N UMERIC EVALUATION FROM CANTILEVER BEAM AND PLATE

As soon as fluid passes over a solid surface vortices are formed at the surface of structure to satisfy the no slip boundary conditions at the fluid-structure interface. Vortices that are formed at fluid-structure interface diffuse in fluid domain. In the present study, the strength of vortices is evaluated on the surface of an oscillating structural element (cantilever beam and plate) that is interacting with oscillating fluid flow. In the analysis, the analysis of structural element is carried out by finite element method and analysis of fluid domain is carried out by panel method. It is assumed that the deformed geometry of the structure at any moment is same shape as the vortex sheet. The structure is replaced by a virtual vortex sheet of uniformly distributed point vortices on the surface. A few numerical examples are presented to show the variation of vortex strength, lift and pressure coefficients for different type flow passes over the oscillating rigid and flexible structural element in fluid medium. The strength of vortex depends on fluid flow characteristics around the structural element and motion of structure in fluid medium.


Introduction
The process of vortex shedding is a classical problem in fluid mechanics and there have been many theoretical and experimental studies on various aspect of this problem. Vortex shedding from a bluff body has been investigated for many decades due to its important role in understanding fundamental fluid dynamics and its close link to field applications. Under particular conditions, the motion of On the other hand when the structure is periodically oscillated by external forcing, the shedding frequency may be modified or shift from its natural shedding frequency to the forcing frequency. Howe (1987) examined the surface pressure fluctuations due to periodic vortex shedding from the blunt trailing edge of a coated airfoil. Guocan and Caimao (1991) studied numerically on near wake flows of a flat plate and calculate the forces on a plate in steady, oscillatory and combined flows by using the discrete vortex model and improved vorticity creation method. Cortelezzi and Leonard (1993) considered a two-dimensional unsteady flow past a semi-infinite plate with transverse motion. The rolling-up of the separated shear-layer was modeled by a point vortex whose time dependent circulation was predicted by an unsteady Kutta condition. Ting and Perlin (1995) experimentally determined a boundary condition model for the contact line in oscillatory flow, for an upright plate, oscillated vertically with sinusoidal motion in dye laden water. Larsen and Walther (1997) Khatir (2004) presented an approach for solving the source /sink boundary integral equation by using an indirect Boundary Element Method. The author implemented this analysis for solving flow over 3-D obstacles to impose impermeability at the wall in conjunction with discrete vortex method. Lin Lin, Ho, Chang, Hsieh, and Chang (2005)   ...
The induced velocity at collocation point i due to unit strength of vortex at point j may be represent by influence coefficient Where, n  is the normal of the surface at collocation point i.

Determination of Influence Matrix
The The equation (7) The equation (9) The coefficient of the matrix equation (10)  Vortex is formed on the solid surface and washed out from the surface as shown in Figure 3. The vortex formed at the trailing edge is known as wake vortex.
For any instant of time the influence of instantaneously wake vortex Γ w also affect the induced velocity. The latest wake vortex is assumed to be born on the Where, .

Boundary Condition for an Oscillating Cantilever Beam in Oscillating Fluid
Flow A cantilever beam is submerged in an oscillating fluid medium as shown in is the total vortex strength within the element of length x.
The normal lift force on the structure due to vortex formation may be written as Total lift force on the structure may be written as  Table 1 to compare the results with the results of Katz and Plotkin (2001). The strength of vortex at five points is denoted by 1, 1, ... 5 and corresponding nondimensional vortex strength is γ1, γ2, ... γ5.
From Table 1, it is observed that the magnitude of vortex strength at each vortex point is well matched with the results of Katz and Plotkin (2001). The vortex strength decreases slowly in the trailing edge compared to the leading edge.