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 …
Author Archives: saemiller
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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Brian Spalding
One last poem by turbulence / numerics researcher Prof. Brian Spalding I shall have no regrets when I am dead. Of deadlines none will matter but my own. Unwritten papers? Hopelessly misled. Inheritors? All claimants I’ll disown. Yet hope, while still alive, there’ll be but few Who think: I was a fool to trust him. …
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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Geometrics and Art
The Renaissance, a period of significant intellectual, artistic, and cultural rebirth, marked the combination of art and science, especially through the application of geometric principles in artistic representation. This era witnessed the pioneering development of linear perspective, a technique that revolutionized the way depth and three-dimensional objects were portrayed on two-dimensional surfaces. The mathematical foundation …
Additional Thoughts on Half-Equation Model of Johnson and King
The Johnson King turbulence model represented a significant advancement in the understanding and modeling of turbulent flows. Introduced amidst the exploration of first and second equation models, the Johnson King model distinguished itself through the innovative concept of a half-equation model, emphasizing the critical role of memory in turbulence phenomena. The early stages of turbulence …
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Hindu-Arabic Numerical System
The decimal number system, an integral part of daily life, traces its origins back to 6th-century India. Characterized by the digits zero through nine, revolutionized numerical computation and record-keeping, setting the stage for advancements in mathematics, science, and commerce. Despite its apparent simplicity and utility to the contemporary observer, the widespread adoption of this system …
Diophantine Equations
Contributions of Diophantus of Alexandria hold a distinguished place. His seminal work, Arithmetica, unveiled in the 3rd century CE, is a key in the study of number theory, particularly in the realm of integers. This ancient text, encapsulating 130 equations, laid the foundation for what are now known as Diophantine equations—equations constrained to integer solutions. …
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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