Explain mathematically how potential theory is applied to mo

Explain (mathematically) how potential theory is applied to model aerodynamic forces and moments for a bluff body. State Kutta-Joukowski theorem and provide theoretical explanation of Kutta condition applied for the finite wing. What assumptions, if any, are needed to validate the condition?

Solution

B. The aerodynamic forces acting on these bodies may be evaluated through the previously described simplified potential flow - boundary layer procedure. Conversely, “bluff bodies” are characterized by a more or less precocious
separation of the boundary layer from their surface, and by wakes having significant lateral dimensions and normally unsteady velocity fields. For these bodies no simplified mathematical treatment is usually possible, and the forces acting on them may be evaluated either from the solution of the complete Navier-Stokes equations or from the results of ad hoc experiments.

C_Fi = Fi/ (1 / 2dU²S) d=density
; C_Mi = Mi / (1 / 2dU²Sl ) d=density

C.

The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics, that can be used for the calculation of the lift of an airfoil, or of any two-dimensional bodies including circular cylinders, translating in a uniform fluid at a constant speed large enough so that the flow seen in the body-fixed frame is steady and unseparated. The theorem relates the lift generated by an airfoil to the speed of the airfoil through the fluid, the density of the fluid, and the circulation. The circulation is defined as the line integral, around a closed loop enclosing the airfoil, of the component of the velocity of the fluid tangent to the loop

As the airfoil continues on its way, there is a stagnation point at the trailing edge. The flow over the topside conforms to the upper surface of the airfoil. The flow over both the topside and the underside join up at the trailing edge and leave the airfoil travelling parallel to one another. This is known as the Kutta condition.

The most important assumption of this method is to assume zero pressure difference at trailing end.