10. Conclusion#
10.1. Results#
This project completed the set of tasks outline in its Roadmap:
It developed a novel Foil geometry specifically to enable simple representations of paraglider canopies.
It selected, implemented and validated a fast-but-accurate theoretical aerodynamics model well-suited to the nonlinear geometries and challenging flow conditions of paraglider canopies, as outlined in the Modeling requirements defined at the beginning of the project.
It developed parametric models to estimate the inertial properties and resultant forces of the components of a paraglider.
It used the parametric components to demonstrate how to produce a complete flight dynamics model of a commercial paraglider wing using only limited technical data, photos, and video of the wing.
It validated the longitudinal performance of the demonstration model against basic flight test data, as well as highlighted some areas in which the accuracy of flight dynamics could be improved.
This final section of the paper will address the last of the Modeling requirements: it will revisit the set of motivating questions that helped guide the design process, and consider the ability of these models to answer them.
10.1.1. Study: drag breakdown#
A common question for curious pilots is how to reduce the drag of their glider so they can improve the glide ratio or top speed of their wing. The natural progression of this curiosity is wonder where all the drag comes from in the first place. One way to answer that question is to plot the drag contributions from each component [50].
Fig. 10.1 Drag breakdown for Niviuk Hook 3 23 with a pod harness.#
Viscous drag includes effects such as the sheer forces produced by the viscosity of the air, and the pressure drag due to flow separation (the “vacuum” that can occur on the downwind side of an object); these forms of drag occur on every surface of the glider, including the lines and payload. Inviscid drag is less intuitive: commonly referred to as “lift-induced drag”, it is the energy lost in the vorticity that the wing sheds into its wake as a side-effect of producing lift.
This diagram provides a satisfying look into the behavior of a wing across the range of speeds. At the low end, pilots understand that the “brakes” will slow the wing by increasing its drag, but may be surprised to discover that the increase in drag is dominated by how the wing produces lift. At the high end, it can be surprising to learn what proportion of the total system drag is produced by the seemingly-negligible suspension lines. Although drag is just one piece of the lift/drag ratio, this sort of breakdown is valuable for estimating how much improvement is possible by (for example) reducing the drag of the payload.
This decomposition is also educational because it offers another perspective of how each component of the wing affects the overall design. Consider the general guideline that paraglider wings are designed to achieve their maximum glide ratio at “trim” (zero controls), which usually coincides with the speed that minimizes the total system drag (as seen here). Now suppose the design was changed; for example, increasing the aspect ratio of the canopy will tend to decrease its lift-induced drag, which in turn requires repositioning the payload at trim. The complete system behavior is a complex interaction of components, and having access to a parametric model such as this is an excellent resource for quickly answering questions about glider efficiency by developing an intuition of how their interactions affect the system behavior.
10.1.2. Study: effects of Reynolds numbers and apparent mass#
There were two questions at the start of this project that affected my modeling choices:
How significant are the effects of apparent mass?
How significant are the effects of accurate Reynolds numbers?
Both contributions to the flight dynamics are typically neglected in paraglider dynamics models without clear justification or discussion of their expected impact on model accuracy. The models developed in this paper can be used to provide insight on those questions. Using the Niviuk Hook 3 (size 23) component models created for the Demonstration, a programming script created multiple instances of the 6-DoF system models, configuring them to either respect or ignore the effects of apparent mass and precise Reynolds numbers (which are normally computed dynamically for each wing section). Pairs of models — one with the full dynamics and the other lacking one or both effects — are put into a figure-8 maneuver starting at that model’s equilibrium state and receiving the same control inputs over a span of 60 seconds. (The maneuver did not use weight shift control to avoid possible issues modeling canopy deformations.) Three simulations were run:
To show the affect of neglecting apparent mass (Fig. 10.2)
To show the effect of neglecting accurate Reynolds numbers by using a constant \(Re = 2 \times 10^6\) (Fig. 10.3)
To show the combined effect of neglecting both apparent mass and accurate Reynolds values (Fig. 10.4)
Fig. 10.2 Figure-8 when neglecting apparent mass#
Fig. 10.3 Figure-8 when neglecting accurate Reynolds numbers#
Fig. 10.4 Figure-8 neglecting both apparent mass and accurate Reynolds numbers#
Fig. 10.5 Figure-8 neglecting both apparent mass and accurate Reynolds numbers, topdown view#
The differences produced by each simplification are similar in this case, and will be discussed jointly. First, the less noticeable difference between the two simulations in Fig. 10.4 is the total altitude loss, where the “fixed Reynolds, no apparent mass” model descended an extra 2 meters. The difference is not visually interesting so no side-view is shown, but the effect is worth noting and should be expected for two reasons:
There is minimal acceleration in the \(z\)-direction so the \(z\)-component of the apparent mass is negligible.
The sections most impacted by the incorrect Reynolds values are at the outside of the span. Since the majority of the lift is produced by the central sections, which are already near the \(Re = 2 \times 10^6\) value, total lift is not greatly affected by assuming a fixed value of \(Re\).
The more significant effect was on the lateral motion of the glider, which is easier to see from a top-down perspective (Fig. 10.5), where the complete model exhibited a turn radius of \(54 \, [m]\) versus \(51 \, [m]\) of the simplified model. (The cumulative horizontal distances traveled were \(522 \, [m]\) at \(8.7 \, \left[\frac{m}{s}\right]\) and \(532 \, [m]\) at \(8.87 \, \left[\frac{m}{s}\right]\), respectively.) Again, the effect is expected for two reasons:
Apparent mass resists changes to the translational velocity, which reduced the complete models centripetal acceleration and prevented it from producing as narrow a turn as the simplified model.
Lower Reynolds values resulted in lower lift coefficients, especially for sections with deflected trailing edges (since their increased curvature magnifies the viscous effects). The lift vectors of sections on the inside semispan are angled into the turn and pull the canopy into the circle, so reducing their lift contributions further reduced the complete models centripetal acceleration.
Because these affects are heavily dependent on the glider design and specific flight maneuvers, this discussion focused on the qualitative nature of these effects. Whether these sources of error are significant depend heavily on the model (the canopy geometry in particular, as well as target airspeed of the glider) and its application. For example, when developing a linearized model to generate an error term for a control model these effects can be safely neglected, but any long-run simulation should review their specific control sequence (because turning magnifies their impact). With this model, checking the impact of such choices is readily available.
10.1.3. Study: indirect thermal interactions#
A reliable way to start a lively discussion on a paragliding forum is to question what happens when a wing encounters a thermal on only one side of its wing. Some pilots will argue that the thermal will pull the wing in; other pilots will argue that the thermal will push the wing away. A grand desire of this project was that the resulting flight dynamics model might be able to shed light on why two seasoned pilots might hold such opposing views.
This final study used the Niviuk Hook 3 size 23 components from the Demonstration with a 6-DoF system dynamics model. The scenario is simple: place a thermal slightly off-center of the path of a paraglider flying straight forward at equilibrium with symmetric brakes. Because the span of the wing is only \(8.84 \, [m]\), the thermal was placed \(15 \, [m]\) to the right with exponential falloff such that the thermal strength was reduced to 5% by the time it reached the center of the canopy with a peak (core) strength of \(3 \, [\frac{m}{s}]\) (extremely strong for such a tight thermal). The effect of the exponential falloff was a peak gradient of \(0.67 \, [\frac{m}{s}]\) from the wingtip nearest the thermal to the center of the canopy as the glider passed the core.
Fig. 10.6 Indirect thermal interaction.#
The first row represents the Euler angles for position, the second row represents the angular velocities, and the third row is the angular accelerations.
These results can be viewed in two ways: quantitatively and qualitatively. From a quantitative perspective the results are disappointing: the absolute angular deviations were on the order of 1°, which seem impossibly small for pilots to argue over. From a qualitative perspective, however, the results are perhaps more interesting. As the wing passes the thermal, the canopy initially rolls to the right (into the thermal), pitches forward (into the thermal), and the adverse yaw twists the wing to the left (away from the thermal); although the angular deviations are tiny it may produce an effect similar to falling, which needs only a small distance to produce a striking sensation. The same logic applies after the initial response, where the accelerates again, but more rapidly, and in the opposite direction: now the wing is rolling away from the thermal while yawing into it. Perhaps the sensation of acceleration holds the key to the argument: whether a pilot is more sensitive to roll or yaw, and whether they’re more sensitive to the initial or secondary accelerations may offer a partial explanation?
Personally I find this argument unconvincing. Despite the potential explanation offered by the qualitative analysis, it seems much more likely that the model has failed to capture one or more of the significant dynamics of the system. One possible cause is the foil aerodynamics model, which is not intended to capture unsteady aerodynamics; despite its accuracy in the wind tunnel testing, it may be inadequate for this level of subtlety in dynamic scenarios. Another possible cause is the quasi-rigid-body assumption imposed on the canopy geometry; real wings would flex and distort, especially in such a strong thermal, and it seems like that such deformations may play a larger roll that anticipated.
All in all, despite the underwhelming results the truth is this was always an ambitious goal, and I hope it demonstrates the theoretical advantages of pursuing flight dynamics models that are capable of capturing the effects of non-uniform wind vectors along the span of the wing, and will serve as a starting point for some future work. Perhaps we will someday have an answer for the forums.
10.2. Future work#
10.2.1. Canopy#
Arc deformations: the design curves that define the foil geometry are not required to be constant functions; they can be functions of control inputs, such as weight shift. The primary difficulty is that the current implementation of the NLLT assumes that the shape of the canopy is constant, but that a practical limitation, not a theoretical one.
Weight shift modeling: the Steady-state turn sanity check of the demonstration model suggests that lateral movement of the mass centroid is not the primary control mechanism for weight shift control. The alternative mechanism is the wing deformations that occur during weight shift. At the outset of this project the assumption was that the canopy deformations during weight shift would be negligible compared to the displacement of payload mass, but the turn radius and sink rate suggest otherwise. It may be fruitful to generate plausible \(yz(s, \delta_w)\) design curves (so the foil arc deforms as a function of weight shift), and consider if the changes to the canopy aerodynamics would explain the inaccuracies in the rigid canopy model. If canopy arc deflections prove to be a significant factor for accurate weight shift predictions, they should probably be implemented as an interaction between \(yz(s)\) and the suspension line model. (Paraglider pilots quickly discover the relationship between chest riser strap width and weight shift control, which strongly suggests that the lines play a dominant role).
Choice of airfoil: the Demonstration chose the NACA 24018 as an example of a conservative guess, but if a few commercial section profiles were measured accurately (including their spanwise variation), all models of commercial paraglider wings would benefit.
Deflected profiles: the demonstration used section Profiles produced by a “two circle” model of trailing edge deflection. That optimistic model was designed to balance the accuracy of profile deformation against the ability to estimate the aerodynamic coefficients with XFOIL. In reality, their unnaturally smooth curvature likely causes them to underestimate flow separation. Future work would benefit from more accurate deflection profiles.
Aerodynamic coefficients: in conjunction with more accurate deflection profiles, another improvement would be is to use more sophisticated methods to estimate the aerodynamic coefficients. One option is RFOIL from Delft University of Technology (a fork of XFOIL that is reported to improve estimates, particularly at high angles of attack), or to apply a complete computational fluid dynamics approach with OpenFoam.
10.2.2. Lines#
The parameters for the brakes are confusing at first glance, and tedious to tune. At the least they would benefit from an automated procedure where instead of having to tune \(s_\textrm{start,1}\) and \(s_\textrm{stop,1}\) to match \(\kappa_b\) (which was in turn limited by the \(\bar{\delta_d}_\textrm{max}\) supported by the aerodynamic coefficient set). It would be much easier to define \(s_\textrm{start,1}\) and \(s_\textrm{stop,1}\) at some hypothetical value of \(\kappa_b\) and have the lines adjust their values based on the true \(\kappa_b\).
10.2.3. Harness#
The spherical model neglects pitch and yaw moments due to angle of attack and sideslip, but because paragliders put their legs out in front those effects seem likely.
The harness model uses constant drag coefficients. [20] developed a model for the harness that accounts for Reynolds numbers, but that model was not tested in this work.
10.2.4. System dynamics#
This paper derived a 9-DoF system dynamics model that modeled the connection between the lines and payload as a spring-damper system, but without flight testing the parameters were difficult to estimate. It would be interesting to review the applicability of the spring-damper model and to estimate suitable parameters. I suspect that the lack of canopy deformations and the inability of the 6-DoF to show payload-relative roll are at least partial explanation of the underwhelming results of the indirect thermal study. The sensation of payload-relative roll and yaw accelerations could definitely play a role in why pilots disagree on the behavior of a paraglider encountering a thermal.
10.3. Open source#
The materials to produce this paper and its implementation [1] are both available under permissive open source licenses. Although this work focused on paragliders, the structure of the models is mirrored in the structure of the code, and should be easily adaptable to other gliding aircraft such as hang gliders or kites. For maximum versatility and approachability, the entire implementation was built on the Python scientific computing stack; despite not producing the fastest implementation, Python made up for the performance cost with value in other areas:
Free (unlike MATLAB, AutoCAD, etc)
Extensive cross-domain usage (aerospace, computer science, etc)
Powerful scientific computing libraries (NumPy, SciPy, Numba)
Easy to integrate into tools with native Python interpreters (such as FreeCAD, Blender, and QGIS)
I am grateful for the work freely shared by those who came before, and hope that this material may provide some value to those who follow.