In this week’s installment we will once again try to unify many different concepts. Our mission (should we choose to accept it!) is to first develop the fundamental notions of a branch of math called Complex Analysis. While it can be a difficult topic to wrap one’s head around, it is incredibly powerful. After all, there is a saying among mathematicians that the shortest path between a problem and its solution usually passes through the complex plane. After studying what complex numbers and complex functions are, we will explore the topics of complex differentiability and conformal maps. From there, we will have the necessary tools to understand how the Joukowski map, which is a specific example of a conformal map, can allow engineers to analyze the flow around airfoils more easily.
To begin our discussion, we ask: What is the square root of -1? In our usual number system, which we call the real numbers, we cannot take the square root of a negative number. Therefore, if we wish to understand expressions involving the square root of -1, we have to extend our number system. We will do this by developing the imaginary number “i,” which is defined to be the square root of -1. We will then define a complex number z to be equal to x + iy, where x and y by themselves are real and i is the square root of -1. We often call x the real part of z, or Re(z) for short. Similarly, we call y the imaginary part of z, or Im(z) for short. If we set y = 0, then z could be any real number. Therefore, all of the real numbers are still part of the complex numbers, only the complex numbers allow us to do even more than the real numbers allow us to do.
The analog of the absolute value for complex numbers is called the modulus, denoted by |z|. If z = x + iy, then |z| = sqrt(x2 + y2). Furthermore, we can graph complex numbers on a plane, where the horizontal axis represents Re(z) and the vertical axis represents Im(z). This complex plane is usually the best way to visualize complex numbers and the transformations they can undergo when a function acts on them.
This brings us to functions of a complex variable. We can recall that a function f(x) on the real number line takes in one input, x, and for each input, gives us one output, often called y. Similarly, a function f(z) on the complex plane takes in one complex number of the form z = x + iy as its input, and returns a complex number w = u + i v as its output. Since the inputs and outputs of complex functions have two degrees of freedom, the complete graph of f(z) would be four-dimensional, and hence very difficult to visualize! It is common to write a complex function f(z) as follows, where x = Re(z) and y = Im(z):
f(z) = u(x,y) + i v(x,y).
This representation is nice because it allows us to analyze what happens to the real and imaginary parts of the output as we change the real and imaginary parts of the input.
Just as we have the notion of the derivative of regular functions, we can define the derivative of a function of a complex variable, df/dz. While it is more difficult to interpret df/dz geometrically, it is essentially a measure of how much f(z) is nudged when we nudge the real and imaginary components of z a tiny bit. Through some mathematical wizardry, we can show that a function f(z) has a complex derivative at a point z0 = x0 + iy0 if and only if the following equalities simultaneously hold at z0.
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
These are called the Cauchy-Reimann equations, and their consequences are so important that it is noteworthy to spend a few sentences describing one of their implications. In particular, if f(z) = u(x,y) + i v(x,y) is differentiable over the complex plane, then ∆u = 0 and ∆v = 0. To observe that u and v have zero Laplacians, we can observe:
∆u = ∂2u/∂x2 + ∂2u/∂y2 = ∂/∂x (∂v/∂y) – ∂/∂y (∂v/∂x)
∆v = ∂2v/∂x2 + ∂2v/∂y2 = – ∂/∂x (∂u/∂y) + ∂/∂y (∂u/∂x).
There is a rule called Clairaut’s Theorem that tells us that, under nice conditions, taking the partial derivative with respect to x and then with respect to y is the same as first differentiating with respect y and then taking the derivative with respect to x. Thus, both of the equations above are equal to zero. The consequence of ∆u = 0 and ∆v = 0 is that the gradients (more on the gradient in a future post) of u(x,y) and v(x,y) can represent fluid flow that has no sources or sinks and is irrotational. By irrotational, we mean that if we throw a stick into the flow, then the stick would not rotate.
Now back to our primary mission. What will be of interest to us in the following paragraphs is that all complex functions satisfying the Cauchy-Reimann equations are conformal maps. This means that they preserve the local angles between curves in the complex plane. To make this more concrete, suppose we had two perpendicular lines defined by Re(z) = 3 and Im(z) = 3. The line Re(z) = 3 consists of every point in the complex plane whose real part is equal to 3, and the line Im(z) = 3 consists of every point in the complex plane whose imaginary part is equal to 3. Then, if we fed all the points on both of these lines into a conformal map, say C(z), then in our output space we would get two curves that intersect each other at a right angle.
The special conformal map that we will consider is the Joukowski map, defined by f(z) = z + 1/z. What this function does is map circles in the input space (which is one complex plane) onto airfoil-shaped curves in the output space (which is another complex plane). These airfoil shapes are, indeed, called Joukowski airfoils. This transformation has made the lives of aerospace engineers easier because it simplifies the process of analyzing fluid flow around a wing. Instead of doing all of our analysis around the airfoil itself, which is quite cumbersome, we can model the flow around a circle, and then feed all of our analysis into the Joukowski map to determine how the flow would behave around an airfoil. This process works because of the map’s nice angle-preserving properties.
As it turns out, I just learned about Joukowski airfoils in my advanced engineering mathematics class. Therefore, there is plenty more research for me to do before I can fully understand their nuances. However, what we have covered so far can illustrate the deep connections between seemingly unrelated concepts in math and engineering. After all, we began our discussion by asking what the square root of -1 is, and we ended our discussion by exploring conformal maps and their applications in the study of airfoils. Next time, we will continue our discussion by studying the gradient of a multivariable function and a concept called potential flow. Until then, take care.