Demistifying Froude number

Froude number is frequently referred to in this forum. For many members, it appears as a very complicated concept beyond their understanding. As it applies to comparing a full size hull to a test tank model, the equivalent test speed is quite simple to calculate:

The similarity has limits, and breaks down at very large rations. For example 1:500

 

Thank You very much Gonzo,

Your topic is an opportunity to disclose my ignorance regarding the use of this ratio, when analysing Surface Piercing Hydrofoils (SPH)
I remenber to have read something about Froude=0 & Froude= infinity, it was well beyond my understanding, at this time.
In the meantime I have been struggling to make a little self-education in CFD, mostly with sail/wing applications in mind. So may be I might catch it better now, but nothing is granted.

Thanks for this topic

EK

 

Less useful for boats with planing hulls or hydrofoils.

 

Actually, it's more useful for hydrofoils than you might think. Weight tends to scale by the cube of the scale factor, while area scales as the square of the scale factor. So how do you scale a hydrofoil to have a similar operating condition? The answer is Froude scaling.

The forces and moments on a hydrofoil scale (mostly) with the square of the speed. If you want the hydrofoil to operate at the same lift coefficient and a similar angle of attack, then

Weight_model/Area_model/(Speed_model^2) = Weight_fullscale/Area_fullscale/(Speed_fullscale^2). ​

Since

Size_model = K * Size_fullscale
Area_model = K^2*Area_fullscale
Weight_model = K^3 * Weight_fullscale​

You get

K^3 * Weight_fullscale/(K^2*Area_fullscale)/(Speed_model^2) = Weight_fullscale/Area_fullscale/(Speed_fullscale^2)
K^3 /(K^2)/(Speed_model^2) = 1/(Speed_fullscale^2)
Speed_model^2 = (K)*(Speed_fullscale^2)
Speed_model = sqrt(K)*Speed_fullscale​

This is exactly what you get when you operate the model at the same Froude number. I think of the square root of speed as the "extra dimension" you're missing when you scale areas up and down, compared to scaling volume or mass up and down. When you scale the speed this way, the hydrofoils on the model can be geometrically similar to the hydrofoils of the full scale boat. The hull will also float on its lines. Both are nice features if you're interested in finding out what the takeoff performance will be

The same argument holds for planing hulls, because the dynamic lift of a planing hull also depends on the square of the speed.

 

To fully demistify this Froude number, it may be helpful to give some explanation to ist physical background without too much maths, my tentative here below :
** let's imagine a quiet bay, where the water particles are at rest, taking advantage of the sunrise
** then a cruel speed boat arrives, its a shock for them, exactly a fluid shock
** the water particles are pushed by the hull body, i.e. pressure forces are in action on the water particles, creating waves, gravity waves due to the existence of the gravity g acting on this water/air interface (the so-called free surface).
>> shock leads to a transfer of energy, here a fraction of the boat kinetic energy (dimensioned as ½ m V2) is spent into potential energy (dimensioned as m g L, where L is a dimension representative of the experience scale).
>> this fraction is unchanged whatever the scale of the boat as long as the ratio of these two forms of energy is kept constant, i.e. ½ m V^2 / m g L = constant > so the ratio V^2 / L to maintain constant, it is the relation given by Gonzo in his introduction.
>> the Froude adimensional number istself, Fn = V / √ gL , appears from this ratio : for a boat, Lwl is usually used and, so defined, the Froude number is consistent to caracterise the various geometries of waves attached to the hull, whatever the scale.
>> For example, it is well known that at Froude 0,4, there is bow wave (~ at fore perpendicular) and a rear wave (~ at aft perpendicular) and the wave drag due to this configuration (the wave drag) is becoming very high.
>> The so-called « Hull speed » given in some sailboat specifications refers exactly to Froude = 0,4. Why this data is given ? Because a normally design sailboat, providing that she is beam reaching and/or downwind with sufficient wind, can reached at least that speed, it is not a big risk for a naval architect to claim this speed reachable.

** But water particles are not only pushed but also dragged in some way by the hull body and then released further, i.e. frictional (tangential) forces are also in action due to the viscosity of the fluid.
> >The similitude law for such forces is different, should respect another adimensional number, the Reynolds number V L / ν ( ν the viscosity) less easy to introduce intuitively
> >so to do scale experiences with the presence of a free surface (when gravity waves can occured to expense energy), you should kept unchanged both the Froude and the Reynolds numbers : unfortunately, it is not possible pratically to reduce the viscosity of the fluid at low scale, so the Reynolds number cannot be kept unchanged.

** This remark to introduce the complementary aspect of Froude contribution, his assumption as regard drag estimation from a model scale experience in a towing tank when only the Froude number is respected (i.e. in partial similitude conditions) : « (to a certain extent), the wave resistance does not depend on the friction, therefore on the Reynolds, and reversely the viscous resistance does not depend on the wave field ».
>> Under this assumption, and in short :

• from the total resistance measured at model scale, one can substract by computation the friction component with using the model scale Reynolds (and the wetted surface of the model),
  
• the residuary drag, put adimensional (i.e. divided by boat displacement Mg), is so assumed to be the same at scale one for the same Froude number.
  
• the frictional drag at scale one is computed with using the scale one Reynolds and add to the scale one residuary drag

** A last word : these two adimensional numbers Froude and Reynolds are actually inside the general Navier-Stokes equation for Fluid Mechanism, appearing when writing this equation in an adimensional way (it is itself an application for a mechanical equation of an adimensional process ruled by the Vashy-Buckingham theorem). Numbers to no confused with some other adimensional parameters used by naval Architect, e.g. the Displacement Lenght Ratio DLR, which are fit for purpose to classify some boat design characteristics and/or model test results.


For more details on Similitude Law in Fluid mechanic, this powerpoint which help me writing this text :
https://uma.ensta-paris.fr/conf/tipe/2012/talks/jeups2012-Ortiz,Sabine-presentation.pdf

 

I think I understand what you're trying to say but there may be some confusion of terminology and concepts. "Shock" in fluid mechanics refers to the very abrupt increase in density and decrease in speed which occurs when a supersonic flow slows to a subsonic flow. The flow around a vessel on water is not supersonic. The analogy of a shock wave for free surface flow is a hydraulic jump which occurs in shallow water with the transition from fast super-critical flow to slow sub-critical flow. This is also not relevant for boats moving on a free surface (except possibly a boat moving very fast in shallow water).

The kinetic energy of a vessel moving at constant speed is constant, so there is not a transfer of the vessel's kinetic energy to the water. (If the vessel accelerates or deaccelerates there is a transfer of kinetic energy to the surrounding fluid in what is usually referred to as "added mass" but that occurs with a fully submerged vessel as well as a vessel on the free surface and is distinct from wave making.)

Splitting wave formation by a boat into a "bow wave" and a "stern wave" can be useful but is not exact. The wave system generated by a boat depends on the shape of the entire length of the hull, and never exactly corresponds to two discrete wave systems, one created at the bow and other created at the stern.

At at Froude number of 0.4 the wave length of the transverse waves equals the reference length, typically the waterline length, used in the Froude number. The resistance due to wave making may increase rapidly at this speed depending on the displacement of the vessel relative to it's length. The larger the displacement relative to length the more rapid the increase in wave resistance will be.

 

Froude number is useful for planing hulls as it is useful for non-planing hulls. Froude number is used in the exactly the same way to predict full size resistance and other hydrodynamic characteristics from scale model tests.

Alternative reference lengths for Froude number which can be useful with planing hulls are the Froude number based on beam (typically the beam at the chines for hard-chine hulls) and the Froude number based on the cubic root of the volumetric displacement.

 

The water "particles" start moving before the boat reaches them and moves away from the boat. The change in motion is smooth without any sudden impacts, jolts, etc.

The energy in the water waves originates from what ever is propelling the boat, travels through the boat hull and into the water in contact with the boat. The energy then flows through the water to water away from the boat. Wave energy has two components, kinetic energy due to the motion of the water, and potential/gravity energy due to the difference of the water free surface height from the undisturbed height. Waves are the result of energy swapping between motion and height.

 

I use "shock" here with just the common sense and in line with the point of view of the water particles who see the boat suddenly arriving on them. Before the boat arriving, they are at rest, after, they are in motion making waves, so they have received energy. If you are in the beach looking at to the boat passing, you can see the generated waves finally arriving and breaking on the beach, and if you have a wave energy device, you can recover a (small) bit of that energy. Ok , "a fraction of" the boat kinetic energy is not good, "a proportion of " is better.
I concede the confusion with the "shocks" that you reminded with reference to specific situations in fluid mechanism, it was not my intention to allude to them.

Here , I referred specifically to Froude 0,4, because it is a well known configuration, often quoted and easy to see by experience in situ. Of course, for other Froude values, the waves field is different in the ship referential.

 

Water starts moving before the boat reaches it, and only a very small portion of the water which makes up the waves ever comes into close proximity with the boat. The change in motion of the water as waves develop due to hull motion is smooth; no sudden impacts, jolts, etc. Nothing resembling a shock, even in the non-technical use of the word.

The energy in water waves has two components, kinetic energy due to the motion of the water, and potential/gravity energy due to the difference in free surface height from the undisturbed equilibrium height.

 

Fluids are either incompressible (liquid) or compressible (gas). Water, a liquid, is incompressible and therefore there is not an increase in density. ( at the deepest part of the ocean where pressures can exceed 15,000 psi, the density only goes up about 2%)

There has to be a transfer of the vessels kinetic energy to the water. The water in the cross sectional area has to be accelerated out of the trough that the boat creates and it is this energy that results in waves being formed. Eventually over time the energy dissipates due to viscous effects within the water.

 

Water is compressible as noted by the comment that the density of water changes with depth due to increasing pressure. Shock waves are possible in water (for example created by an underwater explosions) but are not caused by boats.

Why does there have "to be a transfer of the vessels kinetic energy to the water"? There is kinetic energy associated with water waves as I mentioned above, but that kinetic energy is not transfered from the kinetic energy of the boat generating the waves (assuming the boat is traveling at constant speed.)

 

If a force acts upon a liquid, water in this case, and the liquid begins to move, the water gains energy. In a boat three main kinetic energy control boundaries involved.
The propeller accelerates water out the back and produces kinetic energy in/on the boat. The boat moves develops energy and in order to move the trough of water out of the way, causes the water to
have a velocity and mass, with the creation of waves.

When a mass has velocity, it has energy

While there are 3 main kinetic energies involved above. One can certainly add wind resistance where the movement of the boat moves air out of the way and viscous drag
which cause vortices (angular momentum) which consumes some energy as well.

 

I agree with what you say in your post and I add, even a static mass, at a certain height, has potential energy.

 

Compressible fluids are analyzed by the Ideal Gas Law PV=nRT. You will never see a liquid dealt with by these equations, only gasses.

There is absolutely no requirements that compressible fluids have to follow the ideal gas law.

Compressibility of water can be completely ignored when analyzing flow around a boat. In other words when analyzing flow around a boat the water can be considered to be incompressible. I don't think anyone is arguring otherwise. But that does not mean water is actually incompressible and that the compressiblity of water can always be ignored.

Incompressible fluids, liquids are analyzed by the Newtonian equations.

The same equations are used to analyze the flow of air when the speed is sufficiently below the speed of sound. In those flows compressibility of the air can be ignored, just as it can be for water in similar situations. That does not mean air is incompressible.