aero complex

7/30/2019 Aero Complex

1/12

Chapter 3

The complex velocity

In ch. 2 it was shown that plane ideal flows satisfying

u

x+

v

y= 0 (3.1)

v

x

u

y= 0 (3.2)

satisfy the continuity and Euler equations for a perfect fluid. In ch. 3, we discover a connectionwith the theory of complex variables that provides many solutions to the flow equations.

3.1 Review of complex variables

This material is standard: see, for example, Abbott and von Doenhoff (1959, pp. 4748).

An imaginary number is formed by multiplying a real number y by the imaginary unit i, aconstant with the special property

i2 = 1 . (3.3)

A complex number is formed by adding a real number to an imaginary number: z = x + iy .This gives a one-to-one correspondence between points of the plane and complex numbers;the complex number x + iy is called the complex coordinate of the point (x, y).

We define, for z = x + iy :

z = x (real part), and (3.4)

z = y (imaginary part). (3.5)

Complex numbers can be added and subtracted by adding and subtracting the real andimaginary parts.

(x1 + iy1) (x2 + iy2) = (x1 x2) + i(y1 y2) . (3.6)

Complex numbers can be multiplied according to the usual rules of algebra, augmentedby (3.3):

(x1 + iy1)(x2 + iy2) = x1x2 + i(x1y2 + y1x2) + i2y1y2 (3.7)

= (x1x2 y1y2) + i(x1y2 + y1x2) . (3.8)

29


2/12

30 AERODYNAMICS I COURSE NOTES, 2005

Complex numbers are equal if and only if their real and imaginary parts are both equal.

The polar coordinates in the plane are related to the Cartesian by

x = r cos (3.9)y = r sin (3.10)

and

r =

x2 + y2 (3.11)

= arctany

x. (3.12)

In terms of the polar coordinates,

z = x + iy = r cos + ir sin . (3.13)

De Moivres Theorem:

ei cos + i sin . (3.14)

Therefore a complex number can also be expressed in polar form as:

z = x + iy = rei . (3.15)

Multiplication is easier in polar form:

r1ei1r2e

i2 = r1r2ei(1+2) . (3.16)

The modulus (or magnitude or absolute value) of a complex number is defined by

mod z |z| = |rei| = r . (3.17)

The argument (or phase) of a complex number is defined by

arg z arg(rei) = . (3.18)

The value of the argument is nonunique in that any multiple of 2 can be added, since thecosine and sine functions are periodic.

The complex conjugate z of a complex number z has the same real part and oppositeimaginary part, and same magnitude but opposite phase:

z = z iz = |z|ei arg z . (3.19)

We have:

z + z = 2z (3.20)

z z = i2z (3.21)

zz = |z|2 . (3.22)

The last of these results is useful for rendering the denominator of a complex fraction real:

z1z2

=z1z

2

|z2|2. (3.23)


3/12

The complex velocity 31

3.2 Analytic functions

A complex function of a complex variable is said to be analytic if it possesses a well-definedderivative. Say

f(z) = f(x + iy) = g(x, y) + ih(x, y) ,

where g(x, y) = f(z) and h(x, y) = f(z). Then the differential of f(z) is

df =g

xdx +

g

ydy + i

h

xdx +

h

ydy

and the differential of z isdz = dx + idy ,

and the derivative of f(z) is

df(z)

dz=

gx dx +

gy dy + i

hxdx +

hy dy

dx + idy

. (3.24)

When dy = 0, so that dz = dx, this becomes

df(z)

dz=

g

x+ i

h

x,

and when dx = 0, so that dz = idy, it becomes

df(z)

dz=

gy

+ i hy

i=

h

y i

g

x.

For the derivative of f to be well defined, these two expressions must be equal. Equating thereal and imaginary parts then gives

g

x =

h

y (3.25)

h

x=

g

y. (3.26)

These equations are the condition that a complex function be analytic; they are called theCauchyRiemann equations.

All the elementary functions ofz (powers ofz, trigonometric functions, exponential function,and hyperbolic functions) have the usual derivatives, e.g. dzk/dz = kzk1, dez/dz = ez. Theseare all analytic, except at points where the derivatives are undefined; e.g. for 1/z, the derivativeis

d

dz

1

z=1

z2

which is undefined at z = 0. Isolated points where an otherwise analytic function doesnt havea derivative are called singularities.

3.3 The complex velocity

3.3.1 Analytic functions and plane ideal flow

Consider the analytic functionw(z) = u(z) iv(z) .


4/12


The CauchyRiemann equations require

u

x=

v

y(3.27)

v

x=

u

y, (3.28)

which are seen to be identical to the requirements that the plane velocity field with componentsu and v be divergence-free (3.1) and irrotational (3.2). Therefore, any analytic function defines asolution of the two-dimensional continuity and Euler equations. We call w = u iv the complexvelocity.

3.3.2 Polar form

For a complex velocity field w(z) = u iv, the speed is given by

q(z) = |w| =

u2

+ v2

. (3.29)

The direction of the velocity at a point, expressed as its slope, is

v

u= tan ( arg w) . (3.30)

Thus the complex velocity can also be expressed as

w(z) = u iv = qei arctan v/u = qei , (3.31)

where (z) is the angle of the velocity at the point z to the positive x-axis.

3.3.3 Polar components

The radial and polar components of a velocity field are related to the Cartesian components by

vr = u cos + v sin (3.32)

v = v cos u sin . (3.33)

This can be expressed as (Ashley and Landahl 1965, p. 47)

vr iv = eiw (3.34)

orvr iv = qe

i() . (3.35)

Note that vr iv

, unlike u iv , is not an analytic function.

3.3.4 General component

For any angle , the vector i cos +j sin has unit magnitude and is at an angle anticlockwisefrom the positive x-axis. Therefore, the component of a vector q = ui + vj in this direction is

(i cos +j sin ) q = (i cos +j sin ) (ui + vj) (3.36)

= u cos + v sin . (3.37)


5/12


Now consider the product

eiw = (cos + i sin )(u iv) (3.38)

= u cos + v sin + i(u sin v cos ) . (3.39)

Thus

eiw

= u cos + v sin (3.40)

gives the component of velocity in the direction and

eiw

= v cos u sin (3.41)

the perpendicular component (reckoned positive in the direction rotated a quarter-turn anti-clockwise, i.e. making an angle + /2 with the positive x-axis). These expressions are useful inaerodynamics for finding the components of velocity tangential and normal to a given surface.

In terms of the polar form:

e

i

w = qe

i()

(3.42)eiw = qcos( ) (3.43)

eiw = qsin( ) . (3.44)

3.4 The complex potential

Define W(z) by the differential equation

dW

dz= w (3.45)

where w(z) is the complex velocity w = uiv . The function W(z) is called the complex potential.It is evidently analytic, since it has a derivative: w . Adding any complex constant to the complexpotential has no effect on the aerodynamics; it is sometimes convenient for the mathematics.

Let the real and imaginary parts of the complex potential W be and :

W = + i . (3.46)

Using the definition (3.45),dW

dz=

d + id

dx + idy= w = u iv . (3.47)

Taking dy = 0 so that dz = dx gives

x+ i

x= u iv . (3.48)

Taking dx = 0 so that dz = idy gives

i

y+

y= u iv . (3.49)

Thus

x= +

y= u (3.50)

y=

x= v . (3.51)


6/12


7/12


The vorticity in polar coordinates is (Thwaites 1987, pp. 60, 469)

=1

r (rv)

r

vr , (3.64)

so for w = zk,

=1

r

(rk+1 sin[k + 1])

r

(rk cos[k + 1])

(3.65)

=1

r

(k + 1)rk sin(k + 1) + (k + 1)rk sin(k + 1)

= 0 . (3.66)

3.5.2 Complex potentials

The complex potential for the complex velocity w(z) = zk is

W(z) = zk dz (3.67)=

(k + 1)1zk+1 , (k = 1) ;

ln z , (k = 1) .(3.68)

Notice that the case w = z1 is special. The logarithm of a complex number z satisfies

ln z = ln

rei

(3.69)

= ln r + ln

ei

(3.70)

= ln r + i . (3.71)

That is

(ln z) = ln |z| (3.72)(ln z) = arg z . (3.73)

Thus for w = z1,

= ln r (3.74)

= . (3.75)

3.5.3 Examples

k = 1: corner flow

Set w = z1 = z. The Cartesian components are u = x and v = y. This is illustrated in figure 3.1

with an arrow plot. This representation is similar to that which would be obtained experimentallyby holding an array of thin light short threads of silk in the flow (Prandtl and Tietjens 1957,p. 266), except that in addition to the direction, it also shows the relative magnitude via thelength of the arrows. Since the horizontal component u = x vanishes on the vertical y-axis andthe vertical component v = y vanishes on the horizontal x-axis, w = z represents a flow withimpermeable walls along the x and y axes.

From (3.55), we see that the speed of the flow increases without bound with distance fromthe origin for positive k. We will therefore normally be more interested in nonpositive powers ofz in aerodynamical applications.


8/12


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(a)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(b)

Figure 3.1: The complex velocity field w = z : arrow plot (a) and level curves of the imaginary(solid) and real (dashed) parts of the complex potential W = z2/2 (b).

k = 0: uniform stream

Set w = z0 = 1. Then the Cartesian components are u = 1 and v = 0; i.e. a uniform stream inthe x-direction with unit speed. The field is plotted in figure 3.2.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(a)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(b)

Figure 3.2: The complex velocity field w = 1 : its arrow plot (a) and level curves of the real(dashed) and imaginary (solid) parts of its complex potential W = z (b).

k = 1: source

Set w = z1. Then, referring to (3.34), the polar components are given by

vr iv = eiw (3.76)

= e iz1 (3.77)

= e i

rei1

(3.78)

= r1 ; (3.79)


9/12


i.e.

vr = 1/r and (3.80)

v = 0 . (3.81)

The velocity is purely radial out from the origin with a speed inversely proportional to thedistance. Since the circumference of a circle grows in proportion to its radius, the net flow out ofany circle centred at the origin is independent of the radius. This flow is called a source at theorigin. It satisfies the continuity equation everywhere except at the origin, where the complexfunction z1 is singular. The arrow field is plotted in figure 3.3(a).

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(a)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(b)

Figure 3.3: The complex velocity field w = z1 and its complex potential W = log z .

Since the magnitude of the velocity is unbounded at the origin, the simple arrow plot (fig-ure 3.3 a) is less useful.

k = 2: doublet

Set w = z2 . The Cartesian and polar components of the velocity are:

u =x2 y2

(x2 + y2)2(3.82)

v =2xy

(x2 + y2)2(3.83)

vr =cos

r2(3.84)

v =sin

r2. (3.85)

The flow, called a doublet, is plotted in figure 3.4.

3.6 Modifying complex velocity fields

3.6.1 Multiplication by a complex constant

Say we have a complex velocity field w(z) expressed in polar form w = qei , where q is thespeed and the angle to the positive x-axis. Multiplying an analytic function by a constant


10/12


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(a)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(b)

Figure 3.4: The complex velocity field w = z2 (a) and its complex potential W = 1/z (b) .

gives another analytic function. The velocity field obtained is

w =

Aei

qei

= (Aq)ei() ,

so that the effect of multiplying by the complex constant Aei is to change the speed to q = Aqand the direction to = .

Thus, for example, noting that i = ei/2, multiplication by the imaginary unit i keeps thespeed the same but rotates the velocity vector at each point through a clockwise right-angle.

Example: uniform stream with arbitrary direction

The uniform horizontal (rightward) stream w(z) = 1 becomes, on multiplication by ei/2, w =

iw = i, with components u

= 0 and v

= 1. This is a uniform vertical downward stream.A uniform stream with speed q and angle is therefore obtained by multiplying the complex

velocity w = 1 by the complex constant qei:

qei = (q cos ) i (q sin ) . (3.86)

Example: vortex

Since the velocity of a source is purely radial, multiplying its complex velocity z1 by i rotatesthe velocity at each point through a right-angle to become purely azimuthal:

w = iz1 = ir1ei (3.87)

vr iv = eiw = ir1 , (3.88)

so vr = 0 and v = 1/r . This flow is called a vortex; it is plotted in figure 3.5.

Example: polar components

Another example of multiplying the complex velocity by a constant to rotate the velocity vectorsat a point can be seen in the formula (3.34) for obtaining the polar components. This is becausethe polar components can be thought of as the components relative to a perpendicular unitvectors that have been rotated so that one is aligned with the radius from the origin.


11/12


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(a)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(b)

Figure 3.5: Arrow (a) and complex potential (b) fields for the complex velocity w = i/z .

3.6.2 Linear combinations

Iff(z) and g(z) are analytic functions and and are complex constants, the linear combinationf(z) + g(z) is also analytic.

Example: circular obstacle in a stream

From (3.60) the radial components of velocity for the uniform stream vr = cos and the doubletvr = r2 cos have the same dependence on the polar angle , and are identical on the circler = 1. Therefore, subtracting the complex velocity for a doublet from the complex velocity fora uniform stream

w = 1 1

z2

(3.89)

gives a velocity field

vr =

1

1

r2

cos (3.90)

v =

1 +

1

r2

sin (3.91)

that has zero radial component on the circle. This is as if the circle were an impermeable barrier,and so w = 1 z2 for |z| > 1 is the complex velocity for a stream with a circular obstacle.The field is plotted in figure 3.6. With this interpretation of the velocity field, the velocity insidethe circle is irrelevant; the same velocity field restricted to the exterior of the circle is shown infigure 3.7.

3.6.3 Translating the whole velocity field

If we have an analytic function f(z) and form another by replacing z by z z0, the effect is totranslate the whole velocity field.


12/12


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(a)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(b)

Figure 3.6: The complex velocity field w = 1 z2.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 3.7: The velocity field of figure 3.6, restricted to |z| > 1 .

aero complex

Documents