Hull Speed & Froude Number

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Hull Speed & Froude Number

Hull speed, sometimes referred to as displacement speed, is a common rule of thumb based on the speed/length ratio of a displacement hull, used to provide the approximate speed potential (i.e. maximum speed possible) of the hull. It is the speed of a deep water wave whose wavelength is equal to the waterline length of the hull. The most commonly used hull speed constant is the wave propagation speed for the hull length, and it serves well for traditional sailing hulls. In English units, it is expressed as:

$\mbox{knots} \approx 1.34 \times \sqrt{l \mbox{ft}}$

Or, in metric units:

$\mbox{knots} \approx 2.43 \times \sqrt{l \mbox{m}}$

where "l" is the length of the waterline (LWL) in feet or meters.

Hull speed is typically not a term used by naval architects (they use, instead, a specific speed/length ratio for the hull in question) but is often used by amateur builders of displacement hulls, such as small sailboats and rowboats.

The concept has to do with the effect of drag from the water on the hull. With all else being equal, a longer boat will have a higher hull speed. In yacht racing this is demonstrated by looking at handicap ratings such as PHRF; generally speaking longer boats have higher handicap, although there are other factors.

Hull Speed

1 History
2 Use
3 Examples
4 References

Froude Number

1 Origins
2 Dimensionless form
3 Densimetric Froude Number
4 Uses
5 See also
6 External links

History

The quantification of the speed/length ratio is generally credited to William Froude, who used a series of scale models to measure the resistance each model offered when towed at a given speed. Froude's observations led him to derive the Froude number, which allows experimental observations performed on scale models to be applied to full-scale vessels.

The speed-to-length ratio is traditionally expressed in knots of speed (V) and feet of waterline length (LWL):

$\textrm{Speed Length Ratio} =\frac {V}{\sqrt \textrm{LWL} }$

Use

The speed/length ratio is strictly only useful when comparing different scalings of otherwise identical hulls whose drag is dominated by wave drag. However, for many hulls, a generic speed/length ratio will provide a good general estimate of the speed potential of the hull when it is operating in displacement mode. This is commonly called the hull speed, and this term is commonly found in the boating community and among amateur builders, though it is not used by naval architects or engineers.

The hull speed limit does not readily apply to certain types of hull which are not primarily limited by wave drag. Examples of these craft are:

Very long, narrow hulls such as rowing shells, flatwater racing canoes and kayaks, and multihulls such as catamarans and proas. In these hulls, skin drag is often far greater at the normal operating speeds than the wave drag.
Boats which operate in a semi-displacement mode where the hull shape provides some lift. In these hulls, the lift reduces the displacement, providing a reduction in the quantity of water moved and a corresponding reduction in wave drag.
Small, highly powered boats such as sailing dinghies and personal watercraft, which can easily plane. These hulls quickly and easily surmount their bow waves, and rely entirely on dynamic lift when planing.

Some boats, such as the proa, have both a narrow hull and are capable of operating in a semi-displacement or planing mode. Very large vessels, such as supertankers, are also generally limited by skin drag. This is not due to any special property of the hull, but rather to a low power to displacement ratio, which keeps the vessels operating at speed/length ratios well below the hull speed.

The most commonly used hull speed constant is the wave propagation speed for the hull length, and it serves well for traditional sailing hulls. In English units, it is expressed as:

$\mbox{knots} \approx 1.34 \times \sqrt{l \mbox{ft}}$

Or, in metric units:

$\mbox{knots} \approx 2.43 \times \sqrt{l \mbox{m}}$

In reality, speed/length ratios (in English units) of real hulls vary from as low as 1.18 for blunt hulls such as barges to over 1.42 for long, thin hulls. Also, since hull speed takes into account only the wave making resistance, large hulls (over 200 ft or 60 m) will be more limited by other forms of drag[1].

Examples

Displacement hulls (for example those not planing on the surface of the water) have a maximum speed beyond which they tend to 'dig in', with their bows high and sterns low, and become increasingly wasteful of propulsive power. This is known as their hull speed and it depends mainly upon waterline length. For various displacement boat hulls the following table relates waterline lengths to hull speeds and so gives some examples of usage of the unit knots..

Hull speeds
Waterline length (ft)	Waterline length (m)	Hull speed (kn)	Type of Boat
10	3.0	4.4	Dinghy
18	5.5	5.9	Small fishing or pleasure boat
28	8.5	7.3	Small yacht
36	11.0	8.2	Family yacht
50	15	9.8	Small commercial fishing boat or ferry
200	61	20	Small commercial ship
400	122	28	Typical cruise or container ship

References

A simple explanation of hull speed as it relates to heavy and light displacement hulls
Hull speed chart for use with rowed boats
On the subject of high speed monohulls, Daniel Savitsky, Professor Emeritus, Davidson Laboratory, Stevens Institute of Technology
Low Drag Racing Shells

Froude Number

The Froude number is a dimensionless number comparing inertial and gravitational forces. It may be used to quantify the resistance of an object moving through water, and compare objects of different sizes. Named after William Froude, the Froude number is based on his speed/length ratio.

1 Origins
2 Dimensionless form
3 Densimetric Froude Number
4 Uses
5 See also
6 External links

Origins

The hulls of swan (above) and raven (below). A sequence of 3, 6 and 12 (shown in the picture) foot scale models were constructed by Froude and used in towing trials to establish resistance and scaling laws.

The quantification of the resistance of floating objects is generally credited to Froude, who used a series of scale models to measure the resistance each model offered when towed at a given speed. Froude's observations led him to derive the Wave-Line Theory which first described the resistance of a shape as being a function of the waves caused by varying pressures around the hull as it moves through the water. The Naval Constructor Ferdinand Reech had put forward the concept in 1832 but had not demonstrated how it could be applied to practical problems in ship resistance. Speed/length ratio was originally defined by Froude in his Law of Comparison in 1868 in dimensional terms as:

$\mathrm{Speed Length Ratio} =\frac {V}{\sqrt \mathrm{LWL} }$

where:

v = speed in knots

LWL = length of waterline in feet

The term was converted into non-dimensional terms and was given Froude's name in recognition of the work he did. It is sometimes called Reech-Froude number after Ferdinand Reech.

Dimensionless form

The dimensionless Froude number is defined as

$\mathrm{Fr} = \frac{V}{c}$

where $V$ is an average velocity , and $c$ is the propagation velocity of a shallow water wave. The Froude number is thus the hydrodynamic equivalent to the Mach number.

The wave velocity, $c$ , is equal to the square root of gravitational acceleration times cross-sectional area divided by free-surface width:

$c = \sqrt{g \frac{A}{B}}$

and so the Froude number can often be simplified to

$\mathrm{Fr} = \frac{V}{\sqrt{gd}}$

where $d$ is a depth or length scale.

It is also common in fluid mechanics to use the form

$\mathrm{Fr}=\frac{V^2}{gd}$

which is simply the square of the previous version.^[1] This form is proportional to inertia rather than velocity, and is the reciprocal of the Richardson number.

In the study of stirred tanks, the Froude number governs the formation of surface vortices. Since the impeller tip velocity is proportional to $N d$ , where $N$ is the impeller speed (rev/s) and $d$ is the impeller diameter, the Froude number then takes the following form:

$\mathrm{Fr}=\frac{N^2d}{g}$

Densimetric Froude Number

When used in the context of the Boussinesq approximation the densimetric Froude number is defined as

$\mathrm{Fr}=\frac{u}{\sqrt{g' h}}$

where $g'$ is the reduced gravity:

$g' = g{\rho_1-\rho_2\over {\rho}}$

The densimetric Froude number is usually preferred by modellers who wish to nondimensionalize a speed preference to the Richardson number which is more commonly encountered when considering stratified shear layers. For example, the leading edge of a gravity current moves with a front Froude number of about unity.

Uses

The Froude number is used to compare the wave making resistance between bodies of various sizes and shapes.

In free-surface flow, the nature of the flow (supercritical or subcritical) depends upon whether the Froude number is greater than or less than unity.