Cavitation Notes
ref: PNA pages 181-183
handout
p 0 = uniform_stream_total_pressure
p 1 = pressure_at_arbitrary_point
V0 = uniform_stream_velocity
V1 = velocity_at_arbitrary_point
q=
1
2
⋅ρ⋅V0 = dynamic_or_stagnation_or_ram_pressure
2
p0 +
1
2
⋅ρ⋅V0 = constant
2
p 0 := constant −
1
2
⋅ρ⋅V0
Bernoulli
2
p 1 := constant −
1
2
⋅ρ⋅V1
2
for propeller immersion, measured at radius r, minimum p 0 is obtained from ...
p a = atmosphere
p 0 = p a + ρ⋅g⋅h − ρ⋅g⋅r
h = shaft_centerline_immersion
ρ⋅g⋅r
V0 estimated as (VA^2+(ω*r)^2)0.5
if p1 => pv = vapor pressure, cavitation occurs
σ L = local_cavitation_number =
define:
accounts for minimum when r vertical up
p a + ρ⋅g⋅h − ρ⋅g⋅r − p v
ρ
2 2
2
⋅  V + ω ⋅r 

2  A
and if pressure REDUCTION / q
>= σL cavitation occurs
early criteria (Barnaby) suggested limiting average thrust per unit area to certain values (76.7
kN/m2 = 10.8 psi) for tip immersion of 11 in increasing by 0.35 psi (unit conversions don't
match up)
76.7
kN
2
= 11.124 psi
earlier PNA (1967) stated Barnaby suggested 11.25 psi
m
can calculate pressure distributions around blade so can calculate local cavitation situation
early in propeller design, want blade area to avoid cavitation (more blade area, less pressure
per unit area for given thrust)
Burrill ((1943) "Developments in Propeller Design and Manufacture for Merchant Ships",
Trans. Institute of Marine Engineers, London, Vol. 55) proposed guidance as follows:
limit thrust (coefficient) to a certain value depending on cavitation number at the 0.7 radius
T
τ c = coefficient_expressing_mean_loading_on_blades
T = thrust
τc =
ρ = water_density
AP
1
2
AP = projected_area
⋅ρ⋅VR
2
VR = relative_velocity_of_water_at_0_7_radius
can estimate projected area from
AP
AD
= 1.067 − 0.229⋅
P
D
from Taylor S & P page 91 P/D
from 0.6 to 2.0 elliptical bladed
prop, hub = 0.2 D
and as usual ...
T=
PE
( 1 − t) ⋅ V
PE
η D = quasi_propulsive_coefficient =
PD
this parameter is plotted versus
ρ := 1.0259⋅ 10
ρ = 1.99057
m
σ 0.7⋅ =
2
⋅ρ⋅ VA + ( 0.7⋅π ⋅n ⋅D)
2

(1 − t) ⋅ V
2
=
2
VA + 4.836⋅( n ⋅ D)
PE = effective_power
PE = RT⋅V
ηH =
cavitation number at 0.7*r using relative velocity at
0.7*r and pressure at CENTERLINE
m
VA =
sec
188.2 + 19.62 ⋅h
2
2
D=m
C :=
τ c + 0.3064 − 0.523⋅σ
0.0305⋅σ
0.2
0.2
n = sec
−1
units in PNA (61)
approximation SI
pv apparently ~
0.69 psi (90 degF)
C = cavitation %
σ
− 0.0174
example numbers
0.2
σ := 0.4 τ c := 0.2
C :=
(
τ c( C , σ ) := C⋅ 0.0305⋅σ
0.2
0.2
τ c as above
at 0.7 radius as
above (centerline
immersion)
for C <= 25 (%) or so
τ c + 0.3064 − 0.523⋅σ
0.0305⋅σ
or ...
h=m
in US units
2
Carmichael correlation
σ := 0.1 ,0.11 .. 2
1−t
1−w
3
2026 + 64.4⋅
σ 0.7⋅ =
RT = T⋅( 1 − t)
= η H⋅η R⋅η o
VA + 4.836 ⋅ ( n ⋅ D)

PD = delivered_power
slug
ft
p 0 + ρ⋅g⋅h − p v
1
PD⋅ η D
σ 0.7⋅
3 kg
3
=
C = 8.88
% cavitation
− 0.0174
)
− 0.0174 + 0.523⋅σ
0.2
− 0.3064
see plot below
T
this can be carried further
into ...
τc =
1
2
AP =
1
(
⋅ρ⋅VR ⋅ C⋅ 0.0305⋅σ
2
2
T
0.2
)
(
AP
⋅ρ⋅VR
2
= C⋅ 0.0305⋅σ
− 0.0174 + 0.523⋅σ
0.2
− 0.3064
0.2
)
− 0.0174 + 0.523⋅σ
0.2
− 0.3064
Ap = minimum_area_for_specified_cavitation
1
τ c(30, σ)
τ c(20, σ)
τ c(10, σ)
τ c(5, σ)
τ c(2, σ)
0.1
0.1
1
σ
Carmichael correlation valid only for C <= 25 %. 30 % shown to indicate over estimates
compared with fig 45 page 182 of PNA
for example from prop_design_notes
T := 278000lbf
derived above
VA := 14kt
−1
rpm := min
kt := 1.688
p 0 = 14.696 psi
n := 218rpm
ft
s
p v = 0.694 psi
D := 15ft
P_over_D := 0.8
σ 0.7⋅ r =
p 0 + ρ⋅g⋅h − p v
1
2
σ :=
h := 10ft
⋅ρ⋅ VA + ( 0.7⋅π⋅n⋅D)

2
2
=

188.2 + 19.62 ⋅ h
2
VA + 4.836⋅(n ⋅ D)
2
2
2
⋅ρ⋅ VA + ( 0.7⋅π⋅n⋅D) 


2
consider % cavitation in steps of 5%
D=m
h=m
m
VA = 7.203
D = 4.572 m
s
p 0 + ρ⋅g⋅h − p v
1
m
VA =
sec
188.2 + 19.62 ⋅ 3.048
σ = 0.179
2
7.203 + 4.836⋅( 3.633 ⋅ 4.572)
C := 5 , 10 .. 25
2
2

0.5
−1
h = 3.048 m n = 3.633
= 0.179 using SI
approximation
2
VR := VA + ( 0.7πn ⋅ D)

n = sec
m
VR = 37.234
s
1
s
AP( C) :=
1
2
T
(
AP
)
0.2
0.2
⋅ρ⋅VR ⋅ C⋅ 0.0305⋅σ
− 0.0174 + 0.523⋅σ
− 0.3064
2
AD
= 1.067 − 0.229⋅
P
D
assume AD ~ AE
AP( C)
AE( C) :=
1.067 − 0.229 ⋅ P_over_D
cavitation %
C =
estimated minimum EAR to avoid
AE( C) AE( C) =
=
2
D
23.045 2
m
π⋅
4
18.48
AP( C) =
20.367
5
10
15
20
25
16.333
2
m
13.632
15.425
1.404
11.698
13.236
1.126
10.245
11.592
0.94
0.806
0.706
supercavitating τc σ to the left. σ very low
cavitation % is 100
h = 3.048 m
D = 4.572 m
m
VA = 7.203
s
n = 218
m
VA := 10
s
σ 0.7⋅ r =
2
2
⋅ρ⋅ VA + ( 0.7⋅π⋅n⋅D) 


2
σ :=
(
τ c( C , σ ) := C⋅ 0.0305⋅σ
0.2
0.7⋅π⋅n⋅D = 167.573
p 0 + ρ⋅g⋅h − p v
1
2
to avoid 25% cavitation
0.7⋅π⋅n⋅D = 36.531
min
n := 1000rpm
p 0 + ρ⋅g⋅h − p v
1
1
⋅ρ⋅ VA + ( 0.7⋅π⋅n⋅D)

2
m
s
m
s
−3
2
σ = 8.8 × 10

C := 25
)
− 0.0174 + 0.523⋅σ
0.2
− 0.3064
τ c( C , σ ) = −0.243
off the scale hence
supercavitating propellers
correlation not valid but
trend is ok