3.1 Physical Components of Main Hull Resistance
3.1.6 Dimensional Analysis and Scaling
In addition to the physical approach to the separation of total resistance into identi- fiable components, the methods of dimensional analysis may also be applied [3.31], [3.32]. Physical variables and their dimensions are as follows:
Hull resistance: RT
ML
T2 (force),
Hull speed: V L
T (velocity),
Hull size: L
Fluid density: ρ M
L3 (mass/unit volume),
Fluid viscosity: μ M
LT (stress/rate of strain), Acceleration due to gravity: g M
T2 (acceleration).
According to the methods of dimensional analysis, the relationship between the quantities may be expressed as
f(RT,V,L, ρ, μ,g, αi)=0 or
RT = f(V,L, ρ, μ,g, αi), (3.26) whereαiare non-dimensional parameters of hull shape.
If geometrically similar models are considered, the method of dimensional anal- ysis yields
RT
L2V2ρ =k v V L
x
ã gL
V2 y
, (3.27)
wherev=μρ, and
RT
L2V2ρ =k[RexF ry], (3.28) or sayCT = f(Re,F r), where
CT = RT
0.5ρSV2, (3.29)
where S = hull wetted surface area and S ∝ L2 for geometrically similar hulls (geosims), and 0.5 in the denominator does not disturb the non-dimensionality, where
Re=V Lv is the Reynolds number, and F r= √V
gL is the Froude number.
Hence, for complete dynamic similarity between two hull sizes (e.g. model and ship), three conditions must apply.
1. Shape parametersαimust be the same. This implies that geometric similarity is required
2. Reynolds numbers must be the same 3. Froude numbers must be the same The last two conditions imply:
V1L1 v1
=V2L2
v2
and V1 g1L1
= V2
g2L2
i.e.
L1 g1L1 v1
= L2 g2L2 v2
or g1L31
v1
= g2L32 v2
or v1
v2
= L1
L2 3/2
(3.30) for constantg. Hence, complete similarity cannot be obtained without a large change ingor fluid viscosityν, and the shipscaling problembasically arises because complete similarity is not possible between model and ship.
The scaling problem can be simplified to some extent by keepingReorFrcon- stant. For example, with constantRe,
V1L1 v1
=V2L2 v2
, i.e.
V1
V2 = L2v1
L1v2 =L2 L1
if the fluid medium is water in both cases. If the model scale is say LL2
1 =25, then
V1
V2 =25, i.e. the model speed is 25 times faster than the ship, which is impractical.
With constantFr,
V1
g1L1 = V2 g2L2, i.e.
V1 V2 =
L1 L2 = 1
5
for 1/25th scale, then the model speed is 1/5th that of the ship, which is a practical solution.
Hence,in practice, scaling between model and ship is carried out at constantFr.
At constantFr,
Re1
Re2 =V1L1 ν1
ã ν2
V2L2 = ν2
ν1
ãL31/2 L32/2. Hence with a 1/25th scale,ReRe1
2 =1251 andRefor the ship is much larger thanRefor the model.
The scaling equation, Equation (3.28):CT =k[RexFry] can be expanded and rewritten as
CT = f1(Re)+f2(F r)+ f3(ReãF r), (3.31) that is, parts dependent on Reynolds number (broadly speaking, identifiable with viscous resistance), Froude number (broadly identifiable with wave resistance) and a remainder dependent on bothReandFr.
In practice, a physical breakdown of resistance into components is not available, and such components have to be identified from the character of thetotal resistance of the model. The methods of doing this assume thatf3(ReFr) is negligibly small, i.e. it is assumed that
CT = f1(Re)+ f2(F r). (3.32) It is noted that if gravitational effects are neglected, dimensional analysis yieldsCT
=f(Re) and if viscous effects are neglected,CT =f(Fr); hence, the breakdown as shown is not unreasonable.
However,ReandFrcan be broadly identified with viscous resistance and wave- making resistance, but since the stern wave is suppressed by boundary layer growth, the wave resistance is not independent ofRe. Also, the viscous resistance depends on the pressure distribution around the hull, which is itself dependent on wavemaking.
Hence, viscous resistance is not independent ofFr, and the dimensional breakdown of resistance generally assumed for practical scaling is not identical to the actual physical breakdown.
3.1.6.2 Froude’s Approach
The foregoing breakdown of resistance is basically that suggested by Froude working in the 1860s [3.19] and [3.33], although he was unaware of the dimensional methods discussed. He assumed that
total resistance=skin friction+ {wavemaking and pressure form}
or skin friction+‘the rest’, which he termedresiduary, i.e.
CT =CF +CR. (3.33)
The skin friction,CF, is estimated from data for a flat plate of the same length, wetted surface and velocity of model or ship.
The difference between the skin friction resistance and the total resistance gives the residuary resistance,CR. Hence, the part dependent on Reynolds number,Re, is separately determined and the model test is carried out at the corresponding velocity which gives equality of Froude number,Fr, for ship and model; hence,dynamic sim- ilarityfor the wavemaking (or residuary) resistance is obtained. Hence, if theresid- uaryresistance is considered:
RR=ρV2L2f2
V gL
.
For the model,
RRm=ρVm2L2mf2
Vm gLm
.
For the ship,
RRs=ρVs2L2sf2 Vs
gLs
,
whereρis common,gis constant andf2is the same for the ship and model. It follows that if
Vm
√Lm
= Vs
√Ls
, or Vm2 Vs2 = Lm
Ls, then
RRm
RRs =Vm2 Vs2 ãL2m
L2s =L3m L3s = m
s
which is Froude’s law; that is, when the speeds of the ship and model are in the ratio of the square root of their lengths, then the resistance due to wavemaking varies as their displacements.
The speed in the ratio of the square root of lengths is termed thecorresponding speed. In coefficient form,
CT = RT
0.5SV2 CF = RF
0.5SV2 CR= RR
0.5SV2. S∝L2,V2∝L; hence, 0.5SV2∝L3∝.
CRm
CRs =RRm
m
× s
RRs = m
s
× s
m
=1 at constant √V
gL. Hence, at constant√V
gL,CR is the same for model and ship and CRm=CRs.
Now,CTm=CFm+CRm,CTs=CFs+CRs, andCRm=CRs. Hence,
CT m−CT s=CF m−CF s
or
CT s=CT m−(CF m−CF s), (3.34) that is, change in total resistance coefficient is the change in friction coefficient, and the change in the total resistance depends on the Reynolds number,Re. In practi- cal terms,CTmis derived from a model test with a measurement of total resistance, see Section 3.1.4,CFm and CFs are derived from published skin friction data, see Section 4.3, and estimates ofCTsand ship resistanceRTscan then be made.
Practical applications of this methodology are described in Chapter 4, model- ship extrapolation, and a worked example is given in Chapter 17.
Table 3.2. Resistance of appendages, as a percentage of hull naked resistance
Item % of naked resistance
Bilge keels 2–3
Rudder up to about 5 (e.g. about 2 for a cargo vessel) but may be included in hull resistance tests
Stabiliser fins 3
Shafting and brackets, or bossings 6–7
Condenser scoops 1