1. Aerofoil Theory : Development of Turbine BladeBY
Dr. P M V Subbarao
Mechanical Engineering Department
I I T Delhi
A concept to construct a flying machine ….

2. The Aerofoils

3. Airfoil Geometry
Airfoil geometry can be characterized by the coordinates of the upper and lower surface.
It is often summarized by a few parameters such as: maximum thickness, maximum camber, position of max thickness, position of max camber, and nose radius.
One can generate a reasonable airfoil section given these parameters.

4. The NACA airfoils were created by superimposing a simple mean line shape with a thickness distribution that was obtained by fitting a couple of popular airfoils of the time: y = ±(t/0.2) * (.2969*x *x *x *x *x4) The camber line was defined as a parabola from the leading edge to the position of maximum camber, then another parabola back to the trailing edge.

5. Airfoil Pressure Distributions
The aerodynamic performance of airfoil sections can be studied most easily by reference to the distribution of pressure over the airfoil.
This distribution is usually expressed in terms of the pressure coefficient:
Cp is the difference between local static pressure and freestream static pressure, nondimensionalized by the freestream dynamic pressure.

6. Pressure Distribution around an Aerofoil
Cp is plotted "upside-down" with negative values (suction), higher on the plot. (This is done so that the upper surface of a conventional lifting airfoil corresponds to the upper curve.)

7. Angle of Attack

8. Parts of Pressure Distribution
Upper Surface:The upper surface pressure is lower (plotted higher on the usual scale) than the lower surface Cp in this case. But it doesn't have to be.
Lower Surface :The lower surface sometimes carries a positive pressure, but at many design conditions is actually pulling the wing downward. In this case, some suction (negative Cp -> downward force on lower surface) is present near the midchord.
Pressure Recovery : This region of the pressure distribution is called the pressure recovery region.
The pressure increases from its minimum value to the value at the trailing edge.
This area is also known as the region of adverse pressure gradient.
The adverse pressure gradient is associated with boundary layer transition and possibly separation, if the gradient is too severe.
Trailing Edge Pressure : The pressure at the trailing edge is related to the airfoil thickness and shape near the trailing edge.

9. Uniform Flow
U is real the flow is in the x direction with a speed U. The flow direction can be adjusted by changing real and imaginary parts.

10. Line Source or Vortex
When K is real the expression describes a source with radially directed induced velocity vectors; imaginary values lead to vortex flows with induced velocities in the tangential direction.

11. Doublet
A doublet is formed by superimposing a source and a sink along the x-axis. The doublet strength is given by S dx. The fundamental doublet singularity with the potential shown above is formed by taking the limit as dx goes to zero and S goes to infinity while keeping the product constant. The doublet is commonly used as one of the fundamental singularities in many panel methods.

12. Cylinders
The flow on a circular cylinder may be computed from a uniform stream and a doublet.

13. It is not derived here, but the result follows from the theory of residues, the complex potential, and the incompressible Bernoulli equation. (Or one might just use the momentum equation and compute the net force by far field integrals.)                                           
                             where G is the total circulation and S is the net source strength. In the case of no net source strength, the net force exerted on a collection of sources and vortices in a flow with freestream velocity U is perpendicular to the freestream and proportional to U and the total circulation.

14. Conformal Mapping
Any analytic function of a complex variable satisfies the equation for incompressible, irrotational flow:       
We can, therefore, relate one flow field to another by setting:          
where z' is related to z by an analytic function of z, z' = f(z). (Recall z = x + iy.) The idea behind airfoil analysis by conformal mapping is to relate the flow field around one shape which is already known (by whatever means) to the flow field around an airfoil. Most often a circle is used as the first shape.

15. The problem is to find an analytic function that relates every point on the circle to a corresponding point on the airfoil.                                                          
Joukowski found that the simple function: z' = z + 1/z transforms a circle to a shape which looks a bit like an airfoil.
By taking the origin of the circle at various points, different airfoil-like shapes are produced.

16. Thin Aerofoil Theory
A simple solution for general two-dimensional aerofoil sections can be obtained by neglecting thickness effects and using a mean-line only section model.
For incompressible, inviscid flow, an aerofoil section can be modelled by a distribution of vortices along the mean line.

17. The vortices along the mean line form a continuous vorticity distribution.
The assumed distribution function is shown in the following equation.
This function is Glauert's approximation and is based on Joukowski transformation results and obeys the Kutta condition with zero vorticity at the trailing edge.
The vorticity distribution is given as a function of the angular variable (q ) which is related to chordwise position (x) as follows,

18. where c is the chord length
where c is the chord length. Note that chordwise position (x) is used instead of distance along the mean line (s).
For typical aerofoils with small camber, the difference is negligible.
The magnitude of the vortex distribution strength must be calculated to complete the mathematical model.
For these thin cambered plate models a boundary condition of zero flow normal to the surface is applied in order to create an equation that can be solved for the required strengths.
The vorticity distribution function has two parts.
The first is a constant coefficient (Ao) multiplied by a tangent function which describes the variation due to angle of incidence effects.
The second part is a Fourier sine series which will account for variation due to camber.
Finally, both parts are scaled by multiplying by the freestream velocity.
Given an aerofoil geometry, freestream velocity and angle of incidence, the magnitude of the coefficients (A0,A1,A2,....) is to be found by solving the condition of zero flow normal to the surface.
This condition can be formulated in terms of horizontal and vertical flow velocity components.

19.             thus              
The ratio of vertical to horizontal velocity at the surface must equal the surface (mean line) gradient.
The flow horizontal and vertical velocities are made up of freestream and vortex induced components.
                            and                             
where ui and vi are the horizontal and vertical velocities induced by the vortex distribution.
Both of these components will be much less than the freestream velocity so for small angles of incidence the horizontal vortex induced component can be neglected.

20. If small angle assumptions are made for the incidence, the boundary
condition equation becomes
The velocity induced vertically (vi) at any point on the mean line can be found by summing up the effects of small individual segments (ds) of the vorticity distribution.
where x is the location at which the induced velocity is being calculated and s is the chordwise location of the vortex element.

21. Substituting this result for induced velocity into the boundary condition equation gives
The solution for coefficients (A0,A1,A2,...) can now be obtained from this equation.
The solution is based on Glauert's integral method.
The equation is summed along the chord line to find initially coefficient A0.
It is then scaled by cosine multiples and again summed along the chord.
Each scaled integration will yield one coefficient.

22. Once the vorticity coefficients are found the lift of a small element of the vortex line can be predicted from the Kutta-Joukowski law.
The complete lift is found by summing all elements of lift from leading to trailing edge.

23. Turbine Blade Profiles

25. Geometrical Properties of Blade Profile
In a turbine the fixed and the moving blades are arranged in such a way as to form endless cascades.
Blade cascades are made up of an infinite number of similar blade profiles situated equidistantly from each other.
The distance between points situated at exactly the same place on each of two adjoining profiles is known as the pitch of the cascade, t.
The dimension a shows the width of the interblade passage at exit.
The width of th eblade is b. b a t

26. Blade Profile b

27. Flow Pattern Through Nozzle Blade Cascade
Vni
Vfni
αni
Vuni
Vune
αne
Vfne

28. Flow Pattern Through Moving Blade Cascade
Vri
Vfi βi βe
Vfe

29. Flow Beyond a Blade Cascade