Codex Arundel

While reading Leonardo da Vinci’s Codex Arundel last evening, I noticed that the Codex had less scholars examining it relative to others. The fluid dynamics of da Vinci have been extensively studied, with entire dissertations dedicated to the subject. I came across a curious drawing that exhibited turbulent flow. The text is written backward in Italian, which is his typical style. Recently, Caltech conducted gravity experiments based on da Vinci’s calculations and were able to determine the constant and coefficient of gravity with great accuracy. This is remarkable, and it makes me wonder whether it is possible to replicate or test some of da Vinci’s ideas regarding turbulent flow. This would be an interesting historical project, but acquiring funding for such a project may be a challenge within the university system. Attached is the scan I took of the particular field in the Codex. This is not the usual drawing that people show when they talk about da Vinci’s turbulence.

Additional Thoughts on Pressure

I am obsessed with pressure, particularly the internal pressure of fluids. Unlike viscosity, it is absent of frictional forces, and it is a key driving force in both human behavior and aerospace flows. Pressure is an essential component of a perfect fluid and appears on the right-hand side of the Navier-Stokes equations. Without pressure, fluid flow is boring, and people cannot move forward. Pressure can manifest internally, like a fluid, or externally, on our boundaries. However, I prefer to derive my motivation from within, like intrinsic (motivation) pressure. Think of the poor fluid parcel or person moving only to external forces that they know nothing about (see Laplace).

A simplified semi-empirical model for long-range low-frequency noise propagation in the turbulent atmosphere

My student, Dr. Tianshu Zhang, and myself recently published a modified long range acoustic propagation model that handles turbulence in the atmosphere. The abstract is

We present a semi-empirical long-range low-frequency acoustic propagation model, which accounts for atmospheric turbulence. Ostashev and Wilson’s scattering model is combined with a ray-theory based refraction model to account for turbulent scattering and refraction via a turbulent absorption coefficient. The coefficient is ascertained via integration of scattered energy. The model is formulated, calibrated, and validated via corresponding experiments conducted within the National Science Foundation Boundary Layer Wind Tunnel. The predictions of the newly proposed ‘bridging model’ match the wind tunnel experimental data with an average error of 11.9%. Example predictions are shown to quantify the effect of turbulent kinetic energy and turbulent integral length scale on long-range infrasound propagation. To demonstrate the approach, we present predictions of the propagation of noise from a tornado and a nonlinear wave.

DOI: 10.1016/j.apacoust.2023.109256 [link]

Caverns of the Wind

Silent are the caverns of the wind,
But within them, a secret lies pinned
Like Pandora's box, locked up tight
Holding within, a glimmer of light

For in these tunnels, dreams did reside
And hope for the future, they did provide
But like Pandora's box, they were lost
Their secrets stolen, at a great cost

Deformed fluid parcels, in motion they flow
Deviate from Euclidean laws, their movements aglow
Silent now, in the hidden caverns of the wind
But now there is no way in, the temple of the wind

Stolen by the East, left in their wake
Our dreams destroyed, our future at stake
But in the field, a sign remains
"Here they dreamed in the caverns of the wind"

The Tempest

In the skies above, a tempest swirls,
Its winds, a cascade of power unfurled.
Turbulence, a beast with might so true,
Its roars, a symphony in shades of blue.

The winds, the rain, the lightning’s bright gleam,
A show of strength, a testament supreme.
Turbulence, cascade, and acoustics play,
A performance, unmatched, in skies so grey.

So let us pause, and witness nature’s might,
A performance of beauty, a breathtaking sight.
For turbulence, cascade, and acoustics bring,
A reminder of the power, the majesty of spring.

After listening to Purcell’s The Tempest for years in my office.

A Note on Critical Thinking

Miller, S. A. E., “Note on Critical Thinking,” NASA Alumni Association Magazine, Dec., 2022. pp 5. (one-page)

One might visit any leading university campus in the United States and ask the graduate faculty training future researchers one question, “what is the purpose of educating students?” One of the most frequent answers is to create critical thinkers. An obvious follow-up question is, “how do you create critical thinkers?” More opinions are presented than there are graduate faculty.

We live in a marvelous time where the majority of knowledge is accessible within a minute. Using a pocket computer (cell phone), we can query any question and have the answer almost immediately. We are able to create artwork, essays, poetry, and simple mathematical proofs with emergent machine learning technology.

Often today, when students are faced with problems in the university classroom such as design, mathematics, religious studies, fluid dynamics, economics, art, English, or even creating a poem, students almost unanimously and immediately reach for their pocket computer.

But what does the growing mind do when faced with an ill-posed problem or one that is well-posed but without a solution? In my own experience teaching students, there is often a range of human reactions that have included confusion, frustration, anger, fear, humiliation, and many others. These are emotions to be celebrated, because they represent a reaction from the student of being pushed outside their boundaries and intellectual comfort zone.

Here, students are no longer in the K-12 or early university environment, which lay out lesson plans in carefully constructed curriculums where problems and answers are well-defined. Educated wise minds should be fortunate to be in the position of not knowing or understanding something. As it represents an opportunity to define and solve a problem that challenges us as a people.

If our goal is to create a society where ideas are openly discussed, debated, and used for the benefit of our people, then training critical thinkers are essential. We cannot have a ‘mob’ mentality where ideas are repeated without being criticized.

The computer and Internet are a miracle of our age. These technologies have advanced the world civilization beyond all recent predictions and comprehension. However, we have come to be addicted to these tools as a people. They have created an intellectual handicap and have limited our creativity and critical thinking. It is no wonder that in recent years scores nationwide in mathematics have dropped significantly [1]. As students are using online groups and past homework solutions to ‘ace’ their courses.

I continually ask students in my own research group and classes to perform analyses on their own. They are required to close their laptop, turn off their phone, find a quiet room alone, define the problem, and attempt a solution on a blank piece of paper with a pen. I ask that they write down the laws of motion and examine the variation of a physical phenomenon.

Often a student will use every technique and manipulative emotion to not use their own mind. Instead of presenting their own ideas and analyses, they return to an unfortunate habit of seeking answers online that do not exist.

This is the core beginning of training critical thinkers – to overcome their fear of being wrong, to present their ideas with welcome criticism, and to challenge the status quo. The idea of critical thought is completely foreign to students, as no one has demanded they think critically.

Technology should allow us to enhance critical thinking, but not replace it. We must teach students to use technology in conjunction with their most useful resource, which is their own mind. The solution is simple – first, use our minds to think critically and independently without technology, and use technology for what it is – a tool.

References

[1] Mervosh, S. and Wu, A., “Math Scores Fell in Nearly Every State, and Reading Dipped on National Exam,” New York Times, Oct. 24, 2022.

Viscosity Coefficients

Often viscosity is isotropic, meaning that it is the same in all directions. In such a case, only two of the six components of the strain rate tensor are independent, so two coefficients of viscosity can be used to describe the viscous behavior of the fluid. These two coefficients are related by -2/3. This relationship arises because the strain rate tensor has a simple form for isotropic fluids, and the viscosity is proportional to the magnitude of the rate of deformation of the fluid. Viscosity is multiplied by the strain rate tensor in the Navier-Stokes equations because viscosity is a scalar that describes the internal friction within a fluid and the strain rate tensor describes the rate of change of velocity in the fluid. The use of two coefficients of viscosity in the Navier-Stokes equations is a result of the symmetric form of the strain rate tensor and the isotropic nature of viscosity in many fluids.