On The Flow Characteristics of Non-Slender Delta Wings.
This post gets a bit nerdy with low levels of hyperbole, so to avoid getting a headache, simply read the next sentence and then click back to The Funny Images thread.
On a FF make sure your leading edges are well rounded, not sharp. For the keen masochists: Kato's recent almost inexplicable efforts on an FF28 gave me a reason to revisit the research on non-slender delta wings to search for an explanation. Specifically, the differences between true slender winged deltas and the non-slender delta wings*. I found some good info with investigations done at Reynolds** number in the order of 10^6, this being the range in which windsurfing fins operate. I was lucky enough to be able to make contact with all three authors of the paper below (one of whom is a keen windsurfer!) and they were all extremely generous with their time and thoughts as to the extrapolation of their findings to the behavior of FangyFins.
[JOURNAL OF AIRCRAFT,Vol. 54, No. 2, March-April 2017. Numerical Investigations of Vortical Flow on Swept Wings with Round Leading Edges by Andreas Schutte]
* A non-slender delta is generally regarded as one with a rake less than 60 degrees. ** Reynolds number helps predict flow patterns in different fluid flow situations and is based upon calculations of incompressible flows varying viscosity and velocity. Fun fact - Air traveling at below 100 metres per second is regarded as incompressible!
An illustration of vortical flows from Schutte's paper.
To summarise the research, under the right conditions, the non-slender delta wing has an attached leading-edge vortex, and possible secondary and tertiary vortices depending on the leading edge radius and the curvature of the leading edge. These vortices tend to not interact directly with the wing surface downstream, unlike a slender delta and the hydrodynamics
may be more stable as a result. However, these only develop when the
ratio of leading-edge radius to chord length is less than 0.005. e.g. if you have a chord length of 200mm, the radius of the leading edge will be 1mm.
I noted that a small radius leading edge creates less drag at low angles of attack and conversely, at higher angles of attack, the small radius is more efficient at triggering the vortex flow and produces a more intense pressure gradient than its blunter siblings.
Ideally, I wanted to create a leading-edge with both advantages, one nicely rounded to give attached flow, low-speed lift, and move the vortex away from the board - fin junction pressure wave region. The other a really fine radius to give lift at speed with less drag. Now, whereabouts along the leading edge do I transition from one to the other?
Evolutionary clues on slender delta wings belong to the Swift and I felt this was as good a starting point as any. I copied the layout of a Swift wing as far as having a broad shoulder and arm region, with a rapid transition to the fine radii of the finger region.
In practice, the ability to refine the leading edge radius to a ratio of 0.005 is limited by the desire to ensure the edge is safe to the user, equipment, and environment and not act as a knife. In relative terms, this meant the ratio was initially large at the root, decreased at the transition zone, and then slowly increased toward the tip due to the decreased chord width span-wise. Schutte's studies indicated that a small but increasing ratio toward the tip was not significant.
[ Schutte found that within the parameters he studied, increasing wing sweep, increasing leading-edge radius, and decreasing Angle of attack all promoted attached flow on non-slender delta wings]
The test fin was an FF24. I tested two leading-edge transitions from large to small radii signs. One with the transition complete by about 25% and the other at approximately 50% along the leading edge from the root. The preliminary results from on water were a perception of
a decrease in all-round ability, partially offset by improved top-end speed, with both parameters proportional to the length of small radii leading edge.
My initial conclusions:
At the relatively low angles of attack used in windsurfing, the vortex lift flows are
not a large enough component of the overall lift to be considered a major influence in design.
The physical limitations of achieving small edge radii are such that the amount of the fin leading edge meeting the required physical parameters is relatively small, and therefore any attempt to harness vortical flow is unlikely to be a worthwhile pursuit. (Except perhaps on the FF28 where the dimensions are such that the above conditions can be met over a greater length of the leading edge)
Appendix: A rough formula for calculating the least leading edge radius ratio to ensure attached flow is : LE Radius ratio = 1.12 x t^2 [where, t = maximum thickness as fraction of chord.] Example: On an 8% foil with a chord of 200mm The LE radius ratio = 1.12 x (0.08)^2 => 0.007 (approx) Therefore: Chord length of 200 x LE ratio 0.007 = 1.4mm radius LE required
Parting thoughts:
Interestingly if you do this calculation with a 7% chord thickness foil, the ratio is approx 0.005. On my fins, I find 7% an unstable fin, and I wonder that is due somewhat to the initial vortical flows developing on the fin. Thoughts anyone?
I suspect it's not just the fin that loves LG!
(and quietly, I must confess that i really like the FF20 - I think it's my favourite fin)
