The Myth of Aspect Ratio

Here's my guess as to why "induced drag" is sometimes thought to be Reynold's Number dependent in naval architecture and boat design.

I've been using "induced drag" per the definition given in aerodynamic and hydrodynamic texts, as well as described in books such as PYD and the new "Ship Resistance and Flow" volume of the revised PNA. This "induced drag" is an inviscid phenomena and results from trailing vorticity due to lift, on in the case of a boat or ship, lateral force.

However, when an actual finite span wing, a keel or a rudder is producing "lift" or a boat or ship is making leeway, the resulting increase in drag is due to viscous, frictional effects which will be at least somewhat Reynold's Number dependent in addition to that caused by trailing vorticity. In marine situations with a free surface there may also be increased drag due to wave making.

It appears that it is a common practice to lump all increases in drag/resistance due to leeway together irregardless of the origin, and then call the result "induced drag" or "induced resistance". There is a good pragmatic reason to lump the changes in drag together. If the change in drag/resistance obtained experimentally by force measuement it is essentially impossible to split it into portions due to trailing vorticity, viscous effect, wave making, etc. And there may not be any need depending on the use of the information. For example a VPP just needs how the drag changes with leeway/lateral force; the origin doesn't matter.

To further confuse matters, sometimes the increased drag is assumed to vary with the square of the lateral force, and an emprical adjustment to the classic formula for inviscid induced drag factor based on experimental data is derived.

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An example of lift resulting in increased drag in the absence of induced drag can be seen in any of the CD vs CL plots for airfoils from Abbott and Von Doenhoff's Theory of Wing Sections. The drag increases with increasing CL with the amount of the increase dependent on Reynold's Number and roughness.

 

Would you and Slavi be happy labeling the x scale as Rn on the following plot ?

There are other issues with a much more viscous fluid than air, what about the boundary layer thickness and overall flow, the effective shape of the foil is not the physical shape particulalry in the lift condition (like Jenny's flat plate). Also there's an efficiency factor in the equation for Cdi.

Tests of hydro-foils can be quite illuminating, JE Hardiman has talked several times of real data from flat plate rudders with a properly placed tubular section running vertically through the foil at 1/3 chord or so being indistinguishable from a properly constructed NACA foil. That wouldn't work in air because of Kinematic Viscocity differences.

 

QUOTE=MikeJohns;445444]Would you and Slavi be happy labeling the x scale as Rn on the following plot ?[/quote]

No, I wouldn't be happy labeling the x scale on that plat as Rn because the induced drag behaviour doesn't make sense in the way I've been using the term. Have a look at my post #46 above about apparently different uses of the term "induced drag/resistance." Perhpas you are using the term differently than I am.

Added: See my post #57 below for my revised thoughts on this.

Flow in water (less free surface and surface tension effects) is the same as flow in air (or gasoline, or glycerine, or helium, or ...) at the same Reynold's number, assuming cavitation doesn't occur and the speed in air is less than Mach 0.2 or so.

Induced drag (the standard aerodynamics/hydrodynamics defined induced drag) is not dependent of foil section shape directly. It's dependent on the spanwise lift distribution. It's the same for any speed irregardless of Reynold's number.

Viscous drag due to lift does vary with Reynold's number, but that's not what I'm talking about.

What speed in air and therefore what Reynold's number? At a sufficiently high Reynold's number and small angle of attack a flat plate in water or air behaves almost like a reasonable airfoil. But that is irrelevant to my discussion of induced drag due to trailing vorticity because that is not dependent on the details of the flow over the airfoil; it only depends on the spanwise distribution of lift and resulting trailing vorticity.

We are probably using the term "induced drag" differently.

 

Im not 100% sure what it is you’re trying to say or elude too, or whether it matters to be honest.

The reason why naval architects use Rn a lot is simply because of scaling and its effects on friction.

Rn is determined by the velocity, density and viscosity of the fluid. But since naval architect cannot test ship sizes of 50m or 100m or more especially with little money available, we build small models. These small models conform to the “pi-theorem” of relationships between sets of non-dimensionalised attributes or quantities. In scaling the velocity, using the correct velocity for the scale is important. When Rn is quoted it is because ‘we’ are referring to frictional aspects as this provides a measure of its drag from a “small” model to the “real” ship size; thus what is the velocity of model and ship to gauge the effects on drag?

When we discuss wave making drag/resistance, other factors come into play.

I think it is more of a case of the other way around. Since aeroplanes operate in only one medium, air. Thus separating frictional and wave-making drag is not performed, as there is no “wave making” drag in the naval architecture sense, since this occurs at the interface between two different mediums. Air is one medium not two.

Other than that, im not sure why it matters.

We all accept, i think, that the induced drag essentially comes from the downward velocity over a wing by the wing tip vortices. After that what you wish to "do" with it...does it matter, so long as it is accounted for?...and you know how to take account of it too

 

But you are happy if it is labeled V ? Thanks for the great discussion so far.

 

I know about Reynold's number and the challenges of tow tank test result scaling.

My original post was about induced drag and aspect ratio. Some folks said induced drag also depends on viscosity. I realized there are two different uses of the term induced drag.

An interesting question arises about how the drag increment due to leeway should be scaled between tank test results and full size when some of it is independent of Rn and Fn, some of it is dependent on Rn but not Fn, some of it is dependent on Fn but not Rn, and of course some of it is dependent on both Rn and Fn. And of course there is the question of should the drag be considerd a function of leeway angle or lateral force. But that assumes an understanding of how lateral force vs leeway angle scales. I'm not proposing to even try to answer any of those questions.

 

No, wouldn't like that either. Perhaps if the red curve was labeled frictional drag instead of induced drag then it would make sense to label the x-axis as either Rn or V but the shape of the curves doesn't look right for that either.

Added: See post #57 for my revised thoughts on this.

 

Mike,
The plot you have shown illustrates the drag of a hydrofoil (or a wing) in function of speed. The x-axis is speed and the y-axis is drag. Differently from what has been discussed here, that plot has been drawn for a fixed wing geometry (fixed AR and area, or span). DCockey was instead discussing the case of fixed span but variable AR, which leads to a kind of graph visible in the attachment of my previous post.

Since I don't have time now to derive the equations which lead to your graph, I'll point you to this page, which explains the concepts of induced and profile (or form) drag and shows the same pic with labeled axies: http://www.adl.gatech.edu/classes/dci/aerodesn/dci03aero.html

As about your question, the answer is: yes - I would feel confident to put the Reynolds number in the x-axis, because the Reynolds number is proportional to speed, if everything else is being kept fixed.

I wouldn't enter the discussion of whether the shape of the curve is perfect or not. It can be considered qualitatively correct imho, since it shows a decrease of induced drag (which should be inversely proportional to speed) and increase of form drag (which should be directly proportional to speed^2) - which is what matters for the sake of the discussion.

Cheers

 

Thanks Slavi

Thanks yes I do understand that.

But isn't the energy in those tip vortices (and Di) relative to both the tip chord length AND the velocity of the fluid ?

 

Well yes, since the lift distribution along the span (and, hence the vorticity left in the wake of the wing, and hence the induced drag) depends on both the planform shape and the speed, if you keep the total lift constant (equal to the weight of the craft, for example). And the planform shape means - chord distribution along the span, tip included.
There are also other ways to control the lift distribution along the span, like variable twist and camber. But until now we were talking about rudders and keels here, so these features have not been considered. They do become an option in case of hydrofoils.
Cheers!

 

Discussion is good. I made a post but quickly realized it was incorrect so deleted it.

 

Induced Drag = 1 / [Pi * (0.5 * Density of Water * Speed ^ 2)] *[ (Lift / Span) ^ 2] * (1 + Sigma)

Induced drag with constant lift is inversely porportional to the dynamic pressure and therefore the square of the speed. So the reduction of induced drag as speed increases is correct. However, this reduction of induced drag is NOT due to viscous effects. It's independent of viscosity and therefore Reynold's Number.

Based on the explaination of the chart in Slavi's reference the increase in profile drag in the chart is proportional to dynamic pressure and therefore speed squared. As long as the Reynold's number is sufficiently high the profile drag is generally assumed to be independent of Reynold's Number.

So speed at the bottom Mike's chart is correct. Reynold's Number would be misleading at best. Slavi's reference talks about this chart as a speed chart, not a Reynold's number chart. Change the viscosity but not the density and the chart would be the same vs speed even though the Reynold's number changes.

The argument that Reynold's number is proportional to speed so it's okay to use Reynold's number raises the question of using other non-dimensional coefficients. Why not Mach Number? It is also proportional to speed with everything else being constant. Or Froude Number based on any fixed length? Also proportional to speed.

 

Don't worry Cockey i do that all the time, infact more times than people know about.

 

My understanding is the shape of the load distribution and therefore distribution of the vorticity and relative strength of a concentrated tip vortex is independent of lift, speed, angle of attack and wing CL for an untwisted wing with symmetric sections. It does depend on planform shape including the tip. Assumed is that the aspect ratio is sufficiently high and the sections are sufficiently below stall that the sectional lift vs angle of attack slopes are constant.

A typical aircraft wing with twist and asymetric sections is a different story. In that case load distribution does change with lift and speed unless the changes are such that CL of the wing remains constant.

I don't know enough to comment about delta wings and similar at sufficiently high angle of attack that a vortex sheet is shed from the leading edge which results in lift production.

 

Not entirely related, but another thing some airfoils have either retrofitted, or designed in place are what is known as a Vortex generator, which takes various forms, from "fence" to single slabs at right angles to the foil at the mid section, to winglets. Any thoughts on these in a marine application? Their purpose is to make the laminar layer "stick" better.