Abstract: A framework to create new specific analytical solutions of the equations of motion for hyperbolic boundary value problems is presented. The method relies on a closed-form integral equation for mass density, involving a term that combines sources, geometry, ambient values, and radiation. Products of the density integral result in new more complicated solutions. The …
Category Archives: Physics
Early Rockets and Review Notes
One of the earliest documented uses of rockets was in China. Father Antoine Gaubil, a French Jesuit missionary and historian, described an event in his 1739 writings, “When it was lit, it made a noise that resembled thunder and extended 24 km. The place where it fell was burned, and the fire extended more than …
Hypersonics History of Reentry
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 …
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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