Another physical phenomenon, in addition to the generation of lift force or the effect of viscosity on it, that has not been explained in detail by fluid dynamics (neither theoretically nor computationally), is that related to the stall mechanism that a lifting surface can suffer, for example, on the wing of an aircraft. In general terms, the stall condition is understood as the loss of lift force and an abrupt change in flow regime, going from a relatively stable condition to a completely turbulent one. For reasons of simplification, I will omit to explain the effect on the drag force or on the pitching moment, as well as other details that are assumed to be known by the reader (effects of the thickness and shape of the leading edge in obtaining the aerodynamic forces, nature of the variation or discrepancy of the experimental data, basic knowledge of Potential Flow Theory and/or the vortex lattice method; VLM).
Fig. 1 Airfoil during stall. Source: https://i.pinimg.com/originals/d7/3b/f9/d73bf985a549ea215aca360ff0c0ed58.jpg
The following approach consists of trying to explain the stall mechanism based on the interpretation of published experimental data series for a simple geometry (a quadrangular flat plate laterally aligned with the flow), in addition to computational results obtained through the implementation of purely numerical methods based on vortex lattice [1,2] and on 'particles' of tubes and vortons [3].
Listed below there are 7 identified flow regions based on the angle of attack (AoA) that can be described by the current numerical-experimental analysis:
1. Linear viscous: between 0 and 2 degrees of AoA. The increase in lift force is proportional and can be described by an analytical model or a numerical method, both linear (e.g., standard VLM; single trailing edge wake). For context, for plates with higher aspect ratios, this region can extend to higher AoAs (e.g., around 10 degrees).
2. Non-linear viscous: between 2 and 8 degrees, approximately. It is characterized by the fact that the fluid remains attached to the plate's surface due to the effect of its viscosity. Therefore, its detachment only takes place along the trailing and the lateral edges ('wing tips') of the plate. It can be modeled numerically through a VLM including lateral wakes or solved using the viscous version of the 'tubes and vortons' method (FTVM) [3].
3. Viscous-inviscid transition: located between 8 and 12 degrees, approximately. It is characterized by a very subtle decrease in lift force and its eventual recovery. Within this range, the boundary layer is assumed to be in transition from attached to detached on the plate's surface before finally entering a purely inviscid flow condition. This effect can also be observed in the experimental data series for flat plates with lower aspect ratios (e.g., AR = 0.5). It has not been numerically modeled since the areas with boundary layer detachment cannot be assumed; ideally, it could be modeled by a mixture of attached and detached wakes from the surface or solved by the viscous version of the 'tubes and vortons' method [3].
4. Inviscid-undamped: between 12 and 20 degrees. It is characterized by a sustained increase in lift force. The flow is considered completely detached from the plate's surface, but due to the recirculatory velocity field, it remains practically attached to it (but not due to the effect of viscosity!), including the leading edge vortex (LEV) that some authors have experimentally hypothesized. It can be modeled through the full multi-wake method (FMVLM; inviscid) [1] or solved using the inviscid version of the 'tubes and vortons' method [3] (see Fig. 2 to visualize the 4 regions described above).
Fig. 2 Flow regions (R) for a quadrangular flat plate at different AoA (up to 20 degrees).
5. Inviscid-damped: located between 20 (inflection point) and 35-40 degrees. It is characterized by a smaller increase in lift as the AoA increases, compared to the previous region, until reaching the stall point. This behavior may be mainly due to the shape and direction ('downwash') of the wing tip vortices, since it is considered a region that cannot be simulated under a straight wakes (aligned to the flow) assumption. It can be solved by the inviscid version of the 'tubes and vortons' method [3].
6. Stall: between 35-40 degrees according to most published experimental data. It is the point where the flow condition becomes completely turbulent. The cause of the abrupt loss of lift (due to the plate's sharp edges) may be due to the fact that, at such a point, the LEV ends up attaching to the plate and, once it becomes part of the vortex system of it, then is detached, becoming a conventional separation vortex, similar in nature to the trailing edge and wingtip vortices (Kutta-Zhukovski type). Its 'transformation' could be automatically predicted through an appropriate algorithm, capable of detecting the inversion (in sign) of the LEV (depending on the instantaneous flow conditions) within the development of the 'tubes and vortons' method [3] (see Fig. 3 to show the 2 regions described above).
Fig. 3 Flow regions (R) for a quadrangular flat plate at different AoAs (up to 45 degrees).
7. Post-stall: located from 35-40 degrees. It is the completely turbulent region, characterized by larger amplitude oscillations in the values of the aerodynamic forces due to the flow instability and that begins with an abrupt decrease in the average lift value, which tends to decrease as the AoA increases. It could be accurately modeled (meshless Large Eddy Simulation) or solved (meshless Direct Numerical Simulation) by a future turbulent version of the 'tubes and vortons' method, modified for LEV detachment (see animation of the non-turbulent version).
Fig. 4 Perpendicular flow to a quadrangular flat plate using the non-turbulent version of the tube and vorton method (FTVM, 2022-2023).
Based on the previous analysis, can we still think that the total detachment of the boundary layer occurs around 35 or 40 degrees in a low aspect ratio flat plate with sharp edges when it is known that in a finite wing (with a streamlined shape, rounded leading edge and higher aspect ratio), it occurs around the half of such values?
This article does not intend to show any 'absolute or definitive truth' but rather to serve as a simplified model of reasoning and analysis, with the aim of helping to glimpse one of the most complex phenomena to understand within the field of fluid dynamics.
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