Aerofoil as A Turbine Blade

P M V SubbaraoProfessor

Mechanical Engineering Department

Invention of Basic Fluid Machine…..

Development of an Ultimate Fluid machine

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 3rd 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.

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.

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 Elements in Creation of Thin Aerofoil Theory

Mathematics to Innovate Blade Profile

Creation of A Solid Object to Positively Perturb the Flow ……

THE COMPLEX POTENTIAL

• Invscid Flow field and perturbing solid(s) can be represented as a complex potential.

• In particular we define the complex potential

iW

In the complex plane every point is associated with a complex number

ireiyxz

In general we can then write

zfzizW

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.

z

zfzzf

dz

dfz

0lim

ivudz

dW

Elementary fascination Functions

• 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

UziW

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.

zm

iW ln2

The flow field is uniquely determined upon deriving the complex potential W with respect to z.

Iso lines

Iso lines

zm

iW ln2

A Combination of Source & Sink

THE DOUBLET

• The complex potential of a doublet

zW

2

ma2