On Algorithm

An algorithm, a concept rooted in 9th-century Arabic scholarship, is a methodical procedure for problem resolution, eliminating the need for trial-and-error. This term, reflecting centuries of intellectual endeavor, denotes the evolution from Euclid’s ancient formulations to Al-Khwarizmi’s systematic methods and Ada Lovelace’s 19th-century innovations, highlighting algorithms’ integral role in computational development.


It is my yearly tradition to listen to the opera Salome by Richard Strauss on Valentine’s Day. To celebrate, I treated the department staff to donuts. Working on my research while listening to the opera can be a bit distracting, but it makes for a perfect Valentine’s Day tradition. I first saw the opera while working at NASA Langley Research Center in Norfolk, Virginia, as part of the Virginia Opera Guild.

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, Russia, Ladyzhenskaya’s passion for natural sciences was ignited by her father, a high school mathematics teacher. Tragically, he was executed by the NKVD in 1937, which led to Ladyzhenskaya being barred from Leningrad State University. Despite this, she persisted and graduated with honors from high school in 1939. She eventually studied at Pokrovskii Pedagogical Institute in Leningrad and later, at Moscow State University, graduating in 1947. That same year, she married A. A. Kiselev and joined Leningrad State University for graduate studies under the guidance of S. L. Sobolev.

Ladyzhenskaya’s academic journey continued as she became a postgraduate student under V. I. Smirnov and later led a seminar on mathematical physics and boundary-value problems. In 1953, she defended her habilitation dissertation at Moscow State University and in 1954, joined the Steklov Mathematical Institute in Leningrad. There, she collaborated with notable mathematicians and contributed significantly to the field of mathematical physics.

Her research focused primarily on two areas: the Navier–Stokes equations, and nonlinear elliptic and parabolic equations. In 1951, she proved a fundamental inequality for elliptic operators and explored the convergence of the Fourier method for hyperbolic equations. Her work in the late 1950s on the multiplicative inequality led to the proof of the existence of global unique solutions for two-dimensional Navier–Stokes systems. She also collaborated with A. A. Kisielev to demonstrate global existence in three-dimensional cases for small initial data and external forces. Ladyzhenskaya’s contributions extended to proving global existence of stationary and regular axially symmetric solutions to Navier–Stokes equations.

Her investigations into the regularity and uniqueness of weak Hopf solutions were groundbreaking. She showed that if a weak solution belongs to a specific function space, it is unique and regular. These achievements, along with her other work, were detailed in her monographs on Navier–Stokes equations and in collaboration with other mathematicians on nonlinear elliptic and parabolic equations.

AIAA SciTech 2024 – Parametric Study of the Hypersonic Near-Field and Sonic Boom from Waveriders using a Fully-Parabolized Approach

Citation: King, C. B., Shepard, C. T., and Miller, S. A. E., “Parametric Study of the Hypersonic Near-Field and Sonic Boom from Waveriders using a Fully-Parabolized Approach,” AIAA SciTech, Orlando, FL, Jan. 8-12, AIAA 2024-2106, 2024. DOI: 10.2514/6.2024-2106

Abstract: A parametric study is performed to understand the relationship between volume displace- ment, lift, near-field signature, and sonic boom overpressure for variable wedge angle power-law waveriders. The width of a parametric waverider is varied for freestream Mach numbers from 5 to 7. Both near-field and sonic boom predictions are made with a fully parabolized approach. The Upwind Parabolized Navier-Stokes solver is used to spatially march the hypersonic flow- field in the streamwise direction. The waveform parameter method is used to propagate the hypersonic near-field to the ground from a fixed altitude of 15.85 km. We find the magnitude of the SPL varies with frequency as −19.7 log 𝒇 . There is a positive quasi-linear relationship between near-field and sonic boom overpressures with volume displacement. For 𝑴∞ = 7, a 150% volume increase yields 92.5% and 60.9% rises in near-field and sonic boom overpressures, respectively. The effect of losses due to thermo-viscous effects and atmospheric absorption are quantified. We show that for a waverider of volume ∀ = 4970 cm3 at 𝑴∞ = 7, these losses, predicted by PCBoom using modules PCBurg and enhanced Burgers’ decrease maximum overpressure by 41.6% and 39.5% relative to WPM, respectively.

Priestess of Delphi (Oracle or Pythia)

A twenty year dream came true this December, 2023, as I traveled to Adelaide, Australia to view John Collier’s Priestess of Delphi (1891), the Oracle, or Pythia. I was able to view the painting for two days.

I am not afraid to say that the experience was overwhelming, and I definately had tears in my eyes. I am not a religious person, but it was what I believe people experience when they have religious inspiration or revelation.

People traveled all over the world to consult the Oracle. How is my journey different?

Colors from my camera, and the gallery skylight cast a small glare, but helped illuminate canvas and brushtrokes. Overtime, the sun came and set, casting new reflections and colors, letting me see the painting in new ways. I took 500 high quality photographs of the painting with different light, angles, details, and far-away.

Artist: https://en.wikipedia.org/wiki/John_Collier_(painter)

Painting: https://www.agsa.sa.gov.au/collection-publications/collection/works/priestess-of-delphi/25000/

Thank you to the people and museum in AUS SA at the AGSA for the experience.

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 on the body surface. The shape of the aerodynamic body is defined as a product of generalized functions. A direct solution for the density of the aerodynamic near-field, represented as a function of both space and time, is proposed through the integration of the Navier-Stokes equations in a generalized functional form. Pressure, temperature, velocity, and Mach number are then derived in the near-field. The methodology, being fully nonlinear, surpasses the traditional F-function, impulse, and hypersonic similarity theories originally developed for near-field prediction. The presentation outlines the major steps in deriving the analytical solution and provides predictions from an aerodynamic body in the near-field, along with the associated ground signature. The methodology is focused on aerodynamic bodies operating at high-speeds, ranging from the supersonic to the hypersonic regime. This research is supported by the Defense Advanced Research Project Agency, under Grant Number W911NF-21-1-0342.

Miller, S. A. E., “Analytical Closed-Form Solution of the Navier-Stokes Equations for the Aerodynamic Near-field and Sonic Boom from Axisymmetric Bodies,” Acoustical Society of America, Sydney, Australia, Dec. 4-8, 2023.

Acknowledgements: Research was sponsored by the Defense Advanced Research Project Agency (DARPA) and the Army Research Office and was accomplished under Grant Number W911NF-21-1-0342. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

Remembering Stephen Jurczyk


It seems just like yesterday that I was at NASA Langley working for a short time under then Director Stephen G. Jurczyk. He passed away just recently on Thanksgiving of 2023. He helped propel my career in the later stages of my time at Langley. One thing that I appreciated a lot about Stephen Jurczyk was that he came up through the organization as an engineer. Not all NASA administrators or SES are engineers nowadays. I appreciated this fact about him, because I felt that he could understand the needs of the research staff of the center. Reading books like Engineer in Charge gave me a deep appreciate of the history of NASA, and also gave me a viewpoint that the leaders of NASA should be first and foremost learned people of science. Unfortunately, this is not so much the case today. I’m hopeful that future leaders of NASA come from backgrounds in engineering especially, instead of political appointees with degrees in economics or finance.

Another important lesson is that one should try and depart love ones with kind words, because one never knows when will be the last time someone says goodbye.