Binary numbers were originally used for encryption and communication, a fact recognized as early as the 17th century by Francis Bacon. Bacon used the binary system for encoding the alphabet using strings of binary characters. This laid the framework for subsequent developments in coded communication, such as technologies like the telegraph (Samuel Morse), which relied …

# Category Archives: Mathematics

## Creation of Probability

On chance – Ancient civilizations, despite their engagement in games of chance and divinatory practices, did not formalize the underlying principles of probability. The creation of formal probability theory is linked with gambling and divination, stretching back to antiquity. However, the mathematical formulation of chance / probability remained elusive until the 17th century surrounding the …

## Saffman \(k-\omega^2\)

Saffman’s \(k-\omega^2\) turbulence model, initiated by Saffman’s research, plays a role in the two-equation models dedicated to turbulence research since the time of Kolmogorov in the 1940’s. The basics of Saffman’s model is shown in the portrayal of a statistically steady or ‘slowly varying’ inhomogeneous turbulence field alongside the mean velocity distribution. This model states …

## Baldwin Barth One-Equation Model Reviewed

During the present semester, I reexamined the Baldwin-Barth one-equation turbulence model. This model constitutes a reformulation of the $k$ and $\epsilon$ equations, culminating in a single partial differential equation for the turbulent eddy viscosity, denoted as $\nu_t$, multiplied by the turbulent Reynolds number, $Re_t$. The model’s closure for the Reynolds-averaged Navier-Stokes (RANS) equations was a …

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## 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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## AIAA Journal – Fully Parabolized Hypersonic Sonic Boom Prediction with Real Gas and Viscous Effects

https://doi.org/10.2514/1.J063425 Abstract: We present a methodology to predict the aerodynamic near-field and sonic boom signature from slender bodies and waveriders using a fully parabolized approach. We solve the parabolized Navier–Stokes equations, which are integrated via spatial marching in the streamwise direction. We find that unique physics must be accounted for in the hypersonic regime relative …

## 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 …

## Origins of Complex Numbers

The creation of complex numbers is found in the exploration of square roots of negative numbers, a notion that seemed incongruous within Euclid’s axioms and then present rules governing integers. The problem presented by the square root of negative numbers spurred a significant shift in thinking, leading to the conceptualization and acceptance of “imaginary” numbers, …

## Linear to Nonlinear Relations in Wave Science (Acoustics)

In the realm of acoustics or wave science, the transition from linear to nonlinear physics marks a significant evolution in the understanding of tones and their generation. The foundation of this understanding dates back to Pythagoras, who established a linear relationship between the length of a plucked string and the resultant musical tone. This principle …

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## 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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