1. Advance Fluid Mechanics
Course Number:
Advance Fluid Mechanics
Faculty Name
Prof. A. A. Saati

2. Lecture # 6 Chapter: 3 Irrotational Two-Dimensional Flows
Ref ADVANCED FLUID MECHANICS By W. P. Graebel

3. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
The theory of complex variables is ideally suited to solving problems involving 2-D flow.
Complex variable: means that a quantity consists of the sum of real and an imaginary number.
An imaginary number: is a real number multiplied by the imaginary number

4. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
In many ways complex variables theory is simpler than real variable theory and much more powerful
Complex function F that depends on the coordinates x and y is written in the form:
Where f and g are real functions
Complex function has some of the properties of a 2-D vector.
The above equation has the directionality properties of unit vector
Where i and j are Cartesian unit vectors

5. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
Complex function F can be represented in graphical form and also a spatial position as in the figures
The position vector is written as , where z is a complex number (z here is not 3-D coordinate)
The plane hear is called the complex plane:
F plane & z plane

6. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
The principle of Complex function theory is in that subclass of complex function in this equation
That have a unique derivative at a point (x,y)
Unique derivative: means that, if F is differentiated in the x direction obtaining
And , if F is differentiated in the y direction obtaining

7. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
The results at a given point are the same
For this to be true, the two equations must be equal
and
And it must be that
This equations are called Cauchy-Riemann conditions
Functions whose real and imaginary parts satisfy them are called analytic functions

8. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
Most functions of complex variable z that involve
multiplication,
division,
exponentiation,
trigonometric functions,
Hyperbolic functions,
Exponentials,
Logarithms,
And the like are analytic functions

9. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
Note:
If is a complex number
Its complex conjugate is
If F satisfies the Cauchy-Riemann conditions, will not
Note: a transformation of the form is angle-preserving
Angle-preserving transformations are said to be conformal

10. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
Comparison of equation
With
Shows that the complex function
With as the velocity potential
And as Lagrange’s stream function is analytic function
Since and have already seen the Cauchy-Reimann conditions, w is termed the complex potential

11. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
From differentiation of w find that
The derivative of the complex velocity potential is the complex conjugate of the velocity
Which is thus an analytic function of z

12. Example 1 Complex variable – analytic function
For with a real,
Find the real and imaginary parts of F,
Show that F is an analytic function, and
Decide whether the mapping form z is conformal.

13. Example 1 Complex variable – analytic function
Solution.
Putting into F,
Since
This reduces to
By separation
To study the analyticity of F, using partial derivatives
Thus, F satisfies the Cauchy-Riemann equation , and therefore F is analytic function of z.

14. Example 1 Complex variable – analytic function
Solution.
Since has no singularities for finite z,
and is zero only at z=0,
The mapping from the z plane to the F plan is angle preserving except at z=0.

15. Example 2 Complex variable – analytic function
Report the previous example, but with a real
Find the real and imaginary parts of F,
Show that F is an analytic function, and
Decide whether the mapping form z is conformal.

16. Example 2 Complex variable – analytic function
Solution.
Putting into F, and
Using DeMoive’s theorem, which states that
Find that
By separation
To study the analyticity of F, using partial derivatives
So F is analytic function of z in the entire finite a plan.

17. Example 2 Complex variable – analytic function
Solution.
Since has no singularities for finite z,
is finite and nonzero for all finite z,
The mapping from the z plane to the F plan is angle preserving.

18. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
Comparison of equation with our basic flows in Chapter 2 gives the following
Where an asterisk denotes a complex conjugate.
And the vortex is counterclockwise if , and clockwise if
The expressions above are much more compact and easier to remember and work with than the forms in Ch.2

19. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
The power of complex variable theory is that
Since an analytic function of an analytic function is analytic
We can solve a flow involving a simple geometry and then use an analytic function to transform or map, that into a much more complicated geometry
Note that
because we are dealing with analytic function the fluid mechanics of a flow is automatically satisfied
It is often easier to integrate analytic function than it is to integrate real function
The reason is Cauchy’s integral theorem

20. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
If a function is analytic and single-valued and on a closed contour C, then
Further, for a point inside C.
Here is the nth derivative of evaluated at
This theorem is useful in determining forces on bodies.

21. 3.1 Complex Variable Theory Applied to 2-D Irrotational Flow
Note that the theorem is restricted to single-valued functions and
This mean that for a given z, there is only one possible value for
Example of single-valued function is
Example of a multivalued function is , which is arbitrary to a multiple of

22. 3.2 Flow Past a Circular Cylinder with Circulation
The mapping process showing how the flow past circular cylinder can be transformed into the flow past either an ellipse or an airfoil shape
Form Chapter 2 that the flow of uniform stream past circular cylinder is a doublet facing upstream in a uniform stream, write
The vortex has been added at the center of the circle

23. 3.2 Flow Past a Circular Cylinder with Circulation
Note that:
Since on the cylinder
The complex potential on the cylinder is
Which is real
Also since
Then we can easily find the stagnation point in the flow

24. To put this in a more understandable form
Let giving:
If the stagnation point on the cylinder
if then there are two stagnation points are at

25. If then equation shows that the stagnation point move off the circle

26. 3.2 Flow Past a Circular Cylinder with Circulation
Since the velocity on the cylinder is given by:
By Bernoulli’s equation the pressure on the cylinder is then
Where is the pressure at the stagnation point.

27. 3.2 Flow Past a Circular Cylinder with Circulation
The force on the cylinder per unit distance into the paper is then
Form the definition of G and that

28. 3.2 Flow Past a Circular Cylinder with Circulation
Note three things about the equation
The force is always independent of the cylinder size, being simply the fluid density times the circulation times the stream speed.
The force is always perpendicular to the uniform stream, so it is a lift force.
To have this lift force, circulation must be present.

30. 3.3 Flow Past an Elliptical Cylinder with Circulation
To find the flow past an ellipse, introduce the Joukowski transformation
This transformation adds to a circle of radius b its inverse point to the position z
Note: an inverse point to a circle is the point such that

31. 3.3 Flow Past an Elliptical Cylinder with Circulation
To see what the Joukowski transformation does,
let in the equation
And then divide the resulting expression into real and imaginary parts.
Where a is the radius of the circle
And c is the semimajor axis in
d is semiminor axis in
If is eliminated, the equation of an ellipse is given by

32. 3.3 Flow Past an Elliptical Cylinder with Circulation
The direct solution of equation
We obtain
Then substituting this into equation of flow past circular cylinder with circulation (sec. 3.2)
Then the velocity given by

33. 3.3 Flow Past an Elliptical Cylinder with Circulation
From the above two equation, and using the chain rule of calculus
We can see that the velocity is infinite at the point
Using the expressions for the semimajor and -minor axes, the parameter b in the transformation is given by

34. Section 3.9 & Reading

36. END OF LECTURE 5