Determining the center of lift for a deep vee hull?

If the Munk Moment is not very large, then the Sailboat Balance is:

Lateral Aero Force (A) coincides in longitudinal position with the Lateral Hydro Force (Hy) of the centerboard

and

the Forward Force (F) multiplied by its Lever Arm (h x sin Heel) equals the Rudder Blade Force (R) multiplied by its Lever Arm (r)

and

A = Hy + R

h = Wind Force vertical position - Lateral Water Force vertical position

 

Balance seen in a different way

Roman Balance

R: Lateral rudder force
K: Lateral force of daggerboard
M1: Munk due to Leeway
M2: Munk due to Heel
M3: Munk due to Pitch

Steelyard balance - Wikipedia https://en.m.wikipedia.org/wiki/Steelyard_balance

 

How to Balance a Sailboat

Search Boat Design Net for the concept of Center of Flotation (CF) and learn how to calculate its Longitudinal position (LCF)

Search Boat Design Net for the concept of Center of Buoyancy (CB) and learn how to calculate its Longitudinal position (LCB)

Fit these two Centers and place the Center of Gravity (CG) of the Sailboat a little bit aft, for example 3% of the Waterline, to take into account part of the Sails Force that sinks the Bow.

Obtain a good estimate (Davidson CE) of Aero Dynamic Center of the sails and place the Hydro Dynamic Center (H) of the centerboard more or less below, thus fitting all the centers as shown in the drawing.

 

Make a reasonable estimate of Munk with 3-6 degrees of Leeway.

Size the rudder blade size until the ('no move, no touch, no look' to the rudder) Lateral Rudder Force overrides Munk.

If the rudder blade goes behind the daggerboard (and the daggerboard diverts water to the rudder) then we calculate with half (0.5) rudder

 

The Munk Estimate is a Vector that we place right at the bow.

R x r = m x M

CL3D: Lift Coefficient in 3D
q: Dynamic Pressure = 0.5 x 1025 kg x Velocity^2 (m/s)
RA: Rudder Area (m^2)
f: 0.5-1 (1 if two rudder, 0.5 if One rudder)

Cl2d: Lift Coefficient in 2 dimensions: 0.6 (6 Angle of Attack = Yaw/Leeway)

AR: Aspect Ratio

 

Davidson CE

Davidson Laboratory https://www.stevens.edu/davidson-laboratory

1.7 x Jib/Genoa Area
1 x Main
0.5 x Mizzen

Jib/Genoa Center = geometric Center
Main Center = 0.4 P, 1/2

 

The idea is ...

... If the rudder blade without moving the rudder is able to take over Munk, then by moving the rudder blade a few degrees it can take over the sail force.

Keep in mind that Balance is important whether sailing upwind or downwind.

 

Fine adjustment of daggerboard position

Depending on one's taste, wishes and concerns ... the daggerboard position fine adjustment can be:

(1) geometrical CE on the vertical of the daggerboard leading edge
(2) Davidson CE on the vertical of the hydrodynamic center of the daggerboard (0.20 x chord)
(3) Davidson CE on the vertical of the leading edge

But the big issue is the sailboat Balanced without moving the rudder, or in other words: CLR aft of LCG without moving the rudder, similar to Longitudinal Balance in a passenger or cargo aircraft:

The Munk Estimate is a Vector that we place right at the bow.

if R x r < m x M: unstable Equilibrium
if R x r = m x M: Neutral Equilibrium
if R x r > m x M: Stable Equilibrium

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What happens if the Munk Moment is a huge quantity

Well, you have to make a workaround: move the Sails forward so that the Sail Lateral Force balances the Munk Moment. The sailboat is balanced to sail upwind but it is unstable downwind, and with big waves it is uncomfortable and could even be dangerous.

 

The worst case

The Satanita - Wikipedia https://en.m.wikipedia.org/wiki/The_Satanita

In the worst case the sailboat is uncontrollable even Upwind because of the sum of the Hydro Force of the Hull (M1+2+3: Leeway/Yaw + Heel/Roll + Pitch) and the Air Force (F) of the Sails and its lever arm.

 

In the case of for example a Tiki 21 we could treat the Hull as a very low Aspect Ratio Wing.

We make a Rectangle of say 75% (?) of the Waterline, and the Aspect Ratio is the draft (h) squared divided by the Area.

And place the resulting Force at say 20% (?) of the waterline.

But, there are two hulls, and I don't know if both should be counted.

These questions cannot be thought in the air without empirical data, it is necessary to advance from both extremes: on the one hand, theory and, on the other hand, experience and empirical data.

 

(Hörner, Fluid Dynamic Lift, XVII: Low Aspect Wind)

I don't know, it would be necessary to review this tangled question; but the sharp edge of the Tiki 21 leads me to think that it would be better to consider it a "low Aspect Wing" instead of facing the question as a "slender body".

 

To summarize: in this issue we have five tools in the toolbox:

A) "very Low Aspect Ratio Wing"
B) "slender body"
C) Munk_1 + Munk_2 = professor Nomoto + professor Keuning
D) Munk_3: "cross Flow", Hörner
E) J.W. Slooff, "Aero-Hydromechanics of the Keel Yacht"

 

a Rectangle 0.22 x 4.2 m
AR, Aspect Ratio = 0.052
Angle of Attack (AoA) = 10
Velocity: 2.5 m/s

74 Newtons (as "Low Aspect Ratio Wing")

42 Newtons (as "Slender Body")

Center of Lift = more or less at the Bow

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If the hull had a draft of 30 centimeters, the Force estimate would be 136 Newtons as a Wing with sharp edges (and 79 Newtons as a more or less cylindrical body with rounded edges).

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Another possibility would be a "Delta Wing" formed with the hull up to, for example, half the length: this would result in a force of 138 Newtons and the Center of Lift would be 1.8 (= 0.60 x 3 m) meters from the bow, which is 30% of the length.

Anyway, a good entertainment for a rainy afternoon.

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All in all, for monohulls i think we can make more or less convincing estimates. For catamarans without centerboard, however, i honestly don't understand it.

 

The wild card that is hidden in the sleeve is the so-called non-linear component of the Lift also called "cross Flow", but I honestly do not know if it is justified in this case.

Assuming an Angle of Attack (AoA) of 15 degrees, a Coefficient of 1.8 corresponding to a sharp edge, 30 centimeters of draft (h) and 6 meters of length (LWL) results in a Force of 268 Newtons at the Bow.

That balanced by the Rudder perhaps this could fit with reality. I don't know.



(Hörner, Fluid Dynamic Lift)

Yes, here in this dark cave of the nonlinear component must be the mystery of why these hulls produce so much lift. But I don't know how it is, it's a bit annoying not knowing how to fit the forces of a fxxx Tiki 21.

 

Yes, the enigmatic case of these catamarans without centerboard could perhaps be modeled as follows

It is a very low Aspect Ratio Wing associated therefore with a large Angle of Attack (AoA)

On the other hand, with the heel, a larger hull draft (h) must be taken into account.

A Vortex is created along the sharp edge.

The Force is larger and more backward: 30% (?) LWL to say something in the absence of empirical confirmation.