. Due to several practical implications for many sectors, the number of research related to understanding and controlling these flows has increased. The turbulence effects can be modeled and simulated since emprirical

It is known that even for flows controled by moderate Reynolds numbers, it is not possible to solve directly all frequencies present in a turbulent flow. Reynolds (1894) proposed a decomposition process of the Navier-Stokes equations in a mean and floating part in order to solve the turbulent flow. The decomposition process of the scales yielded two groups of equations for the turbulence, the first being called Reynolds Averaged Navier-Stokes equations, and another called the filtered Navier-Stokes equations (Smagorinsky, 1963). After applying the filtering and the decomposition process and applying the definitions in

> \* *i j* <sup>1</sup> *<sup>j</sup> i i ef <sup>i</sup> j ij ji*

*u u u u p u + =+ + +f t x <sup>ρ</sup> xx xx* 

 

correlations and diagrams up to modern methodology of numerical simulation.

*t tt t t p t*

*F x Vx Vx Vx Vx x* (4)

 

*<sup>F</sup> <sup>F</sup> <sup>F</sup> <sup>F</sup>*

*k T k k k k kk*

*<sup>f</sup> x x x Fx x D S* (3)

*i p v*

(5)

*k*

function.

**(VPM)**

that determines this force is expressed as:

*t* 

*a*

and *Fp* [N] is the pressure force.

**2.2.1 Turbulence equations** 

**2.2 Turbulence model** 

*V x*

*t*

Eqs. (1) and (2), we obtain the following equation:

The Immersed Boundary Method (IBM), due to their capability to deal with problems of complex and mobile interfaces, becomes attractive, especially in cases involving large displacements. In the modeling process of physical problems, the equations that govern the physics of the problem appear naturally. These models can range from those involving only one differential equation to those involving a system of differential equations, which can be fully coupled. However, in most cases, exact solutions can not be obtained and numerical methods appear as a tool to solve these problems. The Immersed Boundary method is used here with the Virtual Physical Model in order to simulate two-dimensional incompressible flows over stationary, rotating and rotationally-oscillating circular cylinders. Different time discretization methods are used: first order Euler scheme and the second-order Adams-Bashforth and Runge-Kutta schemes. The sub-grid Smagorinsky model and a damping function are also used. Considering the existence of a mistaken view about the mentioned numerical methods, their stability analyses are made in the present work. The results are compared with numerical and experimental results obtained from the literature.
