The Myth of Aspect Ratio

But theory states that the shape of the laod distribution is effected by the AoA. Since for a given AoA a lift force on the hydofoil implies that the pressure on the lower surface must exceed that of the upper surface. No AoA no lift. Therefore fluid must escape around the ends of the lower (or under surface) in an outwards manner to the ends (tips) and inwards to the centre on the upper surface. Hence it is not independent of AoA.

Also, looking at the tips of a finite hydrofoil, the pressure differnce between the upper and lower surfaces must be zero, ergo so must the lift force and circulation. Therefore you cannot assume the CL is a constant along the span.

Is that Correct Slavi (this is your field of expertise, not mine)?

 

That's correct, AH.

For the few of you out there who like math , it can be best understood by studying the lifting-line theory of finite wings. It is a too vast argument to be summarized here, but a lots of related resources can be found in internet. One pretty well-done is this one, imho: http://www.aoe.vt.edu/~neu/aoe5104/23 - LiftingLineTheory.pdf

The key equation is at the page 7, where the expression for calculating the Gamma (the circulation around a foil section) of a planar wing is shown.
It states:

Gamma(y) = Pi V c(y) Alpha + Pi w c(y)

where:
y is the spanwise position of the foil section
V is air or water speed
c(y) is the local chord of the foil
w is downwash
Alpha is the aerodynamic AoA.

So these are all parameters which contribute to the spanwise shape of the circulation, which is related to the local (spanwise) lift by the equation:

L(y) = rho V Gamma(y)

The total lift of the wing is then the spanwise integral of this local lift L(y).

I'm attaching a result of an analysis of a simple rectangular wing with NACA 0015 foil sections. The lower part of the pic shows spanwise load distributions for three different AoAs (3°, 5°, 8°). It is clearly visible how the shape of the load distribution changes in the 3 cases.

Cheers

 

The load distribution can be thought of as the chordwise integral of the pressure difference between the lower and upper surfaces. It is not constant along the span.

Trailing vorticity happens not just at the tip but also along the span of the wing proportional to the slope of the load distribution. If the load distribution has a finite slope at the tip which means there is no concentrated tip vortex, there will still be induced drag.

In the physical world or with a math analysis which allows the wake to move, the wake will roll up sufficiently downstream of the hydrofoil and resulting in two concentrated vortices. But that's a ways downstream, somewhat like the Kelvin wave pattern of a vessel's wake.

My understanding from the theory is the shape of the load distribution (not the magnitude) for a hydrofoil with zero twist and symmetric sections does not change with angle of attack (AoA). The magnitude does change with the AoA and is proportional to the AoA. If the load distribution along the span at 1 degree AoA is given by G(y) where y is the spanwise coordinate, then the load distribution at 2 deg AoA is 2 * G(y) at 0 deg AoA it is 0 * G(y), and at -1 deg AoA it is -1 * G(y). The total lift is the spanwise integral of the load distribution.

Again, that's for an untwisted wing with symmetric sections. The load distribution of an arbitrary hydrofoil with twist and/or assymetric sections can be treated as the sum of two load distributions. One is the load distribution at the AoA which results in no net lift, and that AoA can be considered as the zero aerodynamic AoA. For a non-zero aero AoA with lift, the load distribution for a "flat" hydrofoil at the AoA is added. The shape of the total load distribution will change with AoA.

 

If one goes through the lifting line theory they'll find that the load distribution SHAPE (not magnitude) for an untwisted, symmetric wing/hydrofoil is independent of speed or angle of attack.

 

Lifting-line theory assumes that the flow is inviscid, so it won't tell you any effect of Reynolds number directly. However, at different Reynolds numbers, a given foil can have quite different flow separation patterns & therefore different spanwise lift distributions & induced drag.

 

True, Doug. For that purpose a non-linear LLT has been developed, which is based on empirical 2D airfoil sections characteristics.
But at this point, with turbulators and details about LLT methods, perhaps this thread has taken a bit too much side drift?

 

If there is sufficient separation that the load distribution is affected then the lift will be compromised and a considerable drag increase due directly to the separation. In that case there is more to consider than the relatively slight change in induced drag due to the change in load distribution. Also the magnitude of the change in induced drag due to a change in load distribution is generally relatively small compared to the effects of a change in span.

I agree this thread has either drifted or expanded considerably beyond the original intent. It's been a good discussion though.

 

Inviscid theory …..
Perhaps we should talk more about hydrodynamics and where it falls down with an aerodynamic approach?

Water is much more viscous and heavy compared with air. Aerodynamicists work with edit[Thicker not thinner] boundary layers and effects that fade away quickly. e.g. Stir your coffee and it swirls for minutes, stir a cup of air and it dampens in seconds. When can you assume it's all shear? when do you find slip significant? Anyone who's tried teaching basic hydrodynamics will understand how many simple questions suddenly devolve into complex attempts at explanation. It's just such a complex subject.

The boundary layers on a moving foil in water are different to those in air and they are also extremely complex to model mathematically. Disturbances in water also endure for considerably more time, there’s an order of magnitude difference in viscid effects. A lot of aero fudging or simplification of physics to explain lift and drag that works well in air doesn’t necessarily translate as well into hydrodynamics.

And as for math models, most of us if we studied 3D fields know what the math turns into when you applying grad div and curl and it quickly develops into a massively complex and impractical model of complex (imaginary plane components) solution of pde’s.

Even the simplified theories (Euler, Bernoulli, Navier-stokes, Circulation) have omissions, discrepancies, paradoxes and contradictions.
That’s why ‘Practical hydrodynamics’ for Naval architects is the advanced course of the basic fluid mechanics. And it’s not taught as a heavy mathematical physics approach surprise surprise.

But then the equations and explanations used by aerodynamicists are really simplifications too. Different parts of each model explain quite well the different parts of aerodynamic theory. But their assumptions often don’t translate as well into hydro-dynamics. Take out the viscous components and ignore slip and suddenly the theory falls over and the predicted foil performance just doesn't match reality.

Here’s an example to consider that has been mentioned before; a 0012 NACA foil at higher Reynolds numbers in water has a lower drag and higher lift at angles of attack <8 degrees when used backwards, but in air at the same Rn it’s the other way around. These are the sorts of real world observations that tend to throw a spanner in the works for simplified mathematical modeling. Even CFD is quite limited and needs tying to real world data.

Then as I posted before, if I get out of smooth water and into a seaway the practicalities of lower Rn foils in smaller vessels under 100 feet and it's not all that applicable. It does become more applicable for high speed foils.

 

Start another thread on this topic and then we can have a discussion about whether hydrodynamics as currently taught is fundamentally flawed.

" a 0012 NACA foil at higher Reynolds numbers in water has a lower drag and higher lift at angles of attack <8 degrees when used backwards, but in air at the same Rn it’s the other way around." I assume you mean > 8 degrees. Do you have a reference for this? I'm interested in learning more about it. I have a guess as to the cause and it's connected with the stall angle of a 0012 airfoil being around 8 degrees. Simplified math modeling doesn't work well in that situation whether the flow is air or water.

 

Well not fundamentally flawed it's more an imprecise science that can be described with several simplified models with some accuracy.
The learning is knowing which theory has which limitations and why. And I think it's very relevant to this thread because it's easy to start a design spiral for a foil based on extrapolated assumptions backed by inadequate models.

But I'm easy what did you want to talk about?

 

Exactly Mike.

There is nothing ‘flawed’ about it. The problem is with mathematicians and theoreticians. Because many aspects of hydrodynamics which can be observed which then forms a theory, are not so easy to put into a mathematical theorem. As Mike says many ‘models’ exits and to a high degree of accuracy in most cases, but, some aspects are still unable to be fully explained and thus proved a mathematical harmonious proof.

A simple example is that of wave making resistance. Everyone, well naval architects and hydrodynamics, know about wave making resistance and its components. However, as a vessel increases in speed it becomes less precise to accurately calculate (mathematically) the final resistance. To the point of “fudge factors” being used, most notably around ventilated transoms as the Froude number increases further. We can all see it occur we can all repeat it; we can all measure it, we can all tell you why it occurs…but, to provide a mathematical model to precisely account for the phenomenon is not so easy. Many great heavy weights have tried, but only with “fudge factors” to obtain a near complete solution.

Fudge factors do not sit well with theoreticians nor mathematicians. It is an illogical imprecise way of describing something that is ostensibly “logical”. Yet for naval architects, the hydrodynamics are known and the theories (words) well accounted for the “event” that occurs again and again as predicted. However, the exact solutions to precisely describe quantitatively (mathematically) are not known as too many inconsistency and also “unknowns” affecting their precise solutions; despite many years of research and attempts by others.

Thus, there is nothing flawed in hydrodynamics, there are inconsistencies which scientists/hydrodynamacists/mathematicians have yet to account for to provide an exact solution under all conditions.

 

Water is indeed much more viscous and more dense than air. But the only thing which determines the boundary layer thickness/body_size ratio is the Reynolds number. And Reynolds number depends on the ratio of viscosity to density: rho/mu = nu .
Specifically, Re = size*speed/nu

The values are:
nu = 14.52e-6 m^2/s (air at STP)
nu = 1.15e-6 m^2/s (water at 15C)
so for a given speed and body size, a water flow has 12x larger Reynolds number and significantly thinner boundary layers.

And the only thing which makes water flow more complex than low-speed air flow is free surface waves and cavitation. If these are not significant, as is the case in a typical keel, then there's no theoretical or mathematical difference between air and water flows.

The few America's Cup campaigns that I've been involved in did most of their keel testing in a wind tunnel. It's cheaper than tow-tank testing, and unlike in the tow tank, one can get close to the actual Reynolds number by running the wind tunnel at 120 mph or more.

 

Can't testing at too high a fluid speed lead to so-called "anaemic boundary layers"? These can occur when there is insufficient fetch for the BL to fully develop.

The Reynolds numbers of two tests can be identical, but the one with a shorter length and higher speed can have a different c_f and other important quantities.

 

Yes thanks I had the boundary thickness the wrong way around I see it's inversely proportional to the root of Rn.

Reynolds number is a rock solid foundation for sure but isn't an underdeveloped boundary layer one of the significant problems of wind tunnel testing of foils intended for water? edit [I see Leo already mentioned this]

 

I haven't heard the term "anaemic boundary layers" before. Does this have anything to do with using turbulence stimulators?

Same geometry with properly scaled relative surface roughness at the same Reynold's Number will have similar flows. I'm surprised to hear the cf will be different. Is this a matter of proper scaling needed or is scale testing with matched Reynold's number fundamentally flawed? Any references on the subject?

Does some of the skepticism in naval architecture about boundary layer theory and scaling arise from the necessity to test at much lower than full size Reynold's Number and the attempts to deal with that?