High-Order CFD for Validating Analytical Solution of the Navier-Stokes Equations – ‘BlackJack’

The Computational Fluid Dynamics (CFD) code, ‘BlackJack,’ was created with a singular purpose: to generate extremely accurate numerical solutions of the Navier-Stokes equations in support of a broader program to develop new analytical solutions to those same equations. Unlike general-purpose CFD tools designed for industrial applications or engineering approximations, BlackJack CFD is a research code, engineered from the ground up to deliver benchmark-level fidelity for theoretical validation. Its development was motivated by the realization that no existing solver provided the order of accuracy, control over discretization, or modular extensibility required to confirm and refine exact or semi-exact solutions to the full compressible Navier-Stokes equations. Every algorithmic choice in BlackJack from the finite difference stencils to the boundary condition handling, damping strategies, and grid metrics was selected to eliminate numerical artifacts and ensure transparent, reproducible correspondence with analytical theory. It allows one to see what the analytical solutions will be, unlike production CFD codes that yield only relative insights into solutions.

I created the code BlackJack as an extension of my earlier DARPA supported program focused on developing new analytical solutions to the Navier-Stokes equations in high-speed flows. As I began deriving exact and semi-empirical solutions, it became clear that the engineering and applied communities no longer accept mathematical correctness based solely on symbolic insertion into the governing equations. Instead, they now demand that analytical solutions be compared directly to numerical simulations or experimental data, as if analytical solutions require the same form of validation as CFD. This mindset is fundamentally backwards, but it is the reality faced today.

Just as CFD is validated against wind tunnel measurements, engineers now insist that analytical solutions be benchmarked against numerical simulations, regardless of the fact that the solution may already satisfy the governing equations exactly. To confront this, I needed a numerical tool that was capable of producing reference-quality data free of dissipation, dispersion, and algorithmic ambiguity so that it could serve as a baseline for validating these theoretical results. Existing CFD codes, including commercial and open-source tools, were simply not built for this purpose.

This became the motivation for the new code. The goal was to construct a solver from the ground up that prioritizes fidelity over generality, with numerical methods capable of matching analytical solutions point for point in both steady and unsteady flow regimes. I began development nearly two years ago, writing the entire codebase by myself in Fortran. The architecture reflects the demands of research rather than industry: everything is explicitly structured, highest-order accurate, and open to modification.

BlackJack is a high-order structured CFD solver built on a multi-block, sub-block framework. It supports fully three-dimensional simulations on structured grids with arbitrary block decomposition, allowing exact enforcement of interface conditions and strict memory locality across parallel threads. Discretization schemes are user-specified and include a wide range of central and biased finite difference operators, including dispersion-optimized methods suitable for acoustics, shocks, and nonlinear wave propagation. Damping and filtering routines, including high-order sponge layers and localized artificial diffusivity, are tightly integrated into the residual formulation to stabilize underresolved gradients while preserving accuracy in smooth regions. Boundary conditions allow for subsonic and supersonic inflow and outflow, slip and no-slip walls, moving boundaries, periodic conditions, and radiation conditions. Each of these can be independently specified in each spatial direction, offering full control over problem setup and experimental design.

The solver operates on conservative variables, advancing them in time using a suite of advanced integrators, including low-storage and error-controlled methods up to 21st (and higher) order in accuracy. Grid metrics are computed analytically using the full inverse Jacobian formulation, enabling consistent representation of curvilinear geometry and ensuring the fidelity of derivative operations. All numerical routines-including finite difference, damping, boundary enforcement, and thermodynamic evaluation are modularized and can be extended with minimal disruption to the core solver. This modularity is essential, as BlackJack is designed not only to run known problems but to serve as a platform for exploring new equations, new models, and new forms of physical boundary data. The code can easily add additional equations, such as those for electromagnetics with minimal changes.

The code continues to evolve. Current development focuses on enhanced GPU parallelism, support for structured GMSH grid input, and expansion of the analytical validation suite with canonical solutions ranging from nonlinear acoustics to transitional flows and shock interactions. The code is already capable of resolving shock formation in N-wave propagation, reproducing linear and nonlinear acoustic fields, and achieving global accuracies on the order of 21st order or higher with only a handful of points per wavelength.

The name of the code originated as a joke I made with coworkers while I was a professor at the University of Florida. My original goal was to develop a high-order CFD code with spatial accuracy up to 21 points. Since 21 is the critical number in the card game blackjack, I began informally referring to the code as “BlackJack.” I also considered naming it “Gambit,” as a nod to the X-Men series, but BlackJack (or informally Gambit) ultimately stuck. Achieving practical CFD with shock capturing at 21st-order accuracy is not trivial, and the code continues to be pushed toward arbitrary order in both space and time for more complicated problems.

Olaf O. Storaasli

Recently I lost a friend, Olaf O. Storaasli, Ph.D., due to medical complications. I first met him through the NASA Langley Alumni Association after giving a lecture titled Life After Academics. Although we did not overlap during my time at NASA, we connected through shared experiences. Later, he and his wife visited Florida, and I hosted them at the University of Florida. During that visit, I introduced him to several senior design groups, invited him to speak to my class, and arranged a seminar with my research group.

Olaf had spent many years at NASA, primarily in Tennessee and at Oak Ridge, where he was a pioneer in early high performance computing. I believe he was at NASA during one of the space shuttle disasters. Many people from that era carry a lasting sense of responsibility for those events. It raises an important question—whether the hard-won lessons of safety from that time have truly endured, or whether we have started to drift back toward the cowboy mindset of the so-called Golden Age of Aerospace in the 1950s and 60s.

His visit to the university left a strong impression. I remember a dinner I hosted for him and his wife in Gainesville, a moment of quiet conversation and reflection. The students were especially engaged. In one senior design course, he stayed after class answering questions for over an hour, until I finally had to pull him away. Experiences like these remind me how valuable it is for students to engage with experienced professionals. Despite the noise of modern academic life, students remain eager to learn, listen, and absorb the lessons of those who came before them. That, I think, is a good sign.

Univ. Mich Art Exhibit

Today I attended the University of Michigan Art Exhibit on North Campus, which featured thousands of works by prisoners in the state of Michigan. The event serves both as a charity and as a program designed to provide the incarcerated with a creative outlet and a means of connecting with the broader community. The artwork is filled with pain and emotion. Some pieces are remarkably well executed.

https://dcc.carceralstateproject.lsa.umich.edu/s/pcapexhibition29/page/home

New Position

Today I started my new position as the Senior Scientific Technical Manager (SSTM) for Hypersonics at the US Navy’s Naval Air Warfare Center (NAWCAD).

Framework for Analytical Solutions of the Navier-Stokes Equations for Hyperbolic Boundary Value Problems in the Aerodynamic Near-Field

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 density field is used to recover the field variables in the near-field through far-field relative to the aerodynamic body. The aerodynamic body is modeled as a product and combination of generalized functions. Thus, resultant analytical solutions are also a combination of generalized functions. Derivations of time-dependent analytical solutions and example predictions are presented in one and two spacial domains. Cases examined for Euler equations include moving shock waves, oblique shock waves, Prandtl-Meyer expansions, and fields from more complicated bodies. A discussion of limitations and future directions of the methodology is included. The methodology, in a more complicated form, is used primarily for hypersonic sonic boom prediction.

Paper Link [PDF]

Academic Odyssey to Ithaca

Today marks the conclusion of my professorship at the University of Florida. Odysseus’s journey from the Ilion Wars (Trojan Wars) back to Ithaca was an odyssey that defined much of his lifetime. Though separated from his love, it was the journey that made his love meaningful.

Approximately eight and a half years ago, I began as a professor and also created a photographic memoir to document my journey at the University of Florida, titled Academic Odyssey. The memoir was short-lived; however, my tenure was not. Over this period, I earned tenure by working six to seven days a week for six consecutive years.

I had the privilege of working with some of the best students from around the world. I taught thousands of undergraduate students, guided and graduated several Ph.D. students, and collaborated with them to publish journal articles and conference papers. Through these efforts, I helped them develop their research and professional skills, guided by the theme of Technical Excellence originally taught to me by Dr. Charlie Harris of NASA Langley.

Collectively, I delivered approximately 500 lectures, seminars, and classes. These experiences deepened my appreciation for the written word, the English language, and the art of delivering an excellent lecture.

When I began as an assistant professor on the tenure track in 2016, I believed I understood what a professor, student, teacher, leader, university, and scholar were. Over time, I learned at length what it means to produce high-quality research. I engaged with department chairs, deans of leading international universities, colleagues at the University of Florida and abroad, undergraduate students, and distinguished emeritus professors. I also made lifelong friends.

More importantly, I learned who I am. I found my core values as a person. I witnessed others falter when put to moral and ethical tests.

Now, I find myself returning to foundational questions, which I did not understand before: What is a university? What is a student? What is a professor? What is a scholar?

Book Collecting

I have known men to hazard their fortunes, go long journeys halfway about the world, forget friendship, even lie, cheat, and steal, all for the gain of a book. – A.S.W. Rosenbach, Books, and Bidders