# Wave resistance of rudder, using Michlet

Discussion in 'Hydrodynamics and Aerodynamics' started by Remmlinger, Sep 1, 2012.

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### Remmlingerengineer

The rudder of a yacht has a low thickness ratio and operates at chord-Froude-numbers close to 1 and above. The nose radius of the foil is therefore small compared to the wave length. I assume Leo would agree that under these circumstances the wave resistance of a surface piercing foil can be computed by Michlet.
Since it is not convenient to incorporate Michlet into a VPP, a closed solution for the wave resistance is desirable. Attached are equations that approximate the wave resistance of the NACA 00xx family, as computed by Michlet. The diagram shows the comparison between Michlet results and the approximation. Since the rudder resistance is only a small part of the overall resistance, the accuracy seems sufficient to me.

Hope this is of some help for folks who work on VPPs.
Regards
Uli

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### daiquiriEngineering and Design

Interesting work. Did you keep the rudder planform (or lateral) area constant throughout the calculation? What is the definition of the "waterplane area" of a rudder?

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### DCockeySenior Member

Interesting work.

What are the side-view shapes of the rudders - rectangular, semi-elliptical, ???

The results are limited to when the rudder has zero angle of attack and is not producing side force. Turn the rudder to produce side force and the wave making resistance as well as the vicsous resistance will change. Unfortunately the non-zero side force case is considerably more difficult to predict.

I assume the rudder was modeled with out including the hull. In that case the results for a rudder under a hull would questionable at best.

Last edited: Sep 1, 2012
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### Remmlingerengineer

The planform is rectangular. Waterplane area is defined as the intersection of the free water surface and the rudder.
Up til now I used the results by Kuhn and Scragg, published in the 1993 CSYS proceedings. They also show data at yaw angles and the influence of heel. They admit, that their wave drag analysis underpredicts the resistance of the foil. I consider the Michlet-results an improvement for the zero yaw and zero heel case. The influence of yaw and heel needs still to be taken from Kuhn and Scragg.
Uli

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### DCockeySenior Member

Rudder under the hull or transom mounted?

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### DCockeySenior Member

AR = d^2 / Asv (usual definiton of aspect ratio)

Awpl is proportional to TC * Asv / AR for a given side view shape and foil section

c is proportional to (Asv / AR)^0.5 for a given side view shape

where:
AR is aspect ratio
d is the draft of the rudder
Asv is side view area
Awpl is the waterplane area
TC is the thickness ratio

The plots are for non-dimensional Cw and Fn.

Now consider what happens if side view area (Asv), side view shape, and thickness ratio (TC) are kept constant, and Aspect Ratio (AR) and speed are varied.

1) Fn is proportional to AR ^ 0.25 for a given speed. Curves of resistance vs speed will have the resistance peaks spread compared to the Cw vs Fn curves.

2) Awpl is inversely proportional to AR. Remmlinger's results show Cw increasing as AR increases. Corresponding curves of resistance vs speed would the opposite trend with resistance decreasing as AR increases.

A quick look at the Cw vs Fn charts may lead to the conclusion the increasing AR increases drag. That conclusion may well be erroneous depending at what remains constant as AR is changed.

As always, corrections are welcomed.

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### Leo LazauskasSenior Member

I'm very glad somebody finally did the calculations, Uli!

I agree that your results should be valid for the higher
Froude numbers, say > about 0.4.

Attached are 2 pages from Faltinsen's book.
The results of interest are for a thin parabolic strut.
Note that the wave resistance is non-dimensionalised
by beam squared in these plots.
It would be interesting to see how your results compare
with the struts on that basis.

Leo.

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### Remmlingerengineer

Attached is an approximation of a NACA 0005 profile, based on beam (thickness) squared. The profile is obviously different from a parabolic strut. Nevertheless, I think the approximation is acceptable. Except for the lowest aspect ratio of 0.1, which lies outside the range that I used for the determination of my equations. Lowest AR used was 0.25.

Uli

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### DCockeySenior Member

Remmlinger, how close do you think the wave resistance of an isolated strut is to the added wave resistance of a rudder under a hull?

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### Leo LazauskasSenior Member

Thanks, Uli.
I agree with you, but a comparison with a thin strut might give an idea of
the Froude numbers where Michlet estimates for the airfoil break down
because of the bluff bow region.

To simulate a strut under the hull in Michlet, you could add some waterlines
that just contain zeroes to simulate submergence below the surface. (Flotilla
allows you to specify the submergence directly, but I don't have time to do
that quite yet.)

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### DCockeySenior Member

Would adding waterlines which contain zeros only submerge the top of the rudder but not add any hull over the rudder?

As I understand Michlet and the math behind it, Michlet does not model the direct interaction between a rudder and the hull over the top of it. This interaction may be a "higher order" effect, but my guess is the interaction could be significant for estimating the increase of wave drag with the rudder with a sufficently wide hull above the rudder.

Consider two somewhat extreme hypothetical cases of hull geometry over the rudder.

Case I: A flat bottom hull which is much wider than the depth of the rudder below the hull and extends a considerable length behind the rudder. I would expect the added wave drag of the rudder in "reality" (no thin ship assumptions, etc) to be close to zero.

Case II: A hull over the rudder which is much narrower than the depth of the rudder below the hull. I would expect the added wave drag of the rudder in "reality" to be only slightly different than the wave drag of the rudder by itself in the same location.

Most boats with the rudder beneath the hull are somewhere between Case I and Case II. How many boats are close enough to Case II that the Michlet results which neglect the interaction of the hull over the rudder are "good enough"?

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### Leo LazauskasSenior Member

That's correct. It might be a useful exercise to see how the submergence
depth affects the transverse and diverging wave resistance components
of the strut without any inteference from the hull.

Of course, to simulate the effect of a hull and attached submerged strut, you
can just include the strut in the offset table and run Michlet. The program will
not give separate estimates for the wave resistance due to the hull, the strut
and any interactions; it will just give the resistance of the complete body.

Added after original post:

It is possible to investigate interference effects between the hull and the strut
by setting up the problem as a dihull. Hull as one body; submerged strut as the other.
Then Michlet will give separate estimates for each body, and the interference resistance.

Last edited: Sep 2, 2012
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### Leo LazauskasSenior Member

What is needed to incorporate wave resistance estimates in a VPP?

Would a short executable that takes as input a table of offsets and a Froude,
number, and that outputs a single number, Rw, be useful at any level?
I appreciate that you would like to be able to handle a heeled, yawed, pitched
hull, but I am not in a position to do all that in a short code.

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### Remmlingerengineer

Thanks for pointing out this obvious possibility. I am angry with myself that I did not think of that. I will come back as soon as I have results.
Uli

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### Remmlingerengineer

David, you are right, but do you have to offer a better estimate than what I proposed?

The yachts I am interested in have an old fashioned narrow stern and the situation might be close to your case II. Of course a Class 40 yacht is something else but this is not my focus.
Regards
Uli

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