Fundamental Concepts

(This page reproduces some content in the online course that complements the AeroPython Jupyter notebooks.)

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Conservation of mass

To start learning aerodynamics, you need a good foundation: an understanding of the physics and the basic models of fluid flow. You probably have studied this in a general course in fluid mechanics. But let's review.

At the heart of fluid mechanics are the principles of conservation of mass, momentum and energy.

To explain conservation of mass, we use a control volume and state that, since mass cannot be created or destroyed:

where is the mass flow rate (in or out of the control volume).

In words: the rate at which the mass within a control volume changes is equal to the net rate at which mass flows in or out of the control volume.

Derivation of the differential equation of conservation of mass

Watch this video "pencast," walking you through the derivation of the differential equation of conservation of mass. (Recorded some time ago for Prof. Barba's course in computational fluid dynamics.)

https://youtu.be/7UpOXp_i4cc

Conservation of momentum

You remember Newton's second law for a system? The time rate of change of the linear momentum of a system is equal to the sum of external forces acting on the system.

Applying Newton's second law to a fluid, we transfer this statement from a system point of view to a control-volume point of view. All students of fluid mechanics need to master this process, by which we can derive the world-renowned Navier-Stokes equations.

Derivation of the differential equation of conservation of momentum

Watch this video "pencast," walking you through the derivation of the differential equation of conservation of momentum. (Recorded some time ago for Prof. Barba's course in computational fluid dynamics.)

Pencast: derivation of the momentum equation https://youtu.be/_k38xJ_e8UU

Pencast: the Navier-Stokes equations https://youtu.be/aak3XNAuucU

Common Misconceptions in Aerodynamics

Lecture by Doug McLean

Lecture by Doug McLean at the University of Michigan (October 2013), discussing several examples of erroneous ways of looking at aerodynamic phenomena.

Some of the topics covered include:

• Basic physics
• Newton's third law—often erroneously used to explain thrust.
• Lift explanations—full of misconceptions in textbooks and popular literature!
• Vorticity and the Biot-Savart law
• Lift and momentum in 3D

https://youtu.be/QKCK4lJLQHU

Biography of Doug McLean

Doug McLean is a retired Boeing Technical Fellow. At Boeing, he worked on CFD codes for transonic wing design, codes for airplane spanload optimization including the effect of structural weight, novel wingtip devices to reduce induced drag, transonic airfoil technology, swept-wing laminar flow, turbulent skin-friction reduction, and pressure-sensitive paint. He received a B.A. in physics from the University of California at Riverside in 1965 and a Ph.D. in Aerospace and Mechanical Sciences from Princeton University in 1970. He is the author of "Understanding Aerodynamics - Arguing from the Real Physics"(Wiley, 2012),

Airflow across a wing

In this 1-min video released by the University of Cambridge in 2012, we see what happens when airflow moves around the curved surfaces of an airfoil. Pulses of smoke show that the airflow moves faster over the top of the airfoil, and reaches the trailing edge before the airflow under the bottom of the airfoil.

This video was made to help debunk one of the most prevalent and long-lasting misconceptions of aerodynamics: the idea of "equal transit time."

Read more on the Research News section of the University of Cambridge website.

https://youtu.be/UqBmdZ-BNig

Reading

— "How do wings work?" by Holger Babinsky, Physics Education, Vol. 38, 497 doi:10.1088/0031-9120/38/6/001

Flow visualization

Fluid flow can be surprisingly complex. Generations of scientists and engineers have used flow visualization to help them unravel the puzzles of fluids. We need to understand four basic concepts of visualization: path line, streak line, time line and stream line. We will use just the first 13 minutes of this classic video, part of a series made in 1961 by initiative of Prof. Ascher Shapiro of MIT (you can stop watching after that, for the purposes of this lesson).

Watch the video, pay close attention, and attempt the quiz questions!

At the end of this learning sequence, you'll find explanations of streakline, pathline and streamline.

https://youtu.be/nuQyKGuXJOs

Streakline, pathline and streamline

STREAKLINE

If you release some dye continuously into the fluid at a fixed point, the dye spreads into a streak. If you take a snapshot of the instantaneous configuration of the dye you will see a streakline. But if the flow is transient, another snapshot a while later will show a different configuration.

PATHLINE

If you follow a single small parcel of fluid on its actual path over time, that is a pathline. Note that you can't just take a photograph to see a pathline. You would need some long-exposure of a tagged parcel of fluid.

A possible analogy to how you might visualize a pathline is a long-exposure photograph of cars in the night. The path left on the film by the car lights would be analogous to pathlines.

image in the public domain from pixabay.com

STREAMLINE

If you sprinkle small solid particles on the fluid surface, or have a suspension of small visible particles, then take a short-time exposure photograph of the flow, you will capture on the photo short traces of each particle's motion. Mentally, you can imagine the lines that these traces might form, and thus approximately "see" streamlines.

But streamlines are mathematical ideas and you can't directly observe them experimentally. The technique above gives only an approximation. Streamlines are curves that are everywhere parallel to the fluid velocity.

(classic image from M. van Dyke's book – the full book on PDF is available online)

Potential Flow

Velocity potential

Assumptions of potential flow

Potential-flow theory has been a powerful mathematical framework in aerodynamics. Its power derives from the fact that it produces a linear equation that can be solved to get the full velocity field. This is surprising, because the equation of motion for fluid flow is nonlinear.

What's the magic? Assuming that the flow is irrotational.

Irrotational means that the fluid does not carry vorticity, i.e., that the curl of velocity is zero:

$$ \omega= \nabla\times\vec{v}=0$$

This seems like a big restriction. But it turns out that flow fields that are irrotational almost everywhere are very common! So the study of irrotational flow has great practical value in aerodynamics!

That is not the only assumption we need to make, though. We also require that:

• the flow is steady, and
• the velocity remains smaller than the speed of sound (the flow is incompressible).

Circulation

With the assumption that $\nabla\times \vec{v}=0$, we can apply Stokes' theorem1 to get that for any closed curve lying entirely in the fluid:

$$\oint \vec{v} \cdot d\vec{l}=0$$

where $d\vec{l}$ is a unit vector tangent to the integration curve.

If you remember from your undergraduate fluid mechanics course, this integral is the definition of circulation, $\Gamma$:

$$\Gamma = \oint \vec{v} \cdot d\vec{l}$$

Due to Stokes' theorem, if the flow is irrotational, the circulation around any closed contour is zero. Therefore, if we have two paths joining points $A$ and $B$ in the flow such that they form a closed contour, the line integral of velocity going from $A$ to $B$ cancels that going from $B$ to $A$, over any path:

$$\int_A^B \vec{v}\cdot d\vec{l} + \int_B^A \vec{v}\cdot d\vec{l}=0$$

In other words, the line integral of velocity going from $A$ to $B$ always has the same value, regardless of the path. Simply expanding the argument of the integral, we get:

$$\int_A^B \vec{v}\cdot d\vec{l}=\int_A^B u, dx + v, dy + w, dz$$

and the previous statement means that the argument $u, dx + v, dy + w, dz$ is an exact differential of a potential $\phi$, where

$$ u = \frac{\partial \phi}{\partial x}, \quad v = \frac{\partial \phi}{\partial y}, \quad w=\frac{\partial \phi}{\partial z}$$

Or, that $\vec{v}=\nabla\phi$.

The function $\phi$ is called the velocity potential.

Continuity

The continuity equation for incompressible flow is $\nabla \cdot \vec{v}=0$. Applying the definition of velocity potential here results in this beautifully simple equation that governs potential flow:

$$\nabla^2 \phi=0$$

This equation is so lovely because it's linear—we can apply the superposition principle among solutions!

Note:

If you are really rusty with your vector calculus and don't remember Stokes' theorem, you might brush up with Khan Academy's lessons on "Surface integrals and Stokes' theorem."

Visualizing vorticity in a potential vortex

https://youtu.be/loCLkcYEWD4

The theory of lift

Now is the time for a little light reading! We are sharing with you here one chapter of Theodore von Karman's gem of a book, "Aerodynamics: Selected Topics in the Light of Their Historical Development."

• "The theory of lift," Chapter 11. The_Theory_of_Lift.pdf

Magnus effect

https://youtu.be/23f1jvGUWJs