4. Foil aerodynamics#

For the purposes of this chapter, an aerodynamics model provides the instantaneous forces and moments produced on a foil when it moves relative to air. In a rigorous modeling process the aerodynamic forces and moments would be measured experimentally, either in a wind tunnel or with flight tests, but that rigor is time consuming, expensive, and requires physical possession of the wing. Instead, this paper is concerned with estimating the dynamics of commercial paraglider wings from basic technical specifications, and so it must rely on theoretical methods that predict the flow-field surrounding a foil by combining fundamental equations of fluid behavior with the foil geometry.

This chapter suggests performance criteria for simulating paraglider aerodynamics, and selects a theoretical method capable of simulating those dynamics under the typical flight conditions. It presents a derivation of the method, modifies the method to improve its behavior in the context of flight simulation, and validates the modified method by comparing its predictions against wind tunnel measurements of a representative parafoil model from literature.

4.1. Aerodynamics models#

Classical aerodynamics predate the modern computing era, and were forced to prioritize simplifying assumptions that would enable analytical solutions of the governing equations; those assumptions placed heavy restrictions on what geometries could be analyzed and what characteristics of the flow-field must be neglected. These simplifying assumption made the problems tractable in a surprising variety of situations, but — despite their elegance — such analytical solutions are inadequate for analyzing the geometry and flight conditions of a paraglider.

In contrast, modern computational aerodynamics [9] solve the equations numerically, relaxing the need for analytical solutions. As a result, modern methods can analyze significantly more complex foil geometries over the entire set of flow-field characteristics. However, even with modern computers the fluid equations are too difficult to solve in the general case, so simplifying assumptions are still required to produce a tractable system of equations. This modeling process has led to a wide variety aerodynamic models built on different simplifying assumptions regarding the geometry and the characteristics of the flow-field.

4.1.1. Model requirements#

The introduction to this paper established a set of Modeling requirements, which determine the choice of aerodynamics method. Summarizing those requirements here for convenience, the model must account for the following characteristics:

  • Nonlinear geometry

  • Viscosity

  • Non-uniform wind field (different relative wind angles at different sections)

Where “viscosity” is elaborated as a collection of requirements:

  • The model should account for the decreased lift and increased drag due to flow separation across individual wing segments (at least approximately). This requirement is due to paraglider’s tendency to fly at relatively high angles of attack, and for individual sections to experience high angles due to the arc anhedral (especially during turns).

  • The model must demonstrate graceful accuracy degradation approaching stall (but is not required to model post-stall). The goal is not to simulate with absolute accuracy through stall, but the flight simulator should tolerate brief moments near stall.

  • The model should accept empirical corrections to viscous drag to individual wing sections to incorporate experimental wind tunnel results.

  • The model should use section-specific Reynolds values (not a wing average) since the sections of a paraglider canopy can vary from 300k to 2M during a turn (thus spanning the transition regime of Reynolds values)

There was also an optional, but desirable, goal that the method should be fast enough for real-time simulations to support rapid iteration during parameter estimation.

4.1.2. Model selection#

Despite the wide variety of options for choosing a theoretical aerodynamics model, in practice the Modeling requirements makes the selection process rather straightforward. The first requirement — to support nonlinear foil geometries — eliminates the classic LLT. Several authors have developed extensions of the LLT that are able to account for circular arc ([17], [18]), but are unable to model a swept quarter-chord.

The practical answer to nonlinear geometries is to switch to a vortex lattice method or panel method [8], which place the aerodynamic singularities on the nonlinear camber surface, or the profile surface itself, and apply the inviscid flow approximation to reformulate the problem as an instance of Laplace’s equation. Unfortunately, the inviscid assumption necessary to produce those solutions violate another of the modeling requirements: the ability to model viscous effects. Although extended models may apply strip theory to incorporate viscous drag coefficients (through lookups based on the estimated section angle of attack or lift coefficient), the inviscid methods fail to provide graceful accuracy degradation near stall. Because the inviscid solutions rely on linear relationships that are assumed to hold indefinitely, they are incapable of capturing the aerodynamic nonlinearities that arise at high angles of attack.

The next level of aerodynamic models are the computational fluid dynamics [9] methods. Instead of limiting the singularities to points on (or inside) the foil, CFD methods simulate the dynamics of the entire volume surrounding the object. In this way they are able to capture the entire array of flow characteristics such as viscosity, turbulence, and compressibility. Unfortunately, CFD methods have the downside of violating another of the modeling requirements: the requirement for speed. The purpose of this project is to enable a user to rapidly iterate the parameters of a model in order to improve the accuracy of a model. Individual CFD simulations at this level are commonly measured in seconds, if not minutes, rendering the fundamentally unsuitable.

Fortunately, there is yet another category, numerical lifting-line methods, which has progressed sufficiently to introduce a method suitable for wings with arbitrary camber, sweep, and dihedral while also supporting (some) viscous effects.

4.2. Phillips’ numerical lifting-line#

Phillips’ numerical lifting-line method (NLLT) [21] is an extension of Prandtl’s classic lifting-line theory (LLT) to account for the effects of a curved lifting-line.

Unlike the classical LLT, this numerical approach supports the characteristic nonlinear geometry of parafoils by decomposing the foil into discrete wing segments, each with their own scale, position, orientation, and profile. It can also be adapted to non-uniform wind vectors, allowing it to analyze non-uniform, non-longitudinal scenarios involving wind shear and wing rotation.

Unlike pure potential flow solutions, such as traditional vortex lattice and surface panel methods, it is able to approximately account for the effects of viscosity through its use of section coefficients (critical for incorporating viscous drag corrections and approximating flow behavior at high angles of attack).

And unlike full CFD solvers, the implementation is relatively simple, requires minimal manual configuration, and is computationally efficient (a critical point when generating iterated solutions for flight simulation).

4.2.1. Derivation#

For the purposes of discussion, the derivation of Phillips’ NLLT is briefly repeated here using the notation of this paper. Note that to avoid confusion, this derivation breaks the convention of this paper and instead uses Phillips’ convention of a capital \(\vec{V}\) for velocity, and a lowercase \(\vec{v}\) for the induced velocities.

_images/phillips_scratch.svg

Fig. 4.1 Wing sections for Phillips’ method.#

The goal is to establish a system of equations by equating two measures of the aerodynamic force applied to discrete segments of a wing. One uses the 3D vortex lifting law (4.1) and the other uses the local section lift coefficients (4.2):

(4.1)#\[\vec{\mathrm{d}F}_i = \rho \Gamma_i \vec{V}_i \times \mathrm{d}\vec{l}_i\]
(4.2)#\[\norm{\vec{\mathrm{d}F}_i} = \frac{1}{2} \rho_\textrm{air} \norm{\vec{V}_i}^2 C_{L_i} \left( \alpha_i, \delta_i \right) A_i\]

The net local velocity \(\vec{V}_i\) at control point \(i\) is the sum of the freestream relative wind velocity \(\vec{V}_{\infty}\) at the control point and the induced velocities from all the other segments:

(4.3)#\[\vec{V}_i = \vec{V}_{\infty} + \sum^N_{j=1} \Gamma_j \vec{v}_{ji}\]

where \(\vec{v}_{ji}\) are the velocities induced at control point \(i\) by horseshoe vortex \(j\):

(4.4)#\[\vec{v}_{ji} = \frac{1}{4\pi} \left[ \frac {\vec{u}_{\infty} \times \vec{r}_{j_2i}} {r_{j_2i} \left( r_{j_2i} - \vec{u}_{\infty} \cdot \vec{r}_{j_2i} \right)} + (1 - \delta_{ji}) \frac {(r_{j_1i} + r_{j_2i})(\vec{r}_{j_1i} \times \vec{r}_{j_2i})} {r_{j_1i}r_{j_2i}(r_{j_1i}r_{j_2i} + \vec{r}_{j_1i} \cdot \vec{r}_{j_2i})} - \frac {\vec{u}_{\infty} \times \vec{r}_{j_1i}} {r_{j_1i} \left( r_{j_1i} - \vec{u}_{\infty} \cdot \vec{r}_{j_1i} \right)} \right]\]

and \(\delta_{ji}\) is the Kronecker delta function:

(4.5)#\[\begin{split}\delta_{ji} \defas \begin{cases} 1\quad &i = j \\ 0\quad &i \neq j \end{cases}\end{split}\]

Solving for the vector of circulation strengths can be approached as a multi-dimensional root-finding problem over \(f\), where \(f\) is a vector-valued function of residuals, and the residual for each horseshoe vortex \(i\) is the difference between the two measures of section lift, (4.1) and (4.2):

(4.6)#\[f_i \left( \Gamma_i \right) = 2 \Gamma_i \norm{\vec{W}_i} - \norm{\vec{V}_i}^2 A_i C_{L,i} \left(\alpha_i, \delta_i \right)\]

where

(4.7)#\[\vec{W}_i = \vec{V}_i \times \mathrm{d} \vec{l}_i\]

The set of residuals \(f_i \left( \Gamma_i \right)\) represent a system of nonlinear equations that can be solved numerically to produce an estimate of the spanwise circulation \(\Gamma_i\). In order to solve the system, Phillips suggests gradient descent using the system Jacobian \(J_{ij} \defas \frac{\partial f_{i}}{\partial \Gamma_j}\), which expands to:

(4.8)#\[\begin{split}\begin{aligned} J_{ij} =\; &\delta_{ij}\, 2 \norm{\vec{W}_i} + 2\, \Gamma_i \frac {\vec{W}_i} {\norm{\vec{W}_i}} \cdot \left( \vec{v}_{ji} \times \mathrm{d} \vec{l}_i \right)\\ &- \norm{\vec{V}_i}^2 A_i \frac {\partial C_{L,i}} {\partial \alpha_i} \frac {V_{a,i} \left( \vec{v}_{ji} \cdot \vec{u}_{n,i} \right) - V_{n,i} \left( \vec{v}_{ji} \cdot \vec{u}_{a,i} \right)} {V_{ai}^2 + V_{ni}^2}\\ &- 2 A_i C_{L,i}(\alpha_i, \delta_i)(\vec{V}_i \cdot \vec{v}_{ji}) \end{aligned}\end{split}\]

with the effective wind speed in the normal and chordwise directions

(4.9)#\[\begin{split}\mat{C}_{f/s_i} = -\begin{bmatrix} | & | & | \\ \vec{u}_{a,i} & \vec{u}_{s,i} & \vec{u}_{n,i} \\ | & | & | \\ \end{bmatrix}\end{split}\]
(4.10)#\[\begin{split}\begin{aligned} V_{a,i} &= \vec{V}_i \cdot \vec{u}_{a,i}\\ V_{n,i} &= \vec{V}_i \cdot \vec{u}_{n,i} \end{aligned}\end{split}\]

and the effective local angle of attack \(\alpha_i\)

(4.11)#\[\alpha_i = \arctan \left( \frac {V_{a,i}} {V_{n,i}} \right)\]

After solving for the circulation strengths, the 3D vortex lifting law (4.1) is used to compute the inviscid forces at each control point, and the viscous drag and pitching moments are computed as in standard strip theory using the effective angle of attack (4.11):

(4.12)#\[\vec{\mathrm{d}F}_{\textrm{visc},i} = \frac{1}{2} \rho_\textrm{air} \norm{\vec{V}_i}^2 c_i C_{D,i} \left( \alpha_i, \delta_i \right) \hat{\vec{V}}_i\]
(4.13)#\[\vec{\mathrm{d}M}_i = -\frac{1}{2} \rho_\textrm{air} \norm{\vec{V}_i}^2 A_i c_i C_{M,i} \left( \alpha_i, \delta_i \right) \vec{u}_{s,i}\]

4.2.2. Modifications#

Although the original derivation is suitable for simple, static scenarios, it is inadequate for simulating dynamic conditions that commonly occur during paraglider flights. This section presents a number of modifications to improve the usability, functionality, and numerical stability of the method that greatly extend its applicability.

Caution

The material in this chapter up to this point has been a presentation of expert knowledge from literature. What follows is a best-effort attempt on my part as an amateur to identify the limitations with Phillips’ NLLT, and to suggest practical mitigations that allow its use in dynamic simulations of paraglider wings. I am a computer engineering by training, whereas my knowledge of aerodynamics is from reading the materials listed in the related works. As such, these discussions should be viewed with a critical eye.

4.2.2.1. Control point distribution#

The paper recommends placing the control points using a cosine distribution over the 3D spanwise coordinate \(y\), but that recommendation assumes a predominantly flat wing; cosine spacing generates a poor distribution when the wing tips are nearly vertical, which is common with parafoils. Instead, distributing the control points according to the section index \(s\) will maintain spacing along the foil’s \(yz\)-curve regardless of the arc. (Note that although this works well for parafoils, other foil geometries may be better suited to either a different section index, or some nonlinear spacing in \(s\).)

4.2.2.2. Variable Reynolds numbers#

Lifting-line methods typically assume the section coefficient data is an explicit function of angle of attack \(\alpha\), and possibly some sort of control deflection \(\delta\), but assume the coefficients are constant with respect to Reynolds number. For relatively high Reynolds regimes this is reasonable since the airfoil data is essentially constant, but parafoil sections under typical flight conditions experience Reynolds numbers in the range from roughly 150,000 to 3,000,000, spanning the transitional regime where viscous effects can be significant. To verify whether section-local Reynolds numbers have a significant effect on parafoil aerodynamics, the coefficients should be an explicit function of Reynolds number.

4.2.2.3. Non-uniform upstream velocities#

Phillips’ original derivation [21] assumes uniform flow, but [33] relaxes that assumption by replacing the uniform freestream velocity \(V_{\infty}\) with the relative upstream velocity \(V_{rel,i}\) that “may also have contributions from prop-wash or rotations of the lifting surface about the aircraft center of gravity.” (Compare Phillips Eq:5 to Hunsaker-Snyder Eq:5.) The result is that (4.3) is replaced with:

(4.14)#\[\vec{V}_i = \vec{V}_{rel,i} + \sum^N_{j=1} \Gamma_j \vec{v}_{ji}\]

In [33] they are concerned with accounting for propeller wash, but for a parafoil the upstream velocity is simply the local wind velocity at control point \(i\) combined with the velocity produced by the control point \(CP,i\) rotating about the glider center of mass \(CM\):

(4.15)#\[\vec{V}_{rel,i} = \vec{V}_{\infty,i} + \vec{r}_{CP,i/CM} \times \vec{\omega}_{b/e}\]

This change enables the method to approximately accommodate non-uniform wind conditions, such as from wind shear, turning maneuvers, etc. This flexibility should be used with caution, however; see Straight-wake assumption for a discussion.

4.2.2.4. Better solver#

To solve for the circulation strengths \(\Gamma_i\), the Phillips paper suggests using Newtons’ method, which computes the zero of a function via gradient descent. Gradient descent has several practical issues, but the most important problem in this case is that it fails to converge if the gradient goes to zero. For this application, the function under evaluation is the residual error (4.6), and its gradient (4.8) depends on derivatives of the section lift coefficients. When a wing section reaches the angle of attack associated with \(C_{L,max}\) the section has stalled, its section lift slope is zero, and gradient descent will fail to converge. Phillips suggests switching to Picard iterations to deal with stalled sections, but it is unclear whether the target function reliably produces fixed points; a simple prototype failed to converge.

An alternative is to use a robust, hybrid root-finding algorithm that uses gradient descent for speed but switches to a line-search method when the gradient goes to zero. The implementation for this project had great success with a modified Powell’s method, which “retains the fast convergence of Newton’s method but will also reduce the residual when Newton’s method is unreliable” (see the GSL discussion or MINPACK’s hybrj documentation for more information). This method not only mitigates the convergence issues near stall, but it is also significantly faster: it does not depend on fixed step sizes (which must be inherently pessimistic to encourage convergence) and is able to use approximate Jacobian updates instead of requiring full Jacobian evaluations at each step.

4.2.2.5. Reference solutions#

The root-finding algorithm that solves for the circulation strengths requires an initial proposal for the circulation distribution \(\Gamma(s)\). Poor proposals produce large residual errors that can push Newton iterations into unrecoverable states, so it is preferable to use prior information to predict the true distribution. The original paper suggested solving a linearized version of the equations, but that choice is only suitable for foils with no sweep or dihedral. Another common suggestion from related methods is to assume an elliptical distribution; for most foils, an elliptical circulation distribution is a reasonable guess during straight and steady flight, but it is a poor proposal for scenarios that include non-uniform wind or asymmetric control inputs, such as during flight maneuvers. It is clear that generating suitable proposals for nonlinear geometries under variable flight conditions requires a different approach.

For sequential problems, such as the sequence of states in a flight simulator or the points of a polar curve, an effective solution is to use the solution from the previous iteration as the proposal. Provided the time resolution of the simulation is reasonably small then the state of the aircraft should be similar between each timestep, so the proposal will be very close to the target. An added advantage of using a prior solution is an ability to capture hysteresis effects [38].

4.2.2.6. Clamping section coefficients#

A major issue with the method is a tendency to produce fictitious “infinite” induced velocities under certain conditions, causing convergence to fail. This tendency increases as the grid resolution is refined, and is most commonly observed at the wing tips, especially during turning maneuvers. The cause is apparent in equation (4.4), where the induced velocities between bound segments increases as the inverse of their separation distance; as the separation distance goes to zero, the induced velocity goes to infinity. In most cases, the induced velocities from the left and right neighbors of a segment mostly cancel, but if the foil has discontinuities (such as at the wingtips, where the outer segment has only an inboard neighbor) then cancellation may be incomplete, leaving a large imbalance. It can also occur due to numerical issues at very fine grid resolutions.

For parafoils the most significant discontinuities are at the wingtips, where the effect of the induced velocity spike is to dramatically overestimate the effective angle of attack. The NLLT relies on accurate section coefficient data, and if that coefficient data is unavailable (such as at high angles of attack) then the numerical routine cannot continue, causing convergence to fail.

Clearly the lack of coefficient data is not a valid reason to abort, since the large induced angle of attack is fictitious. To mitigate the issue when it occurs at the wingtips, assume the true \(\alpha\) is less than or equal to the maximum \(\alpha\) supported by the coefficient data, and clamp \(C_L\) to its value at that maximum \(\alpha\). In the case where the high \(\alpha\) is fictitious, the \(C_L\) will be incorrect but will at least remain relatively close to the true value, and will allow the simulation to continue. In the case where \(\alpha\) is genuinely large, then the unclamped inboard segments will also lack coefficient data and the method will correctly fail.

It is important to note that this is a practical mitigation, not a theoretically-justified solution. The point is not to “fix” the method, the point is to limit the magnitude of the error and allow the simulation to continue with reasonable accuracy. However, despite lacking a theoretical basis, there are several strong justifications:

  1. If the outer segment is small, then its contribution to the error is expected to be small. For example, if the outer segment represents the last 5% of the wing span means then the error from much less than 5% of the total aerodynamic contributions (since the area of that wingtip segment is very small).

  2. If the outer segment is small, you wouldn’t expect a significant change in alpha from the wingtip to its neighbor, so if the inboard neighbor is in the valid range you can expect that the wingtip alpha is (relatively) close to the valid range.

4.2.3. Limitations#

4.2.3.1. Assumes minimal spanwise flow#

This method argues that the derivation of the 3D vortex lifting law in [39] proves that “the relationship between section lift and section circulation is not affected by flow parallel to the bound vorticity.” In other words, it relies on the fact that the 3D vortex lifting law holds even in the presence of spanwise flow. What this does not account for, however, is the effect of spanwise flow on the section coefficients. Wing analysis using section coefficients relies on the assumption that each wing segment acts as a finite segment of an infinite wing, provided the spanwise flow is negligible ([6], p. 356). Although the 3D vortex law holds in the presence of spanwise flow, solving for the circulation strengths using section coefficients does not.

A similar discussion can be found in [38], who apply a similar NLLT to a flat wing with 45° sweep. They acknowledge that although the sweep introduces significant 3D flow-field effects, the method “shows very good agreement” versus experimental measurements. Their success offers some confidence that the effects of spanwise flow may indeed be negligible, but it is unclear whether the effect has more significance once continuous arc anhedral is involved.

4.2.3.2. Straight-wake assumption#

A common aerodynamic modeling approximation is to assume that vorticity is shed into the wake as a trailing vortex sheet; the strength of the shed vorticity varies with the local variation of lift along the span. In a rigorous analysis, the trailing vorticity should follow a curved path ([6], p. 390), but this produces an intractable nonlinear system of equations. Instead, models apply a further simplification known as the straight-wake assumption: that the trailing wake vortex sheet streams straight back from the lifting-line. The straight-wake assumption is an important step in linearizing the system of equations to allow mathematically tractable solutions.

For a discretized method, such as Phillips’ or Weissinger’s LLT [40], the vortex sheet is lumped into a series of shed vortex filaments whose strength is proportional to the difference in local lift of neighboring segments. Under the straight-wake assumption, the trailing legs of all horseshoe vortices extend from the nodes in straight lines parallel to some freestream velocity direction \(\vec{u}_{\infty}\) (see (4.4)). This is clearly invalid for a rotating wing where a freestream velocity is ambiguous.

Despite this limitation, this project assumes that as long as the rotation rates remain small enough that relative flow angles remain small the method still provides useful approximations. This assumption is made without theoretical justification; instead, this paper relies on the superior aerodynamics knowledge of its sources. First, the use of this method with non-zero rotation is explicitly mentioned in [33]. Also, this assumption is shared with the vortex-lattice model used in AVL [41], although in that method the trailing legs are aligned with the foil \(x\)-axis, regardless of freestream flow. In Phillips’ method the trailing are aligned to the freestream, which for this work is defined as the local upstream velocity \(\vec{u}_{\infty,0}\) of the central section under the assumption that it minimizes average deviation.

For a related technical discussion that incorporates rotation rates into a vortex lattice method, refer to [8] Sec. 6.5; in particular, Eq. 6.33 for aligning the trailing legs with the \(x\)-axis, Eq. 6.37 for accounting by adding it to the flow tangency equations, and Eq. 6.39 for incorporating the rotation rates into the aerodynamic influence coefficients matrix.

4.2.3.3. Reliance on section coefficients#

A significant limitation of aerodynamic methods based on the theory of wing sections their assumption that the section coefficient data is accurate and representative of the flow conditions during a flight. In practice, section coefficient data is notoriously optimistic, relying on idealized geometry, negligible spanwise flow, a uniform flow-field across the segment, steady-state conditions, etc. These assumptions are strong to begin with, and become particularly questionable near stall, especially when using simulated airfoil data.

Not only do these methods assume the section coefficient data is accurate for each individual section in isolation, they also assume the flow conditions of each section will have a negligible impact on the coefficients of neighboring sections. In reality, development of 3D flow-field conditions such as separation bubbles is significantly impacted by such neighboring sections. Part of the interaction can be captured by the induced velocities, but section coefficients are ultimately incapable of modeling effects such as turbulence, 3D separation bubbles, significant spanwise (or “cross”) flow, etc. Such effects seem likely to be even more prominent given the significant arc of a parafoil.

4.2.3.4. No unsteady effects#

This method produces a steady-state (non-accelerated) solution. It does not include unsteady (time-varying) effects, such as ([8], p. 149):

  • Unsteady foil motion

  • Unsteady foil deformation

  • Spatially-varying or unsteady atmospheric velocity field

Thankfully, the (arguably) most important unsteady effect for the purposes of paraglider simulation under typical flight conditions can be accounted for by the simulator itself; see Apparent Mass.

4.2.3.5. Non-unique solutions#

Gradient descent will find a zero of the residual, but it is not guaranteed to be unique, especially given that the numerical solver relies on tolerances instead of exact solutions. Depending on the initial conditions, the solver may converge to different circulation distributions.

4.2.3.6. Sensitive to initial proposal#

This method relies on a good proposal (an initial “guess” of the circulation distribution) to encourage convergence while minimizing optimization runtime. The root-finding problem uses the residual error (4.6) which is likely a non-convex function, in which case a global optimization method such as gradient descent is not guaranteed to find the global minimum for a non-convex function, so the solution is sensitive to the starting point (the initial proposal). In practice this issue is not a major problem when the intended use is flight simulation; solutions are generated iteratively, in which case the previous solution is a natural choice for minimizing the initial residual error (see Reference solutions). As an added bonus, using the previous solution adds the capability of capturing hysteresis effects [38]; for example, in [42] they discuss a wing that demonstrates hysteresis depending on whether data were generated with increasing versus decreasing alpha. Nevertheless, the fact that the method has a tendency to produce different solutions for different proposals mean the method will exhibit hysteresis effects which may or may not be physically accurate.

4.2.3.7. Unreliable near stall#

Phillips suggests that this method can be used up to stall “with caution”. Closely related to the issues of spanwise flow, the development of stall conditions along a wing has a high likelihood of violating the assumptions used to generate the section coefficients. Worse, the flexible nature of a parafoil will exacerbate the effects of section stall, which cause the profiles to deform and wrinkle even more than normal. Nevertheless, this project attempts to apply the method to “near stall” conditions under the belief that, for the purposes of flight reconstruction, it is preferable to get a low-quality estimate as opposed to no estimate at all. It is vital, however, for the filtering architecture to model the increased uncertainty as sections approach stall conditions.