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.