# About the induced drag of sails

Discussion in 'Hydrodynamics and Aerodynamics' started by Mikko Brummer, May 18, 2020.

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### patzefranpatzefran

Also a copy of another paper about sheared flows.................

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• ###### Morita.pdf
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### patzefranpatzefran

Some years ago, I developed a Fortran lifting line code computing wing or sail performance in sheared flow using Morita hypothesis (I had some exchange with T Speer about it)

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### philSweetSenior Member

Regarding the concept of induced drag, and force decompositions in general, as they apply to nonuniform flow conditions—

I believe it all comes down to whether you can set up the scenario beginning with an irrotational potential flow domain far upstream and then perturbing it to get the desired nonuniform conditions.

In the case of a nonuniform flow field, there are severe restraints on what nonuniform domains can be handled. Essentially, you need to start with a uniform flow and perturb it using sources, sinks, doublets, and vorticies. Experimentally, this would be equivalent to placing a lifting body in the uniform flowfield and taking the resultant far-downstream flowfield as the initial conditions for your test volume. Conceptually, you still need the nonuniform flow field to be an irrotational domain.

In simple terms, the only nonuniform flowfield volumes that can be treated using potential flow, and that can be described using the lift and drag decompositions we are familiar with, are ones that have crossections that are themselves a Trefftz plane flowfield of some other upstream object.

On the bright side, you now have an invariant reference - the exact same reference reference system to define your terms - the free stream vector upstream of everything. On the awkward side, you have to figure out what that is by designing an upstream lifting object that converts a uniform flow into the one you actually want. Two immediate results are that the actual reference vector can not appear anywhere in the flow within the test section, and any metrics based on the flow within the test section can not be invariant.

Thus we are left asking what use is all this. It ought to work as a basis for computer optimization of the test object, once the objective function is expressed in terms of the uniform upstream-of-everything flow.

I suspect that the asymmetrical tack angles of a sailboat in sheared wind can be accounted for by this method. The tack angles ought be symmetrical about the reference vector described above.

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### tspeerSenior Member

I wish Morita had shown what the downwash distribution was for his optimum planforms. According to Lissaman, the downwash should have been uniform over the span.

Morita's paper concerns flow that has a nonuniform magnitude, but the freestream is all flowing in the same direction. Lissaman's paper also used a sheared magnitude and single direction. I find the case of sheared apparent wind direction to be tougher to get my head around, because the vortices in the wake would diverge as they head downstream.

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### patzefranpatzefran

PhilSweet, you say Linear sheared velocities as Initial condition may leads to non physical results , as they are not irrotational flows ?
So Lissaman and Morita are wrong.

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### ErwanSenior Member

Well, while the following issue is not exactly about shear flows, but still regards Induced Drag, I ll try to expose it.

Under the Jones assumption, which translates into a linear downwash distribution, It appears that for a given Apparent Wind Speed, and a given Righting Moment, there are an "infinity of solutions".

The case below is an A-Cat rig with AWS = 40.54 ft/s
Using the Induced Drag Application: VORTEX, I computed 4 different Downwash "cases"

1- Downwash foot :2.0 Downwash Head : 0 Induced Drag: 4.632 lb Oswald Coef: 1.873
2-Downwash foot :2.5 Downwash Head : -0.55 Induced Drag: 5.445 lb Oswald Coef: 1.7497
3-Downwash foot :3.0 Downwash Head :-1.09 Induced Drag: 6.447 lb Oswald Coef: 1.6178
4-Downwash foot :6.00 Downwash Head :-4.37 Induced Drag: 16.226 lb Oswald Coef: 1.0264

These 4 "solutions provide the same Righting Moment ( with 4 different lifts and 4 different CoE)

Comparing the Net Driving Force (Useful Lift net of induced drag), using NDF= SIN(AWA) *Lift - COS(AWA)*Induced Drag.
The result are very different:
1 NDF= 72.83 lb
2 NDF=75.80 lb
3 NDF=78.58 lb
4 NDF=91.75 lb The difference between case 4 and case 1 = + 26%.

So the point is where to stop the process of bringing down the CoE increasing the positive Downwash at the foot and the Upwash at the head? If no rules or technologies limits, preclude you to increase the sail foot lenght for instance.

Thanks in advance and Best Regards

EK

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### patzefranpatzefran

Please, Erwan, where can I find the Vortex application you are using. Are the results related to the same AWA. Are the downwash distributions achieved by twisting the same sail geometry or different chord geometries ?
Lifting line codes use small chord/span ratio assumption which limit their accuracy range.
If you send me your detailed hypothesis I could try to run my own code (accounting for apparent wind gradient) to compare with your results

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### tspeerSenior Member

The Vortex application is this lifting line spreadsheet. You can read about its derivation here. Although lifting line analysis does assume high aspect ratios, I believe the induced drag results are accurate for the same spanwise downwash and lift distributions. The planform shapes that result in those same loadings will be different if calculated based on lifting surface methods compared to lifting line analysis.

That is correct. There are an infinite number of combinations that have the same righting moment. However, they do not have the same drag. This figure shows how the induced drag varies as a function of several variables, including the height of the rig and the effect of the gap between the rig and the water surface. The dashed lines all have uniform wake wash along the span. The solid lines have the wake wash going to zero at the head. The dot-dash line has just enough negative wake wash at the head that the chord is zero there for an untwisted planform. The figure shows that by making the rig taller, one can get a reduction in induced drag while still maintaining the same heeling moment.

The paper goes into the tradeoffs in more detail.

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### patzefranpatzefran

Thanks, Tom Speer, I knew your lifting line spreadsheet but I ignored his name ! Basically, my Fortran code use quite the same numerical method as yours, but it accounts for a wind velocity gradient (not direction) using numerical methods from Lissaman and Mojita. I checked with zero gradient, our results were identical, I also checked my results were identical with case having analytical solutions derived by Lissaman !

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### ErwanSenior Member

Patzefran
Yes, an A-Cat in floating mode, sailing @7m/s @ 45° TWA and 6.4m/s TWS which is around 12.38m/s AWS and 21° AWA

The great feature of VORTEX is that as long as you know that the optimum sail plan which optimizes Induced Drag will have a linear distribution of downwash along the span, strangly VORTEX is elaborated to receive as inputs: the downwash value at the bottom and at the head, and in the between, it is a straight line, so pretty linear, isn't it?
For me,the DESIGN page works like a solver and it displays the optimum sailplan which minimizes Di, for the given assumptions.

I just compared different sailplans with same area, same span, same righting moment, just Downwash assumptions are different. Everything else equal, no twist, constant Cl along the span.

You are right, but I think A-Cat Aspect Ratio, especially with decksweepers, is high enough to provide reasonnable results, anyway, I am more interested in "Relative Values" than in the quest of perfect accuracy, imho VORTEX meets smartly the principle of "optimizing the marginal efficiency of the extra complexity".

Yes, but what is striking, is that the increase in driving force, more than offsets the increase in Induced drag when you increase the slope of the Downwash distribution line.

I guess Lifting surface methods is terminology for VLM ?

Cheers
EK

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### ErwanSenior Member

Patzefran
Yes, an A-Cat in floating mode, sailing @7m/s @ 45° TWA and 6.4m/s TWS which is around 12.38m/s AWS and 21° AWA

The great feature of VORTEX is that as long as you know that the optimum sail plan which optimizes Induced Drag will have a linear distribution of downwash along the span, strangly VORTEX is elaborated to receive as inputs: the downwash value at the bottom and at the head, and in the between, it is a straight line, so pretty linear, isn't it?
For me,the DESIGN page works like a solver and it displays the optimum sailplan which minimizes Di, for the given assumptions.

I just compared different sailplans with same area, same span, same righting moment, just Downwash assumptions are different. Everything else equal, no twist, constant Cl along the span.

You are right, but I think A-Cat Aspect Ratio, especially with decksweepers, is high enough to provide reasonnable results, anyway, I am more interested in "Relative Values" than in the quest of perfect accuracy, imho VORTEX meets smartly the principle of "optimizing the marginal efficiency of the extra complexity".

That is right, and it is striking to see how the increase in driving force, more than offsets the increase in Induced Drag, the more you give slope at the Downwash line.

I guess it is the terminology for VLM ?

Cheers
EK

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### patzefranpatzefran

Erwan :
If I have understood, In your study, your inputs are, the sail area, the span, AwA and AWS, downwash at root, downwash at tip.
1) Did you accounted for surface gap effect
2) your downwash parameters variations are arbitrary, did you looked at the corresponding planforms without twist to check the
small chord/span ratio hypothesis ?
Likely, there should be a limit for the maximum chord / span ratio allowable for a lifting line code.
Cheers
PG

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### patzefranpatzefran

Erwan
Remember AR = span²/area = span/mean chord
with unconstrained chord variation, you could get max chord >> mean chord for the same AR

Cheers
PG

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### Doug HalseySenior Member

These days, partial-chord skirts that close the gap under the foot of the sail are becoming more common (Moths, A-Cats, etc.), so it's important to also consider the chordwise extent of the skirt.

This chart shows my initial attempt at such an analysis a couple of years ago, using a Vortex-Lattice Method (VLM) , which is the simplest type of Lifting-Surface Method. This could probably be modeled in a Lifting-Line Method, but I suspect its accuracy might not be as good.

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### patzefranpatzefran

Thanks Doug, most foiling A cats have decksweeper sails and also now some Classic. Does the clew height is relative to the span or the foot length ?
At zero skirt length, your data shows the influence of the gap, but you need a really very small gap to double the geometric AR !

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