We’ll gloss over considerable detail but fill a couple of gaps in the derivations that get us to, “streamline coordinates”. Otherwise, this post is just organizing references in hopes of creating a useful guide on this topic of lift as a result of, “streamline curvature”.

I can’t improve the theory relative to these expert sources. I cannot come close to the genius of Euler and the father-son Bernoulli team! I hope I can clarify it for we mere mortals.

Euler’s Equations

Euler’s Equations for inviscid flow will reveal what causes the lift once we get it into the right form. Inviscid flow is ideal, simplified flow, assumed to slip past the wing with no surface interaction (no shear, so no boundary layer, eddies,  turbulence)…only the shape of the wing affects the flow, not the surface characteristics.

The Equations will reveal where the $V^2$ for the propeller speed-to-thrust relationship comes from. We glossed over this topic in the post with the NASA diagram where we saw the simplified equation with the $V^2$ term.

In the last post we resolved some confusion around the motor+gearbox+propeller model for load on the motor. We cover here the entire point of spinning the propeller: lift from the blades or aggregate thrust along the axis of the propeller.

The following PDF extracts key pages from Fay’s book (my old draft copy) that lead us to, “Euler’s Equation in Streamline Coordinates”. It includes a few pages of Bernoulli equation derivation. It is integrated from Euler’s equation along a streamline. It is a frequently applied formula.

The typically overlooked perpendicular-to-streamline (normal) Euler equation tells us about lift from an airfoil!

James A. Fay: Euler's Equation

Example Application

This tutorial by the late Professor Sonin is excellent. I enjoy his neat sketches and remember them well, as I took notes in his class and attempted to match his detail and neatness.

Sonin: StreamLine Equations of Motion

It’s not easy to digest the mathematical symbols, multivariable calculus, and terminology unless one has an excellent memory of their studies or practices these topics regularly. We’ll fill in some gaps below.

Cylindrical Coordinates

Although it might look daunting on its own, let’s accept Euler’s equation in cartesian (XYZ) coordinates. Then in Fay’s and Sonin’s material above we see we need cylindrical-streamline coordinates to get the equations of motion into some form that will enlighten us on this lift force we so desire.

Step-by-Step Derivation

The key to deriving cylindrical coordinates is to start by representing the unit vectors relative to a Cartesian system.

We start with radial and angular unit vectors in an X-Y plane, and we know Z in the Cartesian system is the same as Z in Cylindrical, so from the above diagram…

\begin{equation}
i_r = cos(\theta(t))i + sin(\theta(t))j
\end{equation}
\begin{equation}
i_{\theta} = -sin(\theta(t))i + cos(\theta(t))j
\end{equation}
\begin{equation}
i_z = i_z
\end{equation}

A big difference between cylindrical coordinates and Cartesian coordinates is the rotation of the ‘r’ and ‘$\theta$’ axes in time, as our, “fluid particle” moves in $\theta$. Hence showing $\theta$ as a function of time in the above equation.

We need derivatives of the unit vectors, and here’s what they are:

\begin{equation}
\dot{i_r} = (-sin(\theta(t))i + cos(\theta(t))j)\dot\theta
\end{equation}

Look at the trig in parenthesis. It’s our term for $i_{\theta}$ above so…

\begin{equation}
\dot{i_r} = \dot\theta i_\theta
\end{equation}

And

\begin{equation}
\dot{i_\theta} = (-cos(\theta(t))i -sin(\theta(t))j)\dot\theta
\end{equation}

Again we see a familiar expression from above so this simplifies to…

\begin{equation}
\dot{i_\theta} = -\dot\theta i_r
\end{equation}

It is these moving axes that give us what look like extra terms in the cylindrical equations of motion. We get to them simply by taking derivatives of position and velocity.

We start with the position of a particle in cylindrical coordinates.

\begin{equation}
P=r\cdot i_r + z\cdot i_z
\end{equation}

By chain rule, we differentiate position to velocity as…

\begin{equation}
V=\frac{dP}{dt} = \dot r\cdot i_r + r\cdot \dot i_r + \dot z \cdot i_z + z\cdot \dot i_z
\end{equation}

But we have terms for the derivatives of the unit vectors from above, and $\dot i_z=0$ so velocity simplifies to…

\begin{equation}
V=\dot r\cdot i_r + r\dot\theta i_\theta + \dot z \cdot i_z
\end{equation}

Then we differentiate Velocity to get acceleration, again by the chain rule…

\begin{equation}
a = \frac{dV}{dt}=\ddot r\cdot i_r + \dot r\cdot \dot i_r + \dot r\dot \theta i_\theta + r\ddot \theta i_\theta + r\dot \theta \dot i_\theta + \ddot z \cdot i_z + z\dot \cdot \dot i_z
\end{equation}

and we again replace those unit vector derivative terms and $\dot i_z=0$ to get, with a bit of rearrangement…

\begin{equation}
a = \frac{dV}{dt}=(\ddot r – r\dot \theta^2)\cdot i_r + (2\dot r \dot \theta + r\ddot \theta) \cdot i_\theta + \ddot z \cdot i_z
\end{equation}

The $2\dot r \dot \theta$ is the, “Coriolis” acceleration term and the $r\dot \theta^2$ is the “centrifugal” acceleration term for a particle. They’re nothing more than a manifestation of the rotating unit vector $i_\theta$, which we can see by following the derivation above from cartesian to polar.

Streamline Coordinates

These are just a slight twist on Cylindrical coordinates. We relate cylindrical coordinates to streamline coordinates as follows…

$i_n=-i_r$    The perpendicular-to-streamline or, “normal” to flow unit vector.

$i_s=i_\theta$          The along-path unit vector, tangent to streamline unit vector.

$i_b=i_z$          By right-hand-rule, normal to the above two unit vectors: The, “up” axis of a cylindrical system. Out-of-page axis here. No fluid dynamics along this axis.

The derivative relationships are

\begin{align*}
\frac{\delta}{\delta n} = -\frac{\delta}{\delta r} \\
\frac{\delta}{\delta s} = \frac{\delta}{r\delta \theta}
\end{align*}

That last denominator might look confusing but it’s just as simple as spanning a distance $ds$ with angular change $d\theta$ operating on a radius $r$.

If you now go back to the Fay PDF and accept Euler’s Equation in Cartesian coordinates on page 90-91 in the PDF above you can use the steps above to get confident with the cylindrical acceleration representation, also on page 90. From there you use the unit vector reltionships to adjust the cylindrical representation to the streamline representation and you get to Fay’s page 105 here below.

Understanding, “lift” from Streamline Coordinates

The key equation for us is

\begin{equation}
\frac{\delta p}{\delta n} = -\rho\frac{V(n)^2}{R(n)}
\end{equation}

V and R being likely functions of n, integrating to estimate a pressure difference would be non-trivial. Sonin’s paper gives us a chance to solve the n-direction equation with models for Velocity and streamline radius as a function of n. See the paper to attempt this derivation for flow over a simple hill.

Fay and Sonin state that this is not a practical method for calculating a lift force. It is over-simplified, and we seldom have representations for V and R as a function of n over a range of s-vectors tangent to the flow over a wing. If we did, we’d still need to integrate them. However, this treatment tunes us into the physics of the problem. Our goal here is physical understanding, even if exact solutions elude us.

The approximation below from some 1995 Sonin lecture notes shows us what we need to know to appreciate why our prop-speed-squared ($\omega^2$) is our lift producer.

Understanding Approximations for Lift

The n-direction equation tells us what we need to know to understand the lift generated by airfoils (thrust from propellers: spinning airfoils). As Sonin’s paper above states, “The n-direction equation states that when there is flow and the streamlines curve, the sum $p+\rho gz$ (which is constant when the fluid is static) increases in the n-direction, that is, as one moves away from the local center of curvature.”

If we imagine integrating from the wing’s surface to far above the wing the streamlines way out at our approximated infinity would have a very large radius while the airspeed is $V_\infty$.

If we just assume airspeed is $V_\infty$ everywhere (a crude approximation because we are speeding it up over the wing) but let the streamline radius get larger from top-surface radius R as we move away from the wing then we can estimate a low pressure on the top of the wing proportional to $\rho v_\infty^2$ as indicated in the lecture notes above.

Remember that if our propeller is spinning with angular velocity $\omega$. At any point along the propeller where we could take a cross-section to sketch an airfoil diagram the “V” is just the propeller rotational rate times the radius from the propeller axis to the location of our cross-section.

\begin{equation}
V_{cross-section} = \omega_{prop}\cdot r_{cross-section}
\end{equation}

At this particular propeller cross-section, we could estimate the free-stream velocity-squared as

\begin{equation}
V_\infty^2 = V_{cross-section}^2 = (\omega_{prop}\cdot r_{cross-section})^2 = \omega_{prop}^2 \cdot r_{cross-section}^2
\end{equation}

There’s our propeller $\omega^2$ term!

We now see how propeller rotational speed squared ($\omega^2$) is the variable responsible for, the “lift” generated by a propeller blade. It is proportional to a simplified, assumed freestream velocity $V^2$ at any point along the blade where we take a cross-section as above. This is a highly simplified model, but it gets us to the essence of a propeller as an airfoil.

Observe this on your next flight

Next time you land or take off in an airplane observe the flaps that extend down from the leading edge and back from the rear of the wings. They create a shorter wing curvature radius and a larger wing surface. This increases lift.

It is required because the plane is going much slower at take-off and landing than when cruising so the pressure difference across the same wing would be less at low speed.

So, the modifiable airfoil extends the flaps, creates a larger surface over which the pressure difference acts, and adds more curvature to increase the pressure difference at low airspeed. This occurs at the expense of drag, however. Given that it is only needed at low speeds, the flaps are retracted for cruising, where $ V_infty$ is higher, so we don’t need the curvature and extra area the flaps provide.

There remains that $\rho$ term for air density, and we know it is higher in the troposphere down near the ground than up at 30,000 feet and higher. We’re going to ignore this because our quadrotor is going to operate near the ground. Planes are more sensitive to “stall” (failure of the lift from wings) at high altitudes due to the rarefied (low-density) air. Airspeed and angle-of-attack are critical factors up there.

Conclusion

We now know that when our quad-rotor main controller asks our 4 motors for more-or-less speed many times per second it is the speed-squared ($\omega^2$) that gives us lift!

I hope the references above and any gap-filling explanations help convey the, “lift” phenomenon!