# Effective Aspect Ratio of Surface Piercing Rudders?

Discussion in 'Hydrodynamics and Aerodynamics' started by Autodafe, Apr 7, 2013.

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### AutodafeSenior Member

I don't follow why you assume that pressure difference falls to zero at the nominal water surface? The models presented here by Leo and others suggest to me that this is not the case.

The surface can bulge up and down on opposite sides of the foil as a result of pressure difference, but it can't flow around the top to equalise as it does at the bottom.

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### daiquiriEngineering and Design

I don't have the paper mentioned by Leo, so I do not know the details of that work. However, let's make a logic reasoning together. In the following lines I'll use the term "interface" for the surface dividing the water and the air.

A surface-piercing rudder has a constant-pressure boundary right above it (the atmospheric air). So, right above the rudder root (the waterplane area which pierces the interface) there is the same pressure both on the dorsal and on the ventral side of the blade. Hence, on the air side (a layer infinitesimally close to the water) of the interface, the pressure difference across the blade has to be zero. It is actually nearly zero, because a blade will produce a small lift in air too - but compared to the submerged part, it will be negligible.

At the water side (a layer infinitesimally close to the air) of the interface another quantity plays a role. It is called "surface tension", and gives water the ability to sustain a slight pressure difference, by curving the interface a little bit. The question is - how much pressure difference can a curved water surface sustain? The mathematical equation which deals with this is called Young-Laplace equation.
For constant-radius curvatures, the Young-Laplace equation gives the following result:
2 Gamma = R dP
where Gamma is the surface tension, R is the curvature radius, dP is the pressure difference. For water at 20° the value of Gamma is around 0.075 N/m (or: Pa m)

If we assume that the radius of curvature is of the same order of magnitude as the foil chord, say 0.2-0.4 m (20-40 cm), the equation tells us that the pressure difference on the curved interface, due to the surface tension, is no more than 1 Pa. If we compare it to mean pressure differences acting on rudder blades at any practical speeds, which is typically well over 500-1000 Pa, we see that the contribution of the surface tension is really insignificant, and we can assume that no pressure difference occurs at the interface.

So what happens is that the water-air interface has to deform and conform (waves are created) in such way to maintain the pressure difference between air and water equal to zero (besides having to satisfy other fluid-dynamics equations and boundary conditions). In other words, no pressure difference at the root of the blade - be it the dorsal or the ventral side of the foil. Hence, the air-water interface acts as the wing tip, halving the effective Aspect Ratio of the rudder.

However, I would love to read that paper cited by Leo. Perhaps it contains some additional aspects which I had failed to notice, and which could prove my reasoning wrong.

Cheers

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### Remmlingerengineer

As already mentioned, the pressure- and hence lift-distribution depends on the Froude number. The attached diagram is from the paper of Kuhn & Scragg. At zero Fr the water plane acts like a mirror. At Fr=1 there is a steep lift decrease towards the free surface. At infinite Fr the decrease towards the free surface is less steep than towards the tip.
Uli

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### daiquiriEngineering and Design

As I said before, I didn't have the honor to read these papers, so can only presume what's behind those graphs. Hence, I presume that the zero-Fn graph in Fig.2 wants to simulate boat speeds which approach zero. That's when surface tension can play a role, as explained in my previous post, but it is not of practical interest in boat design. The Fig.1 is of practical interest, and depicts well my previous explanation.

And by the way, I am not satisfied by simply reading affirmations like "at zero Fn the water plane acts like a mirror", which have appeared a couple of times in this thread. I need to know the physical arguments for these claims, and the only one I can think of is related to the effect of surface tension.

Cheers

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### AutodafeSenior Member

The pressure at the actual water surface will be constant at atmospheric pressure, but it occurs at different levels on either side of the rudder so across any given rudder section a pressure difference exists, or that's my theory anyway.

If the surface level on the low pressure side drops say 30mm and raises 20mm on the high pressure side (just for the sake of argument) then that's a pressure difference of 500Pa.

This is just based on watching rudders, I don't claim theoretical expertise...

Similarly, I think of ventilation as occurring when the surface depression at the low pressure "peak" is deep enough to become unstable and bubbles are sucked down and over the foil, spoiling the flow.

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### AutodafeSenior Member

Thanks, graphs are always good.

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### daiquiriEngineering and Design

I see your point, and I think it is correct. There will be a difference in water levels between the pressure and the suction sides, and it will necessarily give a finite pressure difference between the two, when referred to the static WL.

By the way, if the shape of the vertical pressure profile is a triangle (valid for flat water surface, not sure in this case), then the mean dP between pressure and suction side in your example would be 250 Pa, not 500 P. However, it would be interesting to have the actual data of how much the water level rises/falls in proximity of the blade surface.

IMO, for Fr approaching zero this mechanism cannot explain the use of the image method, which would imply the double effective AR as in Remmlinger's Fig. n.2.

Cheers

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### Remmlingerengineer

From a textbook about fluid dynamics

The Froude number = V / sqrt(g*L) is the relation of inertia force to the gravitational force.

The zero Froude number approximation can be used for very small speed V or very long chord length L. In this case the gravitational force outweighs the inertial force of the flow so much, that there are practically no waves on the surface and the surface can be regarded as rigid - it is a plane of symmetry.

The infinite Froude number approximation is just the opposite, very large speed or extremely short chord. In this case the gravitational force can be neglected, and the water surfaces acts, like Leo explained, as a plane of anti-symmetry. This is hard to imagine, but is is like in space, where there is no gravitational force. When an astronaut creates a wave in his coffee mug, the coffee leaves the mug and distributes itself in the space craft because of its inertia.

On earth the real flows are somewhere in between, the water surface is sort of a very soft "mirror". The attached diagram shows the lift coefficient slope as a function of Fr.
Hope this helps
Uli

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### daiquiriEngineering and Design

This part makes no sense to me, besides being a case of little interest for practical rudder design. Even a trolling-speed gives a chord-based Fn of 0.3 and more. The boundary condition in any case is the constant atmospheric pressure, not the wall tangency. If there are (hypothetically) no waves, it doesn't mean that the flow can support a pressure discontinuity at the air-water interface. It means simply that the flow equations have been (hypothetically) satisfied without the need to deform the interface, i.e. with no need to create the waves.

I just can't see how can a water surface be modeled as a rigid wall, it is imo not physically consistent with the reality.

Another case is the thin-ship theory, where the scope of the calculations is to determine the characteristics of the far-field. In case of the surface-piercing foil, the near-wall lift distribution is under examination, which requires more detailed modeling of the local boundary conditions.

Cheers

P.S.
Just to make myself more clear - I am not saying that you are wrong. I'd just like to read a sound physical explanation, in hydrodynamic terms, which would confirm that what you're saying is right.

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### sottorfmember

Yes there is a big difference between the water levels on the pressure and suction side. The high pressure side will create a wave and spray detaching from the rudder. On the suction side the low pressure will result in the free surface creating a deep through/air-cavity which can extend all the way down the rudder if it has a small aspect ratio. Of course with turbulence and flow separation from the profile at high angles of attack the trough/cavity is not stable and results in an air-water mixture on the suction side.

To reduce these effects of ventilation, surface piercing rudders for high-speed usually have higher aspect ratios that increase the submergence and limit ventilation (as hydrostatic pressure increases the pressure to above atmospheric stopping ventilation).

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### hariboJunior Member

maybe it has half Cdi, but why did you think Cl vs AoA is different?

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### Remmlingerengineer

The zero Froude number approximation, which is a very helpful simplification in the calculation of potential flows, was successfully used at Froude numbers up to 0.1. Since the force-ratios are proportional to Fr^2, the error in the resistance stays below 1%. Even at higher Fr it is helpful for the understanding of the phenomenology of the flow to look at the theoretical extremes.
In the zero Fr-approximation the flow in the vicinity of the surface is parallel to the x/y plane, no surface deformation exists. The surface can be regarded as a plane of symmetry. This is the idea behind the tests of cargo ships in a wind tunnel, that were often conducted before the advent of CFD. The under water part of the hull was mirrored at the water plane and this "double body" model was tested in the wind tunnel to determine the viscous resistance (e.g. papers by Lars Larsson).

It is correct, that the rudder always operates at much higher Fr. Therefore I posted the diagram that shows the lift slope of a surface piercing foil as a function of Fr. It is a good check of the measurements, if the curve at the lower end coincides with the zero Fr-approximation.
Does this help?
Uli

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### Jenny GilesPerpetual Student

Leo Lazauskas sent me a paper by Prof Molland at southampton Uni a few years ago that might clear up some points. it's a bit long to put here but you can get it from http://eprints.soton.ac.uk/43260/

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### Leo LazauskasSenior Member

Thanks, Jen.
I didn't mention that paper because there is now a more up-to-date book that
Molland, Anthony F. and Turnock, Stephen R.,
"Marine Rudders and Control Surfaces: Principles, Data,
Design and Applications".

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### AutodafeSenior Member

Aspect ratio has a large effect on the lift vs AoA curve.
Maximum lift isn't significantly effected, but the stall angle and the required deflection for a given force are, so it's useful to know when doing the detail design of the rudder linkage.

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