Advance Fluid Mechanics Course Number: 804538 Advance Fluid Mechanics Faculty Name Prof. A. A. Saati

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

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

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

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

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

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

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

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

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

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

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.

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.

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.

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.

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.

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.

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

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

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.

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

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

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

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

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

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.

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

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.

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

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

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

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

Section 3.9 & 3.10 Reading

END OF LECTURE 5