Potential Flow Theory for Development of A Turbine Blade

P M V SubbaraoProfessor

Mechanical Engineering Department

A Creative Mathematics…..

THE DOUBLET

• The complex potential of a doublet

zW

2 ma2

Uniform Flow Past A Doublet : Perturbation of Uniform Flow

• The superposition of a doublet and a uniform flow gives the complex potential

zUzW

2

z

UzW

2

2 2

iyx

iyxUW

2

2 2

iyx

yyyxUi

yx

xxyxUW

22

32

22

23

2

2

2

2

22

32

22

23

2

2 &

2

2

yx

yyyxU

yx

xxyxU

222

yx

yUy

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

2220

yx

yUy

yyxUy 2220 2220

yx

yUy

2220 yxU

Uyx

2 22

222

2 R

Uyx

•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.

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

UR

2

•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.

zUzW

2 FlowUniformzUW

z :lim

Study of the Superposed Function

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

UzW

2

22 zU

dz

dW

22222

22

4

2

2 yxyx

ixyyxU

dz

dW

2222222222

22

422

42 yxyx

xyi

yxyx

yxU

dz

dW

ivudz

dW

22222

22

42 yxyx

yxUu

22222 4 yxyx

xyv

222 vuV

2

22222

2

22222

222

442

yxyx

xy

yxyx

yxUV

Equation of zero stream line:

UR

2 222 yxR with

Cartesian and polar coordinate system

sin

cos

ry

rx

sin

cos

Vv

Vu

222 sin4 UV

4

4

2

222 2cos21

r

R

r

RUV

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.

2cos12 22 UV

22max 4UV

UV 2max

Generation of Local Vorticity

2cos12 22 UV

Creation of Pressure Distribution

No Net Up wash as an Effect

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 .

If the closed curve does not include the origin, the circulation will be zero.

ziiW ln2

Uniform Flow Past A Doublet with Vortex

• The superposition of a doublet and a uniform flow gives the complex potential

ziz

UzW ln22

z

zizUzW

2

ln2 2

iyx

iyxiyxiiyxUW

2

)ln()2 2

ziz

UzW ln22

Angle of AttackUnbelievable Flying Objects

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.

z

czzf

2

This gives

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

22

2

1yx

cx

22

2

1yx

cy

11 2

2

2

222

rc

rrc

rr

y

r

x

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

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

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

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.

Flow Over An Airfoil

Pressure Distribution on Aerofoil Surface

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.