19AEE 301 Aerodynamics II Lecture Notes PDF

D’ Alembert Nikolay Yegorovich Lord Rayleigh Zhukovsky D’ Alembert Michael Max Munk Sir Issac Newton...

D’ Alembert Nikolay Yegorovich Lord Rayleigh Zhukovsky D’ Alembert Michael Max Munk Sir Issac Newton Hermann Glauert Ludwig Prandtl Course Instructor: Dr. Rajesh Senthil Kumar. T  Lecture Outline: 1) Circulation 2) Kutta-Zhukovsky lift theorem 3) Kutta Condition for lift Generation (airfoil) 4) Kelvin’s Circulation theorem 5) Helmholtz’s Circulation theorem Dr. Rajesh Senthil Kumar. T  Circulation ▪ A mathematical concept commonly associated with vortex motion is that of circulation. ▪ The circulation, is defined as the line integral of the tangential component of the velocity taken around a closed curve in the flow field. In equation form,  can be expressed as Γ ≡ − ‫𝑠𝑑 ⋅ 𝑉 ׯ‬ 2 , 𝑚 Τ𝑠 The circulation is simply the negative of the line integral of velocity around a closed curve in the flow; it is a kinematic property depending only on the velocity field and the choice of the curve C. Dr. Rajesh Senthil Kumar. T  Kutta - Zhukovsky Lift theorem: Theorem states that “ lift per unit span on a two dimensional body is directly propositional to the circulation around the body. 𝑙 ′ = 𝜌𝑉∞ Γ 𝑉∞ Representation of Circulation loop enclosing the airfoil Dr. Rajesh Senthil Kumar. T  Kutta condition for lift generation: For a given airfoil at a given angle of attack, the value of the circulation must take the unique value which ensure that the flow leaves the trailing edge smoothly. Representation of Circulation loop satisfying Kutta Condition Dr. Rajesh Senthil Kumar. T Finite Angle Trailing Edge Rounded Trailing Edge Cusped Trailing Edge Dr. Rajesh Senthil Kumar. T  Helmholtz’s Circulation Theorem First Theorem: The strength of a vortex filament is constant along its length. Second Theorem: A vortex filament cannot end in a fluid; it must extend to the boundary of fluid or form a closed loop. Third Theorem: In absence of rotational external forces, a fluid that is initially irrotational remains the same. Dr. Rajesh Senthil Kumar. T  Kelvin’s Circulation Theorem Circulation around a closed curve formed by a set of contiguous fluid elements remains same as the fluid element moves through the fluid. D = 0; Dt Dr. Rajesh Senthil Kumar. T