Airfoils to Generate Axial Induction P M V

Airfoils to Generate Axial Induction P M V Subbarao Professor Mechanical Engineering Department I I T Delhi An amazing fluid muscle……

The Strange Natural Geniuses : The Wright Brothers Wilbur Wright on Their Flying Experiments "There is no sport equal to that which aviators enjoy while being carried through the air on great white wings. More than anything else the sensation is one of perfect peace mingled with an excitement that strains every nerve to the utmost if you can conceive of such a combination

The Strange Natural Geniuses : The Wright Brothers Wilbur Wright on Their Flying Experiments "I am an enthusiast, but not a crank in the sense that I have some pet theories as to the proper construction of a flying machine. I wish to avail myself of all that is already known and then, if possible, add my mite to help on the future worker who will attain final success. “

The Strange Natural Geniuses : The Wright Brothers Wilbur Wright on Their Flying "We could hardly wait to get up in the morning. “ "I confess that in 1901, I said to my brother Orville that man would not fly for 50 years. “ "The fact that the great scientist believed in flying machines was the one thing that encouraged us to begin our studies. "

The Strange Natural Geniuses : The Wright Brothers Wilbur Wright on Their Flying "It is possible to fly without motors, but not without knowledge and skill. “ "The desire to fly is an idea handed down to us by our ancestors who. . . looked enviously on the birds soaring freely through space. . . on the infinite highway of the air. “ "Men become wise just as they become rich, more by what they save than by what they receive. "

Development of an Ultimate Fluid machine : The Frozen C Device

The Basic & Essential Cause for Generation of Lift • The experts advocate an approach to lift by Newton's laws. • Any solid body that can force the air downward clearly implies that there will be an upward force on the airfoil as a Newton's 3 rd law reaction force. • From the conservation of momentum for control Volume • The exiting air is given a downward component of momentum by the solid body, and to conserve momentum, • something must be given an equal upward momentum to solid body. • Only those bodies which can give downward momentum to exiting fluid can experience lift ! • Kutta-Joukowski theorem for lift.

Newton’s Third Law for a Lifting Device Any solid body that can force the air downward clearly implies that there will be an upward force on the airfoil as a Newton's 3 rd law reaction force.

Kutta-Joukowski theorem for lift - 1902 • The exiting air is given a downward component of momentum by the solid body, and to conserve momentum, something must be given an equal upward momentum to solid body. • Only those bodies which can give downward momentum to exiting fluid can experience lift ! • This is known as Kutta-Joukowski theorem for lift.

Prandtl’s Experimental Photo of Starting Vortex - 1904

Fascinating Vortex Phenomena : Kutta. Joukowski Theorem

Fascinating Vortex Phenomena : Kutta-Joukowski Theorem The Joukowsky transformation is a very useful way to generate interesting airfoil shapes. However the range of shapes that can be generated is limited by range available for the parameters that define the transformation.

Three Basic Steps in Creation of Thin Aerofoil Theory

The term “potential theory” arises from the fact that, in 19 th century physics, the fundamental forces of nature were believed to be derived from potentials which satisfied Laplace’s equation.

Paul Richard Heinrich Blasius, 1911 • Paul Richard Heinrich Blasius was born on 9 August 1883 in Berlin. • He studied at the universities of Marburg and Gottingen from 1902 to 1906. • He joined as a scientific collaborator and research assistant with Ludwig Prandtl from 1908 at the hydraulics laboratory of Berlin technical University. • He was one of the first students of Prandtl who provided a mathematical basis for boundary-layer drag. • He also showed as early as 1911 that the resistance to flow through smooth pipes could be expressed in terms of the Reynolds number for both laminar and turbulent flow. • He developed another Powerful theorem in 1911.

The Theorem of Blasius • Let a steady uniform flow past a fixed two-dimensional body with bounding contour C be a harmonic flow with velocity potential w(z). • Then, if no external body forces are present, the force (X, Y ) exerted by the fluid on the body is given by

The Complex Potential : Model for A Machine • Inviscid Flow field and force generating solid(s) can be invented as a complex potential. • In particular we define the complex potential In the complex plane every point is associated with a complex variable In general we can then write

Derivatives of Complex Functions Now, if the function f is analytic, this implies that it is also differentiable, meaning that the limit so that the derivative of the complex potential W in the complex z plane gives the complex conjugate of the velocity. Thus, knowledge of the complex potential as a complex function of z leads to the velocity field through a simple derivative.

Functions for Elementary Fascination • To Create IRROTATIONAL PLANE FLOWS – The uniform flow • The perturbation objects – The source and the sink – The vortex

THE UNIFORM FLOW : Creation of mass & Momentum in Space The first and simplest example is that of a uniform flow with velocity U directed along the x axis. In this case the complex potential is

THE SOURCE OR SINK: The Perturbation Functions • Source (or sink), the complex potential of which is • This is a pure radial flow, in which all the streamlines converge at the origin, where there is a singularity due to the fact that continuity can not be satisfied. • At the origin there is a source, m > 0 or sink, m < 0 of fluid. • Traversing any closed line that does not include the origin, the mass flux (and then the discharge) is always zero. • On the contrary, following any closed line that includes the origin the discharge is always nonzero and equal to m.

Diagnosis of Source/Sink Function Iso f lines Iso y lines v. The flow field is uniquely determined upon deriving the complex potential W with respect to z.

A Combination of Source & Sink