An "absurd" numerical method that actually works!

The dynamics of a separated fluid flowing past an object is still a challenging problem in engineering, even in low-speed aerodynamics, where turbulence can be present, making its understanding and solution more difficult. For this reason, some numerical methods have been developed, most of them based on the Eulerian description of the Navier-Stokes equations, by measuring the fluid-flow variables at fixed points defined by a continuous spatial mesh, which in most cases includes several assumptions and empirically based models, to approximate (and artificially close) the governing equations. In the first place, such methods are relatively computationally expensive, since they calculate the variables in the entire surrounding space, even far away from the object, where their measurement is of little value.

Fig. 1 Mesh-based CFD simulation of a parabolic parachute canopy by κ-ω SST turbulence model (Pimentel, 2016).

For several decades, the Boundary Element Method (BEM; e.g., panel methods based on the Potential Flow Theory (PFT)) has been used as an efficient alternative for flow over airfoils, wings, and aircraft in aerodynamics to reduce computational costs, since it allows to obtain a flow solution including the perpendicular component (lift) of the aerodynamic force and pressure distribution with sufficient accuracy by avoiding the computation of the entire surrounding space using one of the most powerful mathematical tools: Green's Theorem. However, such numerical solutions are based on a perfectly attached flow assumption, which should only be valid to approximate such solutions around streamlined bodies at low angles of attack (e.g., a thin airfoil at 5 degrees AoA). Numerical solutions for thicker bodies (where some degree of flow separation naturally occurs) or thin bodies at higher AoAs (e.g. 10 degrees) cannot be properly justified by such methods.

Fig. 2 "Ideal" flow past a simplified airplane by the Vortex Lattice Method. Source: Physics:Vortex lattice method - HandWiki

To avoid the limitations associated with separated flow in PFT-based methods, some authors have proposed different steady-state schemes to account for such a condition. In these approaches, the detached circulation-vorticity from the surface has been the principle to account for the vorticity detachment, avoiding an attached flow assumption and hence a boundary layer (BL) formation. In this way, an ideal (irrotational, incompressible, and inviscid) flow accounts for its separation, which is inherently considered within the solution of the corresponding system of equations, but not as an external 'correction', oblivious to the flow solution. Although most of the results were improved compared to simpler circulation-vorticity models, such nonlinear (in the sense of lift coefficient vs. AoA) schemes are incomplete since they include only some detached vorticity, but not all internal and external vortex wakes as in [1].

Fig. 3 Two schemes for vortex wakes detached from the surface [2].

Although The Full Multi-wake Vortex Lattice Method (FMVLM) [1] follows the same principle as simpler models, it was criticized from the very beginning by most editors and reviewers during its first submissions and peer-review process in top journals (Q1 and Q2) in aeronautics and fluid dynamics. Next, I will paste some of the answers in this sense, of course avoiding to mention authors and journal names. Then I will write my own impressions. This exercise can be seen as an attempt to fight against some flaws in publication decisions, especially when at this time 3 journal articles (and one patent pending) based on the same research have been accepted for publication in other top journals:

1. "Journal A": If you do resubmit, please note that the "Journal A" only publishes articles that advance methods of analysis and/or advances understanding of fundamentals in aerospace sciences and engineering."

In my opinion, this research is precisely about understanding the fundamentals, since it allows to accurately obtain the resultant force and moment in simplified lifting surfaces, under a fully (inviscid) detached circulation-vorticity scheme. What could be more fundamental than that?

2. "Journal J: Work in this area was presented years ago, in the 1970s and 1980s. The current paper does not present a convincing case for its "new" method. What is the physical justification? Why are so many elements needed for the very simple cases presented? Why the poor correlation with other methods and test results?"

So far, I do not know of a single reference to a similar numerical scheme from 40-50 years ago (the novelty criterion for patentability confirms this). On physical justification: if flow is assumed to be inviscid, it cannot a priori adhere to surfaces; it is simple and I think it is physically justifiable, especially if there is a sharp leading edge. On many elements: 64×128 is the finest discretization shown for mesh independence analysis for an AR=5 flat plate to achieve a reasonable level of convergence. So how many elements are sufficient? Perhaps it is possible to compare both linear (single trailing edge VLM) and fully (coupled) nonlinear schemes under the same computational criteria? On correlation: It is clear that expecting the same linear and nonlinear numerical value (outside the low perturbation assumption or the linear region) is...wrong; additional vortex wakes must have an effect on the results, right? Maybe on drag! However, it seems more likely that this reviewer is referring to the comparison of viscous and inviscid results, wherein the low AoA range (quadrangular flat plate case) the viscous values are slightly higher than the inviscid ones, which is physically expected due to the fluid viscosity effect on the lift (see Fig. 4). By the way, an accurate numerical analysis requires not only numerical results based on fine discretizations, but also a correct assessment of what is to be shown and proven. Thus, an additional effort must be made by any reviewer seeking a deeper understanding.

Fig. 4 CL for a quadrangular flat plate at the low AoA range.

3. "Journal C: Unfortunately, a careful examination of the content of the manuscript clearly shows that it does not fit the scopes of the journal. More precisely, the paper does not discuss a new numerical method or the analysis of new flow mechanisms using advanced numerical methods with the required depth for obtaining a paper with archival quality."

What does it mean "advanced"? high-order methods or based on "elegant" and complex equations or schemes? or maybe a novel complex semi-empirical approach (e.g., anooother turbulence model) to enforce numerical results to expected ones...Anyway.

4. "Journal E: Although your paper addresses an interesting topic, I believe it does not fit in the scope of "Journal E". The main reason is that your paper addresses a rather standard problem that is solved with a rather standard method. Although the results may be of interest for a certain group of specialists, their generic value is limited."

A standard method...what does that mean? conventional? On generic value: solving fundamentals cannot be limited.

5. "Journal S: I regret to inform you that your manuscript does not possess the novelty required for articles published in the "Journal S" where we do not do anymore VL methods."

I guess I'm late and outdated...we'll see!

Fig. 5 First conceptual drawing of the FMVLM (November 2020).

The last three journal responses seem to be conditioned by the tendency in modern CFD to try to explain even the fundamentals by bombastic, complex, and trendy methods and techniques, leaving aside efforts to do it more simply; in my opinion, this has been a huge burden in trying to understand the fundamentals of fluid dynamics (and aerodynamics). Yes, fundamentals have not been understood at all. If you do not believe me, look at this: 10+1 common misconceptions about aerodynamics (librepenzzzador.blogspot.com).

While my paper was rejected over and over again, I developed two unsteady versions based on the FMVLM concepts. These results were also openly published. As a result, my first paper was finally peer-reviewed in detail and accepted for publication in a Q1 journal...two years after it was first submitted! But that is another story. By the way, if you are not too sensitive and have a sense of humor, check this out: Why so serious! 🤡 (librepenzzzador.blogspot.com).

[1] The Full Multi-wake Vortex Lattice Method: a detached flow model based on Potential Flow Theory | Advances in Aerodynamics | Full Text (springeropen.com)

[2] H. Schlichting and E. Truckenbrodt (1979), Aerodynamics of the Airplane, McGraw-Hill