Course Number: 804538

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 gare 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 iandjare 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 likeare analytic functions

3.1 Complex Variable Theory Applied to 2-D Irrotational Flow • Note: • If is a complex number • Its complex conjugate is • IfFsatisfies 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,wis 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 areal, • 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 , andthereforeF 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 zplane to the F plan is angle preserving except at z=0.

Example 2 Complex variable – analytic function • Report the previous example, but with areal • 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 zplane 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 asteriskdenotes 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 contourC, 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 transformedinto 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 vortexhas 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 velocityon the cylinder is given by: • ByBernoulli’s equation thepressureon 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 Joukowskitransformation • This transformation adds to a circle of radius b its inversepointto 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) • We obtain • 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

Course Number: 804538