Dynamics of a Body with a Trailing Edge and Embedded Dipoles in an Ideal Fluid

Equations of motion for a body moving through an ideal fluid when the flow is irrotational and incompressible are obtained taking account of embedded dipoles on the boundary and the Kutta – Chaplygin condition. We develop the embedded dipole model from the complex potential of a dipole on the boundary of a body, oriented so as to preserve no-penetration through the body, using a conformal mapping approach. The resulting hydrodynamic force and moment on the body depend on the dipoles’ strength and position along the body. Using the flat plate as a model geometry, we examine the evolution of the resulting system under the conditions of fixed and time-varying circulation with and without embedded dipoles. We assume two embedded dipoles symmetrically positioned about the center point of the plate, finding that the presence of the dipoles reduces the fluctuations of the angle of attack of the plate. We explore conserved quantities for the system and perform a linear stability analysis, which leads to a constraint on the dipole strength for stability of a plate moving at zero angle of attack with either circulation equal to zero or the Kutta – Chaplygin condition applied.