1. Potential Flows Title: Advisor : Ali R. Tahavvor, Ph.D.
Islamic Azad
University Shiraz
Potential Flows
Branch Department of
Engineering .Faculty of
Advisor : Ali R. Tahavvor, Ph.D.
Mechanical Engineering
Year:2012 June
By : Hossein Andishgar
Pouya Zarrinchang

3. FLOWS

4. FLOWS

5. FLOWS

6. STREAKLINES

7. The Streaklines

8. Streamtubes

9. 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/2p)ln(z)
The vortex : A source of energy and momentum.
Complex function for Uniform Flow : W = (ig/2p)ln(z)

10. Velocity field
Velocity field can be found by differentiating streamfunction
On the cylinder surface (r=a)
Normal velocity (Ur) is zero, Tangential velocity (U) is non-zero slip condition.

11. Irrotational Flow Approximation
Irrotational approximation: vorticity is negligibly small
In general, inviscid regions are also irrotational, but there are situations where inviscid flow are rotational, e.g., solid body rotation

12. Irrotational Flow Approximation 2D flows
For 2D flows, we can also use the stream function
Recall the definition of stream function for planar (x-y) flows
Since vorticity is zero,
This proves that the Laplace equation holds for the streamfunction and the velocity potential

14. Elementary Planar Irrotational Flows Uniform Stream
In Cartesian coordinates
Conversion to cylindrical coordinates can be achieved using the transformation
Proof with Mathematica

17. Elementary Planar Irrotational Flows source & sink
A doublet is a combination of a line sink and source of equal magnitude
Source
Sink

18. Elementary Planar Irrotational Flows Line Source/Sink
Potential and streamfunction are derived by observing that volume flow rate across any circle is
This gives velocity components

19. Elementary Planar Irrotational Flows Line Source/Sink
Using definition of (Ur, U)
These can be integrated to give  and 
Equations are for a source/sink
at the origin
Proof with Mathematica

20. Elementary Planar Irrotational Flows Line Source/Sink
If source/sink is moved to (x,y) = (a,b)

23. Elementary Planar Irrotational Flows Line Vortex
Vortex at the origin. First look at velocity components
These can be integrated to give  and 
Equations are for a source/sink
at the origin

24. Elementary Planar Irrotational Flows Line Vortex
If vortex is moved to (x,y) = (a,b)

26. Two types of vortex

27. Rotational vortex Irrotational vortex

28. Cyclonic Vortex in Atmosphere

29. 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):
Complex potential of a sink 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 :

30. 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.

31. Dipole

33. Elementary Planar Irrotational Flows Doublet
Adding 1 and 2 together, performing some algebra, and taking a0 gives
K is the doublet strength

34. 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

35. Doublet 2D

37. Examples of Irrotational Flows Formed by Superposition
Superposition of sink and vortex : bathtub vortex
Sink
Vortex

41. Examples of Irrotational Flows Formed by Superposition
Flow over a circular cylinder: Free stream + doublet
Assume body is  = 0 (r = a)  K = Va2

43. circular cylinder

44. Flow Around cylinder

45. Flow Around cylinder

46. Flow Around a long cylinder

47. Streaklines around cylinder & Effect of circulation and stagnation point

49. 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.

50. 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,

51. Cp distribution of flow over a circular cylinder

53. Combining source and a sink

56. Compare the flow around golf ball& sphere

57. Combination Source +sink + uniform flow

64. Flow around Airfoil

65. Airfoil anatomy

66. Airfoil and boundry layer and sepration

67. Airfoil @ streamlines & stream function

72. JOUKOWSKI’S transformation
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

74. This means that
For a circle of radius r in Z plane x and y are related as:

75. Consider a cylinder in z plane

77. C=0.8

78. C=0.9

79. C=1.0

82. 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.

83. Flow Over An Airfoil

84. Mapping of cylinders through JOUKOWSKI’S transformation : (a) flat plate ; (b) elliptical section ; (c) cymmetrical wings;(d)asymmetrical wing

85. JOUKOWSKI

87. Thank you for your attention
source :
1-Fluid Mechanics M.S.Moayeri 2-Introduction to Fluid Mechanics - Yasuki Nakayama 3-Mechanics of Fluids - Irving Herman Shames 4-White Fluid Mechanics 5th 5-Mansun Mechanical Fluid 6-Wikipedia 7-
Thank you for your attention
Good Luck