section 3.4 Resistance & Propulsion
source: Woud
Chapter 3
3.4.1 Hull Resistance
2
R := c1 ⋅v s
physics
PE := R
R⋅ v s
PE = effective_power
(3.1)
(3.2)
defined
(3.3)
3
PE → c1 ⋅v s
substitution
( )
c1 := y ⋅ c0 v s
physics
(3.4)
y = f( fouling ,displacement_variations , sea_state, water_depth)
essentially time and operations
(3.5)
speed dependency of c1 .... nondimensional resistance coefficient
CT :=
R
1
defined CT = non_dimensional_total_resistance
2
⋅ρ⋅As⋅v s
2
As (ship surface area) not readily available, so use volume proportionality ...
PE
CE :=
3
3
ρ⋅Vol ⋅v s
∆ = ρ⋅Vol
since
PE
CE :=
Ro :=
ρ
CE:=
2
3
 ∆  ⋅v 3
s
ρ 
vs
g ⋅ Le
k
dimensional analysis, physics
(
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2
3
3
(3.8)
3
(3.9)
(3.11)
Fr = froude_number
defined
)
CE = f v s , ∆ , fouling , Hull_form, sea_state, water_depth
PE := R
R⋅ v s
1
(3.10)
Ro = non_dimensional_roughness
Len
PE
ρ ⋅∆ ⋅v s
Re = reynolds_number
η
Fr :=
∆
ρ⋅ 
CE = f( Re , Fr , Ro, Hull_form, external_factors)
ρ ⋅v s⋅Len
(3.7)
PE
3
3
ρ⋅Vol
Vol ⋅v s
Re :=
Vol :=
CE →
2
As ~ Vol^2/3
defined
CE = specific_resistance
2
(3.6)
3
PE → c1 ⋅v s
c1 :=
1
PE
3
vs
(3.12)
1
and from (3.8)
2
3
3
3
PE := ρ ⋅∆ ⋅v s ⋅C
CE
1
2
3
3
c1 :=
PE
1
2
3
3
c1 → ρ ⋅∆ ⋅CE
3
vs
(3.13)
shows dependency of P E on speed and displacement
3
PE := ρ ⋅∆ ⋅v s ⋅C
CE
e.g. if CE and v s are assumed constant ... a change in ∆ from nominal changes effective power
2
3
 ∆  ⋅P
PE := 
E_nom
 ∆ nom 
3.4.2 Propulsion
(3.14)
need to deliver thrust T to overcome
resistance R at speed vs
PE := R
R⋅ v s
(3.2)
N.B. I am assuming one
propeller. Woud uses k p =
number of propellers.
power delivered by propeller in water moving at v A
PT := T⋅vv A
defined
PT = thrust_power
(3.15)
Thrust deduction factor
required thrust T normally exceeds resistance R for two main reasons:
propulsor draws water along the hull and creates added resistance
conversely, the advance velocity is generally lower that the ship's speed, due to operating in the wake
t = thrust_reduction_factor = difference_between_thrust_and_resistance_relative_to_thrust
t :=
T−R
T
=>
R := (1 − t) ⋅ T
T :=
defined
R
1−t
(3.16)
"The term thrust deduction was chosen because only part of the thrust produced by the propellers is
used to overcome the pure towing resistance of the ship, the remaining part has to overcome the added
resistance: so going from thrust T to resistance R there is a deduction. The term is somewhat
misleading since starting from restance R the actual thrust T is increased." page 55
Wake fraction
propeller generally in boundary layer of ship where velocity is reduced; v A is then < vs
w :=
vs − vA
v s
w = wake_fraction
defined
(3.17)
w = difference_between_ship_speed_and_advance_velocity_in_front_of_propeller_relative_to_v s
"(Note that as a result of the suction of the propeller, the actual water velocity at the propeller entrance is much
higher than the ship's speed: the advance velocity, however is equal to the water velocity at the propeller disc
area if the propeller would not be present In other words it is the far field velocity that is felt by the propeller
located in the boundary layer of the hull.)" page 56
thus ...
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v A := (1 − w) ⋅ v s
2
Hull efficiency
redefine with these two factors the thrust power does not equal
the effective power. The ratio of effective power to thrust
power is defined as the hull efficiency.
R
R
T :=
v A := (1 − w) ⋅ v s
1−t
Propeller efficiency
Po := Q⋅ω p
since ...
η o :=
η H :=
η H :=
R vs
R⋅
ηH →
T⋅ v A
PE
E
(3.18)
PT
1−t
(3.19)
1−w
to deliver the required thrust at a certain ship's speed, power must be delivered to
the propeller as torque Q and rotational speed ωp.
defined
ω p := 2⋅π⋅nn p
PT
defined
Po
o
(3.20)
Po = open_water_power
Po := Q⋅ω
ωp
Po → 2 ⋅ Q⋅ π ⋅ n p
η o = open_water_efficiency
ηo →
1
2
⋅T⋅
vA
(3.21)
Q⋅π⋅n p
"In reality, i.e. behind the ship, the torque Mp and thus the power delivered Pp actually delivered to the propeller
are slightly different as a result of the non-uniform velocity field in front of the propeller." page 58
PNA vol II page 135 says: " Behind the hull, at the same effective speed of advance V A, the thrust T and revolutions
T
T⋅
VA
(34)
n will be associated with some different torque Q, and the efficiency behind the hull will be η B :=
2⋅π⋅n⋅Q
The ratio of behind to open efficiencies under these conditions is called the relative rotative efficiency, being given
by η B :=
T
T⋅ VA
η o :=
2⋅π⋅n⋅Q
T VA
T⋅
η R :=
2⋅π⋅n⋅Qo
ηB
ηo
ηR →
1
⋅Q
Q
o
(35)
Thus we define P p as power delivered. (per propeller)
Pp := M p ⋅ω
ωp
(3.22)
Pp → 2 ⋅M p⋅π⋅n p
and ... the ratio between open water power and actually delivered power is
Po
o
Q
η R :=
ηR →
Pp
Mp
Propulsive efficiency
η D :=
PE
PD
(3.23)
combining all these effects .. looking forward to design/evaluation at model
(open water) scale
defined
rewriting ...
η D
=
ηD =
effective_power
power_delivered
=
PE
for kp = 1
(3.24)
Pp
PE PT Po
PE PT Po
⋅
⋅
=
⋅
⋅
Pp PT Po
PT Po Pp
using definitions of efficiency from above ...
ηH =
PE
PT
=
1 − t
1−w
η o :=
PT
Po
η R :=
Po
η D := η H⋅η o ⋅η R
Pp
(3.25)
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3
η D :=
1−t
⋅η ⋅η
1 − w
o R
(3.26)