Naval Architecture

This page is for establish some principles of naval architecture for peoples that want initiate in this beautiful art.

When the boat sail some physical phenomena occur:

Wave formation

One very important is wave formation. The boat do waves that go with the hull. The principals waves are formed by bow and stern, are transverse and longitudinal waves.

These systems interact together. When the bow system crest of wave coincide with the stern system wave trough we have a benefit and if crests or  trough of waves coincide we have a prejudice. The benefit and prejudice we can see on the graph of resistance:

Jegue D'Água

Jegue D’Água

In this graph we have for Y axis the propulsive force (kN) to move the boat at the corresponding velocity (m/s)in X axis.

Also we can see the Y axis force as resistance force acting on the hull at the correspondent velocity.

You see that the magenta line, the total resistance, the white and green lines, wave transverse  and total wave resistances are not continuous, they have in some points an quick  increase in resistance, the above prejudice, and in others points they are nearly horizontal or decreasing, with an increase in velocity without an appreciable increase in resistance, the above benefit.

In some boats these humps are lesser like in Xique-Xique:



The Cp – prismatic coefficient is the manner that we take in count this phenomenon in design. In reality we do not have a direct connection with Cp and this phenomenon but in some researches, this coefficient is attached to it. Cp reflects the distribution of areas along the immersed length of the boat. If Cp is lesser signifies that the sectionals areas is concentrated in one point, the point with maximum transverse immersed area,and bigger Cp signifies that the sectionals areas is more distributed  along immersed length.

The waves formed is function of this sectional areas distribution. Sectionals areas are linked with immersed volume distribution: sectional areas concentrated signifies concentrated volume (low Cp), sectionals areas distributed –  distributed volume (higher Cp).

The Sectional area Curve is the manner that we have to visualize this distribution. See the Sectional Area Curve for Xique-Xique and Jegue D’Água:

Xique-Xique Sectional Areas

The curve showed represents each immersed cross sectional area in each station.  This curve have two great properties: the area under it has the same numerical value that immersed volume. So, when we want know the immersed volume is only do the Sectional Areas Curve and calculate the area under it. The other property is that the longitudinal center of gravity of the area under the curve is in the same point that the boat longitudinal center of buoyancy . The Xique-Xique Cp is 0.55.

Jegue D’Água Sectional Areas

Here we have Jegue Sectional Areas Curve, how both are IOM boats they have the same immersed volume, that is 0.004 m³ and by this reason both areas under  curves are numerically  identical but they have different shape. The Jegue area seems more fat signalizing that the volume is more distributed in  length and that the maximum sectional area is lesser then that of Xique-Xique. The Jegue Cp is 0.574. Unhappy this drawing is from Delftship software and the sectional areas values are not show correctly because the little values of sectionals areas.

The wave formation by the hull is associated with Froude number.

William Froude (1810 – 1979) was an English engineer, hydrodynamicist and naval architect.

Fn = V/√g.L

V – hull velocity

g – gravitational acceleration

L – design waterline length

Two boats with same Fn have same wave formation.

A simplified Fn normally is used since g is constant and it’s name is Velocity Ratio  Vr = V/√L

The wave formation and their velocity ratio (metric units) :

Wave formation according Fn

Figure from the book : Dessiner son voilier – Claudio Diolaiti

We can see that according to Vr we have some  crests along the hull. With Vr =0.9  we have three crests and with Vr = 1.2 we have two crests, one on the fore end and the other in the after end. Vr = 1.2 is the upper limit in wave formation for displacements hulls. For Vr > 1.2 the hull form must have flat bottom areas like the motor boat. For streamlined hull the maximum velocity is V = 1.34*√ L  (V in knot; L in feet)

To take the wave formation in consideration when doing the hull design we need select a Cp appropriate for design velocity.

Skene’s Elements of Yacht Design – book from Francis Kinney suggests Cp values:

V/√L            Cp

1.0           0.52
1.1           0.54
1.2           0.58
1.3           0.62

(V – knots; L – ft)

We can mathematically define Cp as:

Cp = Immersed volume / Am * L

Am – maximum immersed cross sectional area for that volume

L – the waterline length corresponding to that immersed volume

We also can see the Cp as a volume ratio between immersed volume and a prism do by the cross maximum sectional area having the length L.

I prefer see Cp as a volume distribution along L visualizing a curve of cross sectional areas, less Cp indicating more concentrated areas distribution, better for medium velocities, and higher Cp indicating more distributed areas, better for maximum limit velocity.


Figure from the book : Dessiner son voilier – Claudio Diolaiti

Another physical phenomenon that occur is Drag.


Drag or resistance of an object in a fluid environment such as air or water is composed by skin friction and form drag.

Skin friction arises from the friction of the fluid against the “skin” of the object that is moving through it. Skin friction arises from the interaction between the fluid and the skin of the body, and is directly related to the wetted surface, the area of the surface of the body that is in contact with the fluid. (Wikipedia)

Form drag, profile drag, or pressure drag, arises because of the form of the object. The general size and shape of the body is the most important factor in form drag – bodies with a larger apparent cross-section will have a higher drag than thinner bodies. Sleek designs, or designs that are streamlined and change cross-sectional area gradually are also critical for achieving minimum form drag.

Profile drag (Pxp): depends on the longitudinal section of the body. A diligent choice of body profile is more than essential for low drag coefficient. Streamlines should be continuous and separation of the boundary layer with its attendant vortices should be avoided.  (Wikipedia)

If the object is a wing or a keel or some other that produces lift then we have more two  other drag forms, induced drag and interference drag.

Induced drag is the drag caused by Lift force when is not in perpendicular direction of the movement. Is the component of the Lift force in the movement  direction.

Interference drag is caused by intersecting bodies like wing and fuselage, keel and hull.Particular geometric characteristics on aircraft often show how designers have dealt with the issue of interference drag. A prime example is the wing-body fairing which smooths the sharp angle between the wing and the fuselage. (Wikipedia)

Drag equation

Fd = 1/2*ρ*V²*Cd*A

Fd – Drag force

ρ – mass density of the fluid

V – fluid velocity

Cd – drag coefficient

A – reference area

Drag coefficient:

Cd = Fd/(1/2*ρ*V²*A)

Cd is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water.

The reference area change in definition for differents bodies.

The reference area for a car is the frontal area of the vehicle for a boat  hull is the wetted area, for wing is planform area. By this reason compare a car drag coefficient with a wing drag coefficient do not  make much sense.

Some drag coefficients:

Sphere – 0.47

Long cylinder – 0.82

Cone – 0.50

Streamlined body – 0.04

However in hull boats we study the resistance as a sum of two principals resistances:

hull resistance = skin friction resistance + residuary resistance

The residuary resistance is a sum of various types of resistance.

This approach is due to William Froude that establish the process to achieve the hull resistance knowing the total model (in reduced scale) resistance.

The skin frictional resistance formula is:

Rf = 1/2*ρ*V²*S*Cf

ρ – density of water

V – hull velocity

S – wetted area

Cf – friction coeefficient

Cf = 0.0075/(log(Rn) -2)²

Rn = (V*L)/ν

L – body length

ν – kinematic viscosity

For boat hulls usually we use for L = 0.7*Lwl, for keel and rudder L is average chord length.

Normally for residuary calculations we use some formulas obtained from regression analyses do in tank tests results like Delftship Systematics Hull Series or some theoretical approach like do in Michlet software.

But these calculations do not cover all boats and when use this formulas and softwares we need see the range of formulas and softwares validity.

The manner to consider the frictional resistance in boat design is minimize wetted area.

Another preoccupation that we need have in mind is avoid boundary layer separation. Boundary layer separation increase drag and the solution is have streamlined lines without chines or cuts.

Froude method to achieve boat resistance knowing model resistance is very simple.

With model total resistance for some model velocity we subtract the model friction resistance for this velocity by the above formulas. So we have the residuary resistance for that velocity.

Model residuary resistance = Total model resistance – Friction model resistance

With model residuary resistance we can have the total boat resistance using the model scale:

Boat residuary resistance = model residuary resistance / scale ³

If model scale is 1/10 = o.1  the boat residuary resistance will be one thousand more.

Naturally if the water tank testing conditions of temperature and salinity is different from the boat water conditions we need do corrections.

This resistance is for same model and boat Fn and we can achieve the boat velocity for this resistance.

Fn model = Fn boat   –>  Vmodel / √L model = V boat / √L boat

V boat = (V model/√L model) * √L boat or

V boat = Vr model * √L boat

With boat velocity for this resistance we can achieve the boat frictional resistance with the formulas above and adding this  result to the boat residuary resistance we have the total boat resistance for that boat velocity.

Despite the unknowing drag theory is not so important for atual boat hull resistance calculations it is good for understand the phenomenons involved in the keel and to do hydrofoil boats.

The keel in sailboat is for introduces an force opponent to the lateral sail force that tends the boat goes in the wind direction. With this keel force we can go against wind.

The force in the keel is obtained by water passage along the keel foil shape. This passage along the foil generate lift but also introduce all types of drag resistance.

The great understanding importance is know that with Reynolds number variation we need have a optimum foil form.

What will be a IOM Reynolds number?

V = 2.32 knot (V Max) = 4.2 km/h = 1.17 m/s

L = 1 m

Re = 103 *1.17 * 1 / 1.03 * 10-3 = 1.13 * 106

Re IOM = 1.13*10⁶

For a IOM keel

Considering a 0.1 m mean chord:

Re = 10³*1.17*0.1/1.03*10-³

Re IOM keel = 1.13*10⁵

And for  RG 65 ?

V máx = 1,34  *  √ L (m) / 0.3048)

V = 1.34  *  √ 0.65 / 0.3048)

Vmax = 1.96 knot  =  3.5 km/h  = 0.98 m/s

Re = 103 *0.98 * o.65/ 1.03 * 10-3 =  6.18 * 105

Re RG 65 = 6.18*10⁵

For a RG 65 keel

Considering a 0.065 m mean chord:

Re = 103 *0.98 * o.065/ 1.03 * 10-3 =  6.18 * 104

Re RG 65 keel = 6.18*10⁴

And for a boat with 10 m Lwl?

L = 10 m

V max = 7.7 knot = 13.86 km/h = 3.85  m/s

Re = 103 * 3.85 *0.7* 10 / 1.03 * 10-3 = 2.618 * 107

Re boat = 2.618*10⁷

Reynolds number is connected with the flow description.

A flow can be considered as laminar or turbulent, see the video:

For a flow over a flat plate the transition between laminar and turbulent come with Reynolds = 5*10⁵

The keel

Graham Bantock say us that for RC sailboats we need use a 6% camber for keel foil instead a 15 % normal real keel boats.

The motive for this information is just the differents Reynolds number from our models to the real boats.

The IOM keel boat has a Reynolds number 1.13*10⁵ and the RG 65 has 6.18*10⁴.

These Reynolds number (Rn) are  much smaller than real keel boats and by this reason ours foils need has a lesser chord/length ratio, like 6%, that are more adequate to these Rn.

So we can use the NACA 63-006 or NACA 0006

NACA 63-006 ordinates

Nose radius 0.287 % of chord

X % of chord                  Y % of chord (+ or -)

0,000                                            0,000
0,500                                             0,503
0,750                                              0,609
1,250                                               0,771
2,500                                              1,057
5,000                                              1,462
7,500                                               1,766
10,000                                            2,010
15,000                                            2,386
20,000                                           2,656
25,000                                            2,841
30,000                                            2,954
35,000                                            3,000
40,000                                            2,971
45,000                                            2,877
50,000                                            2,723
55,000                                            2,517
60,000                                            2,267
65,000                                             1,982
70,000                                             1,670
75,000                                              1,342
80,000                                             1,008
85,000                                              0,683
90,000                                              0,383
95,000                                              0,138
100,000                                            0,000

NACA 0006 ordinates

Nose radius 0.4 % of chord

X % of chord                     Y % of chord (+ or -)

0,000                                                 0,000
0,500                                                  0,947
1,250                                                    1,307
2,500                                                   1,777
5,000                                                   2,100
7,500                                                    2,341
10,000                                                 2,673
13,000                                                  2,869
20,000                                                 2,971
30,000                                                 3,001
40,000                                                 2,902
50,000                                                 2,647
60,000                                                 2,282
70,000                                                  1,832
80,000                                                  1,312
90,000                                                  0,724
95,000                                                  0,403
100,000                                                0,063

NACA 0006

NACA 0006

The symmetric keel foil do lift only when the flow pass with a angle in relation to its chord, this angle is called angle of attack.

According with angle of attack the attachment of the flow in the foil is modified. See the figure below:


Initially the lift increase with more angle of attack but when the angle pass the point of maximum lift, the lift goes diminishing and comes to zero according the increase of angle of attack.

Increasing the angle the flow begins to take off the foil and reach a point that the foil do not make more lift .

R is the resultant of the forces caused by the passage of fluid on the foil, Lift is the component of R perpendicular to the movement and drag is the component parallel to the movement. (Genericaly R can be also called Lift )

So, lift is against the sail lateral wind force and is the cause that the sailboat can go against the wind. The keel has two functions, generate lift and carry the ballast.

This drag is called induced drag.

Similarly with Drag Formula,

Lift formula

L = 1/2*ρ*V²*Cl*A

A – planform area

Lift coefficient

Cl = L / (1/2*ρ*V²*A)





Mauricio Dantas is a Brazilian living in Texas – USA which has helped a lot this blog. Thanks to him, Eric Rosembaun allowed us to publish several articles on the blog. Some time ago he sent me some pictures of a spectacular and very simple tool to find the longitudinal center of gravity of RC sailboats. It’s very simple, see the photos:

To use it just put the boat on top, taking care that the design waterline (you need drawing it on the hull) should be horizontal with the boat balanced on top of two rods. There, the distance from the stern to the rods is the longitudinal center of gravity of the mass of the boat, referred to the stern. The center of gravity should be exactly in the same position of the longitudinal Center of Buoyancy,  to meet design conditions.

Maurice told me it is used to find model air planes center of gravity.

Maurice sails (and well, won some races  in Texas) a class not known in Brazil , the Victoria, as well as IOM and RG 65.

This tool is so important.

Thanks Mauricio!






14 Responses to Naval Architecture

  1. Johann Adolf Wegmann Morovic says:

    Dear Gentlemen, appreciate very much all you did,I am retired,so bored that I decided to design a Yacht Schooner,If you can I need help:1.-Fundamental Equation (cubic in the Beam),
    2,-Analytical math. method for Curve cross sectional areas(Cp).3.-Reccomend me please a
    FEA code for composites hull calculation,epoxy and fibre reinforcement,4.-Best Standards
    for Yacht building as NV,ABS, so on,
    5,.F.H. Chapman shape vessels method, sadly I lost it,as far I see is simple and easy.Thanks
    Johann Wegmann, I hope to reach Fidji and Tahiti next year. Best Regrds Johann

  2. Dan Newland says:

    Hi Fred

    I sail big boats and also the T37 RC sailboat from Tippeecanoe Boats, I have also built a lot of boats, keels, centerboards and rudder professionally. The T37 is a built it yourself kit from precut plywood, the mast is carbon fiber tube as are the booms, the sail is 3/4 oz ripstop Nylon.

    I was discussing design features with another builder and since we make the keel and rudder from two pieces of 1/8″ plywood, the section shape is up to the builder. He liked squaring the trailing edge rather than a point. I do that for my big boat counterparts but not the RC boats figuring at their Reynolds numbers, the sharper the better. What do you think?

    He also did not like fillets at the keel/hull nor at the keel/bulb. I use them to reduce interference drag and they are pretty small, around .188″ or so radius. Thoughts?

    And finally for a boat like this with a pretty low aspect keel, what do you think is the ideal section shape and thickness? The keel cannot be less than .25″ thick so with the keel base at 6″, that’s 4.17% thick. The keels are typically flat sided with a rounded leading edge and a sharp trailing edge but as you can imagine, there is a tremendous variation from one foil to another. And we can go thicker than .25″. I went to an NACA 0008 section.


    Dan Newland

    • Fred Schmidt says:

      Hi Dan
      I do not have much real experience in these matters, I always try to study it.
      In RC I have seen keels with constant rectangular section with the edges rounded and walk well.
      The same is true in rustic or small real boats.
      It is very difficult to evaluate, if not in towing tanks, measuring all forces involved.
      I once read an article by Australia designer, winner of the America’s Cup, about his keel.
      If you look at the keel of the Australia you see fleeing the normal standards. The detail, to me most strange is that the bottom of the keel is well wider than the top.
      And he said it was the keel that gave better performance to the boat.
      My theory is that the keel and the hull must have a symbiosis, as Dave Hollom said in a letter to Bob Wells.
      Upright and when heeled.
      See, here on the Blog in Heeled Hull Form.
      That being so each boat is a case where the pattern might work better or not.
      In order to know how best keel to a boat should be done several tests with different shapes in a towing tank.
      Otherwise, any choice is a lottery.
      Good in RC is that being cheap to search for a better keel is easier, but always the best will depend on the hull shape, both vertically as heeled.
      Each shell may have its better keel.
      They live in symbiosis.

  3. Jan Theron says:

    Thanks, Fred. I am in the early stages of building a wooden 6ft model sailboat of my own design and found your website very informative. Keep up the good work!

    • Fred Schmidt says:

      If you need to exchange ideas about the design or construction I will be here.

  4. Jan Theron says:

    Your comment on the keel where you quote Graham Bantock surely refers to thickness, not camber? Camber implies an asymmetric airfoil…

  5. Xen says:

    you can by editing the size attributes in the img tag on the html. god spee

    • Fred Schmidt says:

      Thanks for the information. I hope that one day I will have time to change, but the news we will use.



  6. Pingback: Footy design – Freeship II | IOM, RG 65 & Footy

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  8. Don Case says:

    Hi Fred
    Would it be possible to increase the size of the graphs? At the moment I’m just looking at the first two(hull resistance). When I click on them they come up the same size and I can’t read them.

    • Fred Schmidt says:

      Hi Don
      Unfortunately I could not, I think that is the wordpress definition, but I am sending the document by e-mail. Is perfect.
      If anyone else wants to have the original just ask.



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