# From Circular Cylinder to Tilted Droplet Wing

## 1. Circular Cylinder

To understand the generation of lift and drag of a wing it is useful to study potential flow around a circular cylinder with focus on the attachment of the flow at the frontal part of the cylinder as a model of the attachment at the leading edge of a wing and the flow over the top of the cylinder as model of the flow over the crest of the wing.

The picture below shows the pressure (left) and velocity/streamlines (right) in a section of potential flow (from left to right) around a circular cylinder:

We see that the flow is fully symmetric so that the pictures would be the same if instead the flow was from right to left. We see that the pressure is high (red) in the front of the cylinder, but also in the back giving zero drag (Alembert’s paradox), while the pressure is low on top and bottom (blue) giving also zero lift. We see that the velocity is low (blue) in the high pressure zones and high (red) in the low pressure zones, in accordance with Bernoullis’s principle.

The separation of the flow from the cylinder in the rear is of crucial importance: Potential flow separation can be described as 2D irrotational slip separation with line stagnation (along a line on the cylinder surface parallel to the cylinder axis), noting that separation and attachment have the same pattern. In d’Alembert’s Paradox we show that potential flow is exponential unstable at separation (but not at attachment) and therefore develops into a different quasi-stable separation pattern as 3D rotational slip separation with point stagnation, in which the high pressure of 2D irrotational slip separation with line stagnation is replaced by free stream pressure with non-zero drag as illustrated here:

## 2. Wing

The flow around a wing can roughly be thought of as the flow around the upper frontal quarter of the cylinder which generates drag and lift, with the rest of the flow as around the upper half of the cylinder extended back to a trailing edge as indicated in the following principal sketch:

The secret of flight is hidden in answers to the following questions:

• Why does the flow not separate on the crest of the wing but follows the downward direction of the upper wing surface and thus creates downwash?
• How can the flow can separate so smoothy from the trailing edge without high/low pressure?

You cannot fly with circular cylinder wings, you can fly with great effort on upper half cylinder wings, and you can fly with elegance on frontal half circular cylinders smoothed and extended to titled droplets.

## Analytical Solution Formula for Circular Cylinder

We now give the analytical formula for potential flow around a circular cylinder of unit radius with axis along the $x_3$-axis in 3d space with coordinates $x=(x_1,x_2,x_3)$, assuming the flow velocity is $(1,0,0)$ at infinity in each $x_2x_3$-plane, which in polar coordinates $(r, \theta)$ in a plane orthogonal to the cylinder axis is given by the potential function

• $\varphi (r,\theta )=(r+\frac{1}{r})\cos(\theta )$

with corresponding velocity components

• $u_r\equiv\frac{\partial\varphi}{\partial r}=(1-\frac{1}{r^2})\cos(\theta )$
• $u_s\equiv\frac{1}{r}\frac{\partial\varphi}{\partial\theta}=-(1+\frac{1}{r^2})\sin(\theta )$

with streamlines being level lines of the conjugate potential function

• $\psi\equiv (r-\frac{1}{r})\sin(\theta$.

Potential flow is constant in the direction of the cylinder axis with velocity $(u_r,u_s)=(1,0)$ for $r$ large, is fully symmetric with zero drag/lift, attaches and separates at the lines of stagnation $(r,\theta )=(1,\pi )$ in the front and $(r,\theta )=(1,0)$ in the back.

By Bernoulli’s principle the pressure is given by

• $p=-\frac{1}{2r^4}+\frac{1}{r^2}\cos(2\theta )$

when normalized to vanish at infinity. We see that the negative pressure on top and bottom  (-3/2)  is 3 times as big in magnitude as the high pressure at the front (+1/2). We see that the pressure switches sign for $\theta =30$.

We also compute

• $\frac{\partial p}{\partial\theta}=-\frac{2}{r^2}\sin(2\theta )$
• $\frac{\partial p}{\partial r}=\frac{2}{r^3}(\frac{1}{r^2}-\cos(2\theta ))$

and discover an adverse pressure gradient in the back with unstable retarding flow.

Potential flow around a circular cylinder shows exponentially unstable 2d irrotational separation and quasi-stable 2d attachment, and is a useful model for both attachment at the leading edge of a wing and separtion at a rounded trailing edge.