Downwash equation for Bell-Shaped spanload distribution

Sorry, the Smiley is unintentional, just a mistake, I am not digital native

 

Um, no, that isn't right. It's not even close to being right. You are going to have to specify a relative heading (AWA10 @ 10m), a wind speed gradient = f(height), a boat speed/TWS10 Speed Ratio, and a Cl/Cd curve right upfront. AWA = f(h,TWS10,TWA10,SR) and AWS = f(h,TWS10,TWA10,SR). First solve for the TWS10 and TWA10 using AWA10, h=10, and SR, then compute the AWA(h) and AWS(h) at each height. Then you can find the appropriate Cl and Cd as a function of height for this one condition. Then you can compute the side force, heeling moment, thrust force, and trim moment as a function of height and cord for this one condition. See figure 5 in OPTIMISATION OF SPAN-WISE LIFT DISTRIBUTIONS FOR UPWIND SAILS by Peter Richards et al. Now, and only now, do you have an RM for the elliptical case that you can use for comparison. (And when making that comparison, do you want to use TWA10 or AWA10 for equivalence?)

Once you have that, it is only a little bit harder to incorporate some hull hydro numbers (which adds hull trim, RMmax as a function of heel angle,adjusts h as a function of heel angle and includes a nonconstant Speed Ratio) and produce a primitive VPP. This lets you compare setups with different TWA directly in terms of VMG.

The terms listed are all significant, and the rigs of even primitive craft have controls that provide huge variability in span loading. Optimizing for one operating point is fun, and on point for a record attempt in Namibia, but designing a rig that has enough versatility to always get you home is more relevant most of the time. I guess with wind surfers, you can just take ten different ones with you, but probably not with an A-Cat.

<edit> I seem to have left out the bit about where the elliptical loading comes from. After solving for the appropriate Cl for each height, find the cord that maps this to the target elliptical lift distribution. For want of a better approach, I define the target vector everywhere as aligned with the total aero lift vector, then back into the twisted local lift vector. Short of preserving higher order terms in the Trefftz plane, that's the best I've come up with so far. So it's an iterative approach because the total aero vector changes as you recompute the cord.

 

The spreadsheet and the planform paper, along with many of the other articles on my website were the product of homework when I was off work for 40 days. My time not working turned out to be quite productive!

 

As all your time on this forum. I'm sure everyone agrees that we owe you for that!

 

According to the the NASA Workpaper, the bell-like load distribution results in a quadratic (not linear) variation of the downwash.

What am I missing?

 

You are right The linear downwash variation (Jones) gives the least drag for a given lift and root bending moment, with span unconstrained.
There are not exactly the same hypothesis as Prandtl, the lift curve are not exactly the same, but close ! They have both a span higher than the elliptic solution for the same lift.

 

Looking at the NASA paper, it looks like the given downwash angle , (3/2)* (x²-1/2) is far downstream, twice the downwash at the lifting line.
For those interested, after a lot of long and laborious trials, I succeeded to find the exact value of the upwash outside the lifting line for x>1 or x<1 which is given by
DW = - (3/8)*(x-sqrt(x²-1))²

 

Sorry, for x > 1 or x<-1 !

 

My contention is in nearly every case where Prandtl's bell-shaped lift distribution is considered, the design would be better off with a linear downwash distribution. The slope of the downwash distribution can be adjusted to achieve the moment objective - whether that be bending moment for structures, heeling moment for sail rigs, or pitching moment for swept flying wings - with minimum induced drag.

BTW, Jones' solution is not just for unconstrained span. Even when the span is constrained, the linear downwash distribution produces minimum induced drag for the given span and moment. Munk's constant downwash/elliptical lift distribution is just a special case of Jones' more general solution, where the moment itself is unconstrained.

 

There were two different constraints used by Prandl and Jones. Jones only considers the root bending moment, Prandtl considered the integral of the bending moment over the span, which is more representative of the structural weight of wing spars. Theoretically, sailboats need to consider both - root moment to balance RM, and rig weight because of it's effect on reducing RM. Unfortunately, the needed versatility of rigs makes it difficult to concoct a viable expression for the integral constraint, and standards for rig strength (weight) are based on the hull, with only a cursory nod to rig details. If you were planning a freestanding solid wooden mast, it might be worth revisiting the idea.