MAE 252 course notes Normal, axial, lift, and drag force coefficients: Lift Kutta-Joukowski theorem states that the lift per unit span is directly proportional to the circulation. K-J theorem can be derived by method of complex variable, which is beyond the scope of this class. 1 MAE 252 course notes Example. Consider a flow over a semi-infinite body as discussed in Section 3.11 and as sketched below, 3 2 Assume that V∞=20 m/s, ρ=1.23kg/m , and p∞=101325 N/m . Find 1) the pressure at point A, 2) the source strength Λ, and 3) the velocity at point B. Solution: 1) Point A is the stagnation point. 2) Apply the conservation of mass to the control volume sketched below, 2 MAE 252 course notes Note that the source strength is the rate of volume flow from the source, 3) Need to find the coordinates for point B, which is located on the dividing streamline. a) Find the dividing streamline equation Stream function: Velocity field: Therefore, stagnation points: The equation for the dividing streamline: 3 MAE 252 course notes b) The coordinates for point B: c) The velocity at point B: D’Alembert’s Paradox There is no drag for the inviscid, incompressible flow over a closed twol-dimensional body. 4 MAE 252 course notes Chapter 4. Incompressible flow over airfoils 4.1-4.3 Self-reading 4.5 The Kutta-condition An infinity number of potential flow solutions are possible for the lifting flow over a circular cylinder. The same situation applies to the potential flow over an airfoil Kutta condition: Among the infinite possible flows around an airfoil, (mathematically), the one unique solution (physically) is determined by the requirement that the flow leave the trailing edge smoothly. 5 MAE 252 course notes Finite TE angle Cusped TE 6 MAE 252 course notes Mathematically, Kutta condition states that the vorticity at trailing edge is equal to zero 4.6 Kelvin’s circulation theorem and starting vortex For a potential flow, the circulation Γ remains always constant for a curve enveloping a given set of fluid elements. 7 MAE 252 course notes Apply Kelvin’s theorem to an airfoil Airfoil at rest airfoil moving in a stationary fluid with some velocity and α Kelvin’s theorem is applicable only for a given set of fluid elements 8 MAE 252 course notes sheet 9 .