On innovation and other hoaxes: first disagreements (Part 3)

Go to the first part: On innovation and other hoaxes: a true story at university (Part 1)

After numerically understanding the effects of adding lateral wakes to flat plates, especially for low aspect ratio (LAR) configurations, a more complex steady-state scheme was proposed by my advisor. It consists of including internal detached wakes to account for flow separation, similar to Gersten's vortex model (see Fig. 1, up). Therefore, I first explored a simpler model, the lifting line method (based on the Lifting Line Theory or LLT), by including detached horseshoe vortices instead of bounded vortex rings as in the VLM. Such a scheme is not new, in fact some authors have proposed similar approaches in the past to account for flow separation in the context of Potential Flow Theory (PFT), improving the obtained results by far. Furthermore, I found that in the 90s, Prof. D.A. Durston of NASA published a similar vortex model (LinAir code) to account for flow separation by including trailing vortices from each bounded element, more similar to a VLM with multiple wakes along the chordwise direction (Fig. 1, down).

Fig. 1 Detached flow schemes in the context of the Potential Flow Theory. Sources: Aerodynamics of the Airplane (Schlichting and Truckenbrodt) and NASA Technical Memorandum 108786 (Durston).

During this time, I wondered how VLM and LLT are related, or in other words, if it is possible to get a VLM scheme from an LLT one, under the premise that both models are ultimately based on the same principles: vortex filaments. After a few weeks, I developed a systematic way for an m by n flat plate to go from the LLT to the standard or single wake VLM, where the circulation remains attached to the surface. To accomplish this task, it must be assumed that for both models the straight wake must be aligned with the flat plate, but not with the free stream velocity; those who have worked with the LLT or VLM know that this scheme is often used for high AR configurations, and gives satisfactory results for such cases. In the end, this exercise gives the expected results: both models are numerically equivalent by performing the correct operations, which include systematically inverting the direction of some horseshoe vortices. This particular task is not just a curiosity, since it helps to understand in detail how bounded vortex filaments, and hence bounded vorticity, are related on the lifting surface, and how a detached circulation-vorticity model is justified, where in such a case the upstream detached circulation is not summed to the downstream one. It provides the key to understanding the further modeling of the leading edge wake.

Fig. 2 Numerical equivalence between vortex horsehoes and vortex rings (2 bounded elements case).

When I remotely presented the previous scheme to my advisor and co-advisor, the latter tried to refute it based on an "arbitrary" inverted horseshoe criterion to make both results compatible. Even he sent me some kind of formal justification why this scheme was wrong from his point of view. In my defense, I can say that numerical results do not fool anyone. In his defense I can say that I probably should have presented a simpler example, since I showed one with 3 bounded vortex elements (Fig. 3) instead of 2 (Fig. 2), which increases the complexity of the operations a bit, but in the end is completely systematic and well justified. From the previous example it is clear that no arbitrary inverted horseshoe criteria are applied, but only by numerical justification. This situation could be considered the first major discrepancy between us after two and a half years. However, since this task is an intermediate step between the attached and detached schemes based on VLM, I did not pay more attention to it. The real discrepancies were yet to come...

Fig. 3 Numerical equivalence between vortex horsehoes and vortex rings (3 bounded elements case).

So, based on the current state of the art, I start to develop my own detached flow model for a flat plate, quite similar to the one shown before. However, unlike those models, such a development must be completely numerical, avoiding, for example, the use of the semi-empirical Polhamus suction analogy to account for the leading edge flow separation implemented in the LinAir code. This requirement was based on a logical and well-justified idea, since the main objective of such a project is not to solve partially separated flow problems, but massive ones, such as occur behind a parachute canopy. With this in mind, and after verifying the simplest multi-wake model with some that my advisor sent me (Fig. 4), I proceeded to include all lateral (along the spanwise) wakes, including both internal and external (wingtips). Again, at the next opportunity, this model was criticized by both the advisor and the co-advisor on the grounds that such wakes were not necessary since the flow was aligned with the plate. It is clear that at this point I was looking for a general detached flow model where even sideslip cases could be tested, verified and validated, since as far as I know no canopy is flat and never remains aligned with the free flow. At this point I began to suspect their lack of vision about the project.

Fig. 4 Simpler multi-wake scheme coded by my advisor (MATLAB code).

The biggest discrepancy between us is in the modeling of the leading edge vortex (LEV) wake. Assuming that in the current development vortex wakes are also detached from the trailing and lateral (wingtip) edges, the leading edge must also detach its own wake, mainly due to the fact that it models a sharp edge but not a rounded one. It is clear that in the case of a perpendicular (square) flat plate to the free flow, all external detached wakes have the same sign for rotation, or in other words, all external wakes roll back towards the plate; this fact is logical and needs no further explanation. On the other hand, as the angle of attack (AoA) decreases, an abrupt increase in lift (loss of lift in the opposite direction) occurs around 40 degrees at the stall point (Fig. 5), so there must be a major change in the flow condition, and it is certainly not related to the fluid viscosity, since for streamlined configurations as typical wings (with a higher aspect ratio and a rounded LE) such a point is typically around 15 degrees or even less. So it is absurd to think that a small value of fluid viscosity, typical in aerodynamics or hydrodynamics, or even zero in the hypothetical inviscid case, will allow an attached vortex sheet (or boundary layer in the viscous case) to be maintained on a LAR flat plate with sharp edges up to 40 degrees! Any forced attached flow towards the plate should be related to strong wing tip vortices (and vortex lift) and their developed velocity field. As will be shown, the abrupt sign reversal of the LEV is the cause of such abrupt behavior, in agreement with experimental tests where the LEV remains attached to the plate under this particular operating condition (pre-stall region).

Fig. 5 According to experimental data, the stall point for a square flat plate is about 35-40 degrees. Figure: own.

To be continued...