One dimensionless ratio - the Reynolds number - has already been introduced above. This section discusses this and one additional common dimensionless number: the Froude num-ber. These are both commonly found in o¤shore hydromechanics.
Dimensionless ratios are used to characterize families of di¤erent conditions in an e¢cient way. Reynolds used a single number above to characterize ‡ows of di¤erent ‡uids through di¤erent diameter pipes and at di¤erent velocities. His Reynolds number conveniently combined the three independent variables to form a single one. As long as a consistent set of units is used - it makes no di¤erence if the velocity has been measured in meters per second, feet per minute, or even leagues per fortnight!
For example, by using the Reynolds number to characterize the ‡ow in a pipeline, it is possible to calibrate a venturi or ori…ce meter using a ‡ow of water even though the meter may later be used to measure a ‡ow of something more exotic like supercritical steam or even molten sodium.
A second use of dimensionless numbers is to correlate (or convert) measurements made on small physical models to equivalent values for a full-sized or prototype situation. This allows experiments to be carried out on a series of relatively inexpensive physical models rather than having to build as many full-sized ships (or other objects) instead.
Dimensionless numbers will be used for both purposes later in this chapter. The remainder of this section is concerned primarily with physical models, however.
4.3.1 Physical Model Relationships
Physical model experiments require some form of similarity between the prototype and the model:
- Geometric similarity: The model must have physical dimensions which are uni-formly proportional to those of the prototype; it must have the same shape.
- Kinematic similarity: Velocities in the model must be proportional to those in the prototype.
- Dynamic similarity: Forces and accelerations in the model must be proportional to those in the prototype.
These three similarities require that all location vectors, velocity vectors and force vectors in the coincident coordinates of the scaled model and the prototype have the same direction (argument) and that the magnitude of these vectors (modulus) must relate to each other in a constant proportion.
If ® - a number larger than 1 - is used to denote the ratio between the prototype (subscript p) quantity and the model (subscript m) quantity, then:
Item Scale Factor Relationship
With these, the scale factors for the areas S, the volumes r, the masses M and the mass moments of inertia I, are respectively:
®S = ®2L ®r= ®3L ®I = ®½¢ ®5L
®M = ®½¢ ®r= ®½¢ ®3L (4.4)
The velocity of a body or a water particle is de…ned as a displacement per unit of time, so the scale factor for the time becomes:
®T = ®L
®V (4.5)
The acceleration of a body or a water particle is de…ned as an increase of the velocity per unit of time, so the scale factor for the acceleration becomes:
®A= ®V
®T
= ®2V
®L
(4.6) According to Newton’s law, the inertia forces are de…ned as a product of mass and accel-eration, so the scale factor for the inertia forces (and the resulting pressure forces) works out to be:
Then, the relation between the forces Fpon the prototype and the forces Fmon the model is: From this, it is obvious that one can write for these forces:
Fp= C ¢1
2½pVp2¢ L2p and Fm= C ¢1
2½mVm2 ¢ L2m (4.10)
in which the constant coe¢cient, C, does not depend on the scale of the model nor on the stagnation pressure term 12½V2.
Viscous forces can be expressed using Newton’s friction model as being proportional to:
Fv ® ´V
LL2 (4.11)
while the inertia forces (from above) are proportional to:
Fi ® ½L3V2
L = ½L2V2 (4.12)
The ratio of these two forces is then - after cancelling out some terms:
Fi
Fv = ½
´V ¢ L = V ¢ L
º = Rn (4.13)
The Reynolds number is thus a measure of the ratio of these forces. Viscous forces are predominant when the Reynolds number is small.
Gravity forces are simply proportional to the material density, the acceleration of gravity and the volume:
Fg ® ½gL3 (4.14)
The above information can be used to help design physical models. One can deduce that various forces are represented to di¤erent scales. It is therefore impossible to represent all model forces with the same relative importance as in the prototype. A choice is therefore often made, instead, to maintain the ratio between the two most important forces in the prototype when a physical model is built. This can result in several scaling laws as outlined in appendix B. Two of the more important forms are discussed here, however.
4.3.2 Reynolds Scaling
Reynolds scaling is used when inertia and viscous forces are of predominant importance in the ‡ow. This is the case for pipe ‡ow (under pressure) and for wake formation behind a body in a ‡ow. Reynolds scaling requires that the Reynolds number in the model be identical to that in the prototype. Using the basic relations above:
VmLm that matter) is very close to 1.0, so that:
®T t ®2L (4.18)
This last result, that the time scale is the square of the length scale, would mean that time would pass quite rapidly in the model. Also, one can make further substitutions to discover that ®F is equal to unity; forces in the model would be just as big as in the prototype -again very impractical!
4.3.3 Froude Scaling
Gravity forces become important when a free surface of a liquid is involved. This will be true, whenever a water surface or waves are present. Since inertia and pressure forces are nearly universally important, this makes it appropriate to keep the ratio ofinertia or pressure forces
gravity forces
the same in the model as in the prototype. Scaling based upon the square root of this ra-tio is called Froude scaling, after Robert Edmund Froude (as distinct from his father William Froude from the model resistance extrapolation to full scale, treated in a following section) who has …rst used it.
Working this out from the basic information above and in a way very analogous to that used for Reynolds scaling yields:
Since it is especially di¢cult (or at least very costly) to change the acceleration of gravity for a model involving free liquid surfaces, one may safely set ®g = 1. Continuing as was done above, one …nds that:
®T =p®L (4.20)
This is a lot more convenient to handle than Reynolds scaling!
4.3.4 Numerical Example
As an example, suppose a ship with a length Ls = Lpp = 100 meter, which sails with a forward ship speed V of 20 knots in still seawater with a temperature of 15±C. Resistance and propulsion tests will be carried out in a towing tank with a 1:40 scale physical model (®L = 40). The temperature of the fresh water in the tank is 20oC.
The density and the kinematic viscosity of fresh water, salt water and dry air as a function of the temperature are listed in appendix A. Values needed here are for sea water: ½ = 1025:9 kg/m3 and º = 1:19 ¢ 10¡6 m2/s. The relevant values for fresh water are now: ½ = 998:1 kg/m3and º = 1:05 ¢ 10¡6m2/s.
The length of the ship model is:
Lm= Ls
For practical reasons, the speed of the model will be obtained using Froude scaling:
Vm= Vs
p®L = 10:29
p40 = 1:63 m/s (4.23)
As a consequence, the Froude numbers for the ship and model are obviously identical and equal to 0:329:
A consequence of this scaling is that the Reynolds numbers will di¤er:
Rns = Vs¢ Ls
In order to obtain equal Reynolds numbers, the ”model water” needs a kinematic viscosity which is 1/223 times its actual value; this liquid is not available!
Reynolds versus Froude scaling will be picked up again later in this chapter. First, however, the next sections of this chapter will discuss the ‡ow around and hydrodynamic forces on a slender cylinder in a constant current.