NO whirls-NO force, NO force-NO lift-NO drag
Around two years ago, I launched this blog with the initial goal of demystifying my doctoral research, translating complex concepts and terminology into simple language accessible to a general audience. While many early articles focused on straightforward explanations (mostly in Spanish), the content naturally evolved to address deeper technical inquiries from engaged readers. This entire process has been a valuable exercise for organizing my thoughts and sharpening my scientific communication skills. Since beginning this project, I have successfully authored and published one paper in a Q1 journal [1] and co-authored a second paper available on ArXiv [2] as a preprint. Furthermore, I utilized this knowledge to successfully prepare and file by myself the corresponding patent application [3], saving hundred of dollars in an attorney. My research is fundamentally open. I offer my findings for review and use by the community, having moved past the need to convince any particular audience. While others continue to rely on simplified models with empirical corrections to tackle complex phenomena like turbulence, I remain confident that the limits of such approaches will inevitably confront the constraints of physics.
'Why do airplanes fly?' That was the central question of my very first article. This is a classic query in aeronautics that has been addressed in numerous ways, particularly by aeronautical engineers and, concisely, by Computational Fluid Dynamics (CFD) specialists: The answer lies in the Navier-Stokes Equations (NSE). The NSE are a set of partial differential equations (still lacking a general analytical solution) that describe fluid motion from two distinct perspectives: the Eulerian (based on velocity and pressure) and the Lagrangian (commonly described in terms of velocity and vorticity; see Fig. 2). Since both formulations are mathematically equivalent, logically, neither should be superior to the other, right? Nevertheless, the Eulerian description has been established as the standard for approximating fluid dynamics using numerical methods. Let us consider, for example, the incompressible flow past an object—a prototypical case where the NSE should work flawlessly. But, in practice, do they truly deliver? Let's analyze the limits of this paradigm.
Within mesh-based CFD, the Eulerian description has prevailed due to its intuitive nature. It is intrinsically linked to physical measurement: just as a Pitot tube reads pressure at a fixed point in a wind tunnel, the discretized mesh calculates the value of a variable at a determined spatial element. In contrast, the Lagrangian approach focuses on tracking fluid parcels (or whirls) that carry the flow properties, eliminating the need for an external discretized domain. This purely dynamic description is the most natural way to represent fluid motion, concentrating computational power on the truly relevant regions of the flow. While the Lagrangian approach is more complex to analyze or measure directly, its fidelity to fundamental physics is undeniable. Most of our readers are already aware of these facts. So, why is the Lagrangian formulation—the one closer to the actual fluid dynamics—not the computational standard? That is where the real problem lies.
Logic dictates that problems of extreme complexity—such as hypersonic aerodynamics, where thermal, compressibility, and ionization effects are involved—must be broken down into their simplest components and then studied individually first in order to understand the whole problem. My research has deliberately focused on the most fundamental step: incompressible viscous flow past a body. This approach stems from a clear conviction: if the foundations are not clearly understood and explained, the entire structure built upon them is vulnerable to collapse. This caution is justified when observing that the fundamentals of fluid dynamics remain a subject of ongoing debate among the true experts: the experimentalists and theoriticians. The understanding of aerodynamics cannot be achieved from the comfort of a desk; it requires a constant connection with reality through hands-on experience, observation, and testing. Ultimately, aerodynamics is inherently an experimental science. Theories and numerical simulations always seek to match physical measurements, never the other way around.
As I mentioned at the start, the NSE are the cornerstone for addressing fundamental questions in fluid dynamics. While the viscosity term in the NSE introduces significant non-linear complexity, paradoxically, it also smoothes the solution and makes it more manageable by causing the diffusion of vorticity and a less chaotic phenomenon. Following the principle of simplification mentioned above, if we remove the non-linear viscosity term from the NSE, we arrive at the Euler equations, which describe the motion of a non-viscous flow. Solutions based on the Euler equations are intrinsically unstable, given the absence of viscosity's damping effect. Nevertheless, it is essential to fully understand the equations governing this inviscid flow, as they constitute the theoretical foundations upon which all fluid dynamics is built. To ignore these basics is to overlook the problem's very origin.
One of the most persistent misconceptions among experts is equating the inviscid flow described by the Euler equations with the inviscid fluid defined by the Laplace equation within Potential Flow Theory (PFT). I have extensively addressed this topic—comparing experimental and numerical results, analyzing the underlying mathematics of the Euler equations [4], and employing simple observational logic [5]. For those who remain skeptical, numerical simulations conclusively support the original hypothesis: vorticity can indeed be generated in a non-viscous medium. This leads to the fundamental conclusion that titles this discussion: No whirls → no force; no force → no lift-no drag. Therefore, strictly speaking, the NSE are not the ultimate answer to flight, but a far simpler concept is: the vorticity field. Without the existence of vortices, there can be no induced velocity field, and without the induced velocity field, no aerodynamic force can be generated. The pressure field is merely the final link in this dynamic chain and does not, in fact, cause the phenomena typically attributed to it [6].
Although the initial intent for this article's title was to frame another technical explanation, I realized during writing that all necessary technical aspects had already been discussed in previous posts, requiring only cross-referencing. Ultimately, the true purpose was to underscore a fundamental principle: I have never been interested in being right; I am solely committed to the pursuit of truth. The entirety of the content written on this blog over the past two years is not an argument against an imaginary opponent, but a consequence of enriching technical discussions*—both online and offline. This independent journey, lacking institutional or university support for the past three years, has presented significant challenges. It has resulted in missed opportunities, including invited contributions to write two book chapters, invitations to speak at international (no fake) conferences, securing intellectual property protection in other territories besides the US, and continuing research as a Postdoctoral Fellow. Crucially, all of these facts are related to the denial of my right to obtain my PhD degree. By the way, this last remains an open issue that I will continue to address publicly as required [7]. Despite these obstacles, the quality and integrity of the research shared here stand as proof of my dedication to advancing the field, and why not, teaching to think based on evidence.
The educator's obligation lies in fostering intellectual autonomy. The era of absolute knowledge in the classroom is over. True pedagogical value does not reside in replicating the textbook on the board, but in awakening the student's capacity to question and develop their own critical thinking.
*including Prof. Joseph Katz (San Diego SU), Prof. Matthew Juniper (Cambridge), Prof. Adinel Gavrus (INSA Rennes), among other researchers.
The style of this text has been improved by IA (Google Gemini Flash).
[2] [2506.18719] The Full Nonlinear Vortex Tube-Vorton Method: the post-stall condition
[3] WO2024136634 FULL-SURFACE DETACHED VORTICITY METHOD AND SYSTEM TO SOLVE FLUID DYNAMICS
[4] On the ROTationality of an inviscid flow: Laplace /= Euler
[5] A four-question quiz: on fundamentals of fluid dynamics
[6] Which came first: the chicken or the egg? velocity or pressure?
[7] On innovation and other hoaxes: a true story at university (Part 1)
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