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Potential Flow Theory : Incompressible Flow
P M V Subbarao
Professor
Mechanical Engineering Department
IIT Delhi
A mathematical Tool to invent flow Machines.. ..
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Basic Elements for Construction Flow Devices
• Any fluid device can be constructed using following Basic elements.
• The uniform flow: A source of initial momentum.
• Complex function for Uniform Flow : W = Uz
• The source and the sink : A source of fluid mass.
• Complex function for source : W = (m/2ln(z)
• The vortex : A source of energy and momentum.
• Complex function for Uniform Flow : W = (iln(z)
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THE DIPOLE • Also called as hydrodynamic dipole.• It is created using the superposition of a source and a sink of
equal intensity placed symmetrically with respect to the origin.• Complex potential of a source positioned at (-a,0):
•The complex potential of a dipole, if the source and the sink are positioned in (-a,0) and (a,0) respectively is :
)ln(2
azm
W
• Complex potential of a sink positioned at (a,0):
)ln(2
azm
W
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Streamlines are circles, the center of which lie on the y-axis and they converge obviously at the source and at the sink. Equipotential lines are circles, the center of which lie on the x-axis.
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THE DOUBLET
• A particular case of dipole is the so-called doublet, in which the quantity a tends to zero so that the source and sink both move towards the origin.
• The complex potential of a doublet
is obtained making the limit of the dipole potential for vanishing a with the constraint that the intensity of the source and the sink must correspondingly tend to infinity as a approaches zero, the quantity
zW
2
ma2
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Uniform Flow Past A Doublet
• 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
iyxiyx
iyxiyxUW
2
2 2
22
2
2
2
yx
iyxiyxUW
22
2232223
2
222
yx
iyxxyiiyyixyixxyxUW
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22
2232223
2
222
yx
iyxxyiiyyixyixxyxUW
22
3223
2
2
yx
iyxiyyixxyxUW
22
3223
2
22
yx
yyyxUixxyxUW
22
3223
2
22
yx
yyyxUixxyxUW
22
32
22
23
2
2
2
2
yx
yyyxUi
yx
xxyxUW
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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
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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.
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Note that one of the streamlines is closed and surrounds the origin at a constant distance equal to
UR
2
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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 the x-axis with the cylinder, the velocity will be found to be zero.
These two points are thus called stagnation points.
zUzW
2
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Velocity components from w
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To obtain the velocity field, calculate dw/dz.z
UzW
2
22 zU
dz
dW
2
1
2 zU
dz
dW
22222
22
4
2
2 yxyx
ixyyxU
dz
dW
2222222222
22
422
42 yxyx
xyi
yxyx
yxU
dz
dW
ivudz
dW
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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
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Cartesian and polar coordinate system
sin
cos
ry
rx
sin
cos
Vv
Vu
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4
4
2
222 2cos21
r
R
r
RUV
On the surface of the cylinder, r = R, so
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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.
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Pressure distribution on the surface of the cylinder can be found by using Benoulli’s equation.
Thus, if the flow is steady, and the pressure at a great distance is p,
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Cp distribution of flow over a circular cylinder
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Development of an Ultimate Fluid machine
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Anatomy of an airfoil
• An airfoil is defined by first drawing a “mean” camber line.
• The straight line that joins the leading and trailing ends of the mean camber line is called the chord line.
• The length of the chord line is called chord, and given the symbol ‘c’.
• To the mean camber line, a thickness distribution is added in a direction normal to the camber line to produce the final airfoil shape.
• Equal amounts of thickness are added above the camber line, and below the camber line.
• An airfoil with no camber (i.e. a flat straight line for camber) is a symmetric airfoil.
• The angle that a freestream makes with the chord line is called the angle of attack.
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Conformal Transformations
P M V Subbarao
Professor
Mechanical Engineering Department
IIT Delhi
A Creative Scientific Thinking .. ..
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INTRODUCTION
• 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 the expression above to get
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This means that
For a circle of radius r in Z plane x and y are related as:
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Consider a cylinder in z plane
In – plane
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C=0.8
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C=0.9
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C=1.0
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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.
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Flow Over An Airfoil
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