Lately, I have been examining the entire history of hypersonics research and technology, with a particular focus on the re-entry problem and ablation for small vehicles, such as those from ballistic missiles. While reviewing the writings of Wernher von Braun, I was amused to find that he joked about using frozen balsa wood as a …

# Category Archives: Physics

## Navier-Stokes Equations and Practicality

Because an effort is likely impossible and impractical does not mean it is not worth attempting. The Navier-Stokes equations and turbulent flow represent the last great classical problem in physics. Since the time of Leonard Euler and Jean-Baptiste le Rond d’Alembert, many have devoted much of their lives to working on these problems. Although they …

## Gödel and Time

$\mathrm{LL}$ cosmological solutions with non-vanishing density of matter known at present ${ }^1$ have the common property that, in a certain sense, they contain an “absolute” time coordinate, ${ }^2$ owing to the fact that there exists a one-parametric system of three-spaces everywhere orthogonal on the world lines of matter. It is easily seen that …

## Earth Density and Cavendish, 1798

Newton knew that the force of gravity causes falling objects near the Earth’s surface (such as the famous apple) to accelerate toward the Earth at a rate of 9.8 m/s2. He also knew that the Moon accelerated toward the Earth at a rate of 0.00272 m/s2. If it was the same force that was acting …

## Reflections on Spalart-Allmarus Turbulence Model, 2024

The Spalart-Allmaras turbulence model, a one-equation turbulence model, was a response to the inadequacies observed in zero-equation models, particularly their lack of predictive accuracy in complex flow scenarios such as wakes, shear layers, and shock wave boundary layer interactions. The creation of the Spalart-Allmaras model was influenced by multiple prior works, including the Baldwin Barth …

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## Pendulum, Time, and Stokes’

In 1582, an observation by Galileo Galilei at the Pisa Cathedral marked an important moment in understanding of oscillatory motion. Galileo, noting the constant period of a swinging lamp despite diminishing amplitude, laid the foundation for the study of pendulums. This led to his discovery that a pendulum’s oscillation period is directly proportional to the …

## Returning to Ludwig Prandtl’s One-Equation Model

In my turbulence class this semester, I recently reviewed Prandtl’s one-equation model, which was developed over 20 years since the time of boundary theory in the early 1900s. The major paper by Ludwig Prandtl was published in the early 1940s. He presented the first one-equation turbulence model for the closure of the boundary layer equations, …

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## An Improbable Life by D.C. Wilcox, and the $k-\omega$ Model

I just finished reading the autobiography of D. C. Wilcox. He wrote a number of books that were published through his own company. One of the most popular is on fluid dynamics. A less known book is on turbulence modeling. He was famous for a particular two-equation turbulence model in the form of $k-\omega$. It …

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## Remembering Professor Olga Alexandrovna Ladyzhenskaya Twenty Years On

It has been approximately twenty years since Professor Olga Alexandrovna Ladyzhenskaya passed away. An eminent mathematician and member of several Academies of Science, passed away in January 2004. Her distinguished career was marked by significant contributions to partial differential equations, particularly the Navier–Stokes equations and nonlinear elliptic and parabolic equations. Born in 1922 in Kologriv, …

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## Analytical Closed-Form Solution of the Navier-Stokes Equations for the Aerodynamic Near-field and Sonic Boom from Axisymmetric Bodies

I completed my Acoustical Society work and returned to the United States. Abstract: An analytical closed-form solution is presented for the aerodynamic near-field and ground signature from an axisymmetric body with a low thickness ratio. The Navier-Stokes equations are formulated as a boundary value problem that incorporates the incoming ambient flow-field and the aerodynamic properties …