1. Potential Flow Theory for Development of A Turbine Blade
P M V Subbarao
Professor
Mechanical Engineering Department
A Creative Mathematics…..

2. THE DOUBLET
The complex potential of a doublet

3. Uniform Flow Past A Doublet : Perturbation of Uniform Flow
The superposition of a doublet and a uniform flow gives the complex potential

4. Find out a stream line corresponding to a value of steam function is zero

5. There exist a circular stream line of radium R, on which value of stream function is zero.
Any stream function of zero value is an impermeable solid wall.
Plot shapes of iso-streamlines.

6. Note that one of the streamlines is closed and surrounds the origin at a constant distance equal to    

7. Study of the Superposed Function
Recalling the fact that, by definition, a streamline cannot be crossed by the fluid, this complex potential represents the irrotational flow around a cylinder of radius R approached by a uniform flow with velocity U.
Moving away from the body, the effect of the doublet decreases so that far from the cylinder we find, as expected, the undisturbed uniform flow.
In the two intersections of  with the x-axis with the cylinder, the velocity will be found to be zero.
These two points are thus called stagnation points.

8. To obtain the velocity field, calculate dw/dz.

9. Equation of zero stream line:
with

10. Cartesian and polar coordinate system

12. Surface V2 Distribution of flow over a circular cylinder
The velocity of the fluid is zero at = 0o and = 180o. Maximum velocity occur on the sides of the cylinder at = 90o and = -90o.

13. Generation of Local Vorticity

14. Creation of Pressure Distribution

15. No Net Up wash as an Effect

16. THE VORTEX
In the case of a vortex, the flow field is purely tangential.
The picture is similar to that of a source but streamlines and equipotential lines are reversed.
The complex potential is
There is again a singularity at the origin, this time associated to the fact that the circulation along any closed curve including the origin is nonzero and equal to g.
If the closed curve does not include the origin, the circulation will be zero.

17. Uniform Flow Past A Doublet with Vortex
The superposition of a doublet and a uniform flow gives the complex potential

21. Angle of Attack
Unbelievable Flying Objects

22. Transformation for Inventing a Machine
A large amount of airfoil theory has been developed by distorting flow around a cylinder to flow around an airfoil.
The essential feature of the distortion is that the potential flow being distorted ends up also as potential flow.
The most common Conformal transformation is the Jowkowski transformation which is given by
To see how this transformation changes flow pattern in the z (or x - y) plane, substitute z = x + iy into above expression.

23. This gives
For a circle of radius r in z plane is transformed in to an ellipse in z - planes:

25. Flow past circular cylinder in Z-plane is seen as flow past an elliptical cylinder of c=0.8 in z – plane.

26. Flow past circular cylinder in Z-plane is seen as flow past an elliptical cylinder of c=0.9 in z – plane.

27. Flow past circular cylinder in Z-plane is seen as flow past an elliptical cylinder of c= 1.0 in z – plane.

28. Translation Transformations
If the circle is centered in (0, 0) and the circle maps into the segment between and lying on the x axis;
If the circle is centered in (xc ,0), the circle maps in an airfoil that is symmetric with respect to the x axis;
If the circle is centered in (0,yc ), the circle maps into a curved segment;
If the circle is centered in and (xc , yc ), the circle maps in an asymmetric airfoil.

29. Flow Over An Airfoil

30. Pressure Distribution on Aerofoil Surface

31. Final Remarks
One of the troubles with conformal mapping methods is that parameters such as xc and yc are not so easily related to the airfoil shape.
Thus, if we want to analyze a particular airfoil, we must iteratively find values that produce the desired section.
A technique for doing this was developed by Theodorsen.
Another technique involves superposition of fundamental solutions of the governing differential equation.
This method is called thin airfoil theory.