Potential Flows and Conformal Maps

Introduction

In this post we will create the potential flow and conformal map interactive visualization shown above. The goal here is to gain some familiarity and visual intuition of how conformal maps work and how they are used in the theory of aerodynamics. The flows will be rendered using WebGL in an observableHQ notebook, following the approaches of Ricky Reusser and Graham Pullam.

Potential Flow

We will focus on fluid flows whose velocity fields can be written as gradients of scalar potentials, $ \mathbf{u} = \nabla \phi$. Conservation of mass for incompressible fluids, $\nabla \cdot \mathbf{u}= 0$ implies that the scalar potential satisfies Laplace’s equation $\nabla^2 \phi = 0$. Thus, we can create potential flows using solutions of Laplace’s equation, called harmonic functions, and which are mathematically well understood. Harmonic functions are also linearly superimposable, meaning that their linear combinations are also valid solutions to Laplace’s equations. We will exploit this fact to create flows which are relevant to the theory of aerodynamics, using three fundamental building blocks: 1) the uniform flow, 2) the point vortex flow, 3) and the doublet flow. A uniform flow is characterized by its freestream velocity, $U_{\infty}$, and its flow angle, $\alpha$. A point vortex flow is characterized by its circulation, $\Gamma$, and a doublet flow is characterized by its doublet strength, $\kappa$, and (for simplicity) the same flow angle, $\alpha$. The complex velocity induced by each of these flows are,

where $\zeta \in \mathbb{C}$ is the position in complex space. The interactive plot at the beginning of the post plots the streamlines as well as the pressure field induced by each of the above flows. The dropdown menu allows us to toggle between the different flows as well as their linear combinations. For example, choosing Uniform + Vortex Flow plots $W_{U} + W_{V}$. The coefficients $U_\infty,\alpha,\Gamma,\kappa$ can be adjusted using the sliders next to the plot. I encourage you to play around with the settings we discussed so far and see how they change the flow pattern.

Flow Over a Cylinder

An interesting flow arises when we add the Uniform and Doublet Flows. You might have noticed that one of the streamlines for this flow forms a closed circle which varies in size as the freestream velocity and doublet strength change. By definition, the flow velocity is parallel to a streamline and never perpendicular to it. This implies that a closed streamline divides the flow into two regions that cannot communicate. That is, the fluid must flow around that circular streamline and not through it. This is the identical effect that a solid body would have if immersed in a potential flow. We can consider that streamline to be the boundary of a cylindrical solid body whose radius we can compute analytically as a function of the flow variables. The radius ends up being $a = \sqrt{\frac{\kappa}{U_\infty}}$. We can then visualize the body by replacing the flow inside the streamline with whitespace. Checking the option Add body under the plot does just that. Remember, this does not change the flow at all. It just helps us see the boundary between the flow inside and outside the dividing streamline.

An important characteristic of this flow is that we can add a vortex of any strength, $\Gamma$, at the center of the circle without violating this boundary condition. We can see this by choosing Uniform + Vortex + Doublet Flow and moving the $\Gamma$ slider. While the streamlines change quite a bit, they never cross the boundary of the circle. Mathematically, this means that the solution to this problem is non-unique, or that there are several (in this case infinite) valid solutions.

Conformal Maps

Perhaps the most powerful idea in the theory of potential flows is that of a conformal map. Conformal maps allow us to transform solutions of simple problems to solutions of difficult problems. For example, there exists conformal maps that can take the solution for the flow over the cylinder shown above, to the flow over a body of any shape. A conformal map of interest to the theory of aerodynamics is the Joukowsky transform. This transform takes the flow over a cylinder and maps it to the flow over some very reasonable looking airfoils! The transform is defined as,

The map takes every point in the complex $\zeta$-plane to a point in the a new complex z-plane. Checking the option Apply Joukowsky Transform applies the transform to the visualization. While the transform does some interesting stuff to each of the flows, the body never quite looks like an airfoil. To get an airfoil shape we need to move the center of the circle away from the origin and apply the same transform. The sliders Circle x-coordinate and Circle y-coordinate do just that. You can play around with the settings until you have something that looks like an airfoil, or you can just press the Get Reasonable Airfoil button. I encourage you to fiddle around with how each of the parameters changes the shape of the airfoil or the characteristics of the flow around it. For example, changing the Circle y-coordinate adjusts the camber distribution of the airfoil, while Circle x-coordinate adjusts the thickness distribution

The Kutta Condition

Conformal maps guarantee that the flow will not cross through the body’s surface, so long as the flow does not cross the pre-transformed body’s surface. Hence, the flow over the airfoil inherits the non-uniqueness of the flow over the circular cylinder. This means that we can set the circulation, $\Gamma$ to any value without affecting correctness of the solution. A natural question to ask is what is the “physical” circulation value, or the one that would occur in practice? One reason this is important is because the Kutta-Joukowski theorem tells us that the lift produced by the airfoil is proportional to its circulation, $L = \rho U_\infty \Gamma$. How can we determine the physically occurring circulation? The answer is provided by the Kutta condition, which states the circulation must be such so that the flow leaves the trailing edge of the airfoil smoothly. The circulation necessitated by the Kutta condition is,

We have previously defined all of these parameters except $\beta$, which is the angle of the line connecting the center of the circular cylinder with the cylinder’s x-intercept, and is effectively a function of airfoil’s camber. Pressing the Apply Kutta Condition button sets the circulation to the correct value. We can visually confirm that flow indeed leaves the trailing edge smoothly. We can then change the flow parameters and reapply the Kutta condition. We have now covered each of the options/buttons of the calculator.