UFPA Ship Hydrodynamics Professor P A Wilson University of Southampton September 15, 2015 Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 1 / 27
Dec 05, 2015
UFPA Ship Hydrodynamics
Professor P A Wilson
University of Southampton
September 15, 2015
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 1 / 27
General Modelling & Scaling Laws
1 Dimensionless numbers
2 Similarity requirements
3 Derivation of dimensionless numbers used in testing
4 Froude scaling
5 Hydroelasticity
6 Cavitation number
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 2 / 27
Dimensionless numbers
1 Without dimensionless numbers, experimental progress in fluid
mechanics would have been almost nil;
2 Due to the beauty of dimensionless numbers Cf of a flat smooth plateis a function of Re only
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 3 / 27
Applications
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 4 / 27
Types of similarity
What are the similarity requirements for model testing?
1 Geometrical similarity
2 Kinematic similarity
3 Dynamic similarity
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 5 / 27
Geometrical Similarity
1 The model and full scale structures must have the same shape−→ All linear dimensions must have the same scale ratio: λ = Lf /Lm
2 This applies also to:1 The environment surrounding the model and ship2 Elastic deformations of the model and ship
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 6 / 27
1 Similarity of velocities: −→ The flow and model(s) will havegeometrically similar motions in model and full scale. Examples:
1 Velocities in x and y directions must have the same ratio, so that acircular motion in full scale must be circular in the model scale.
2 The ratio between propeller tip speed and advance speed must be thesame in model and full scale.
VF
nF (2πRF )= Vm
nM(2πRm)or VF
nFDF= VM
nmDM−→ JF = JM
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 7 / 27
Dynamic Similarity
1 Geometric similarity, and,
2 Similarity of forces,
−→ Ratios between different forces in full scale must be the same in
model scale
−→ If you have geometric and dynamic similarity, you will also havekinematic similarity
3 The following force contributions are of importance:1 Inertia forces, Fi
2 Viscous forces, Fv
3 Gravitational forces, Fg
4 Pressure forces, FP
5 Elastic forces in the fluid (compressibility), Fe
6 Surface forces, FS .
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 8 / 27
Inertia Forces (mass forces)
Fi ∝ ρdU
dtL3 = ρ
dU
dx
dx
dtL3 ∝ ρU2L3
1 ρ is fluid density,
2 U is a characteristic velocity,
3 t is time,
4 L is a characteristic length (linear dimension)
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 9 / 27
Gravitational Forces
Fg ∝ ρgL3
−→ Just mass × acceleration.g is the acceleration due to gravity.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 10 / 27
Viscous forces
FV ∝ µdU
dtL2 ∝ µUL
where:µ is the dynamic viscosity and a function of temperature and type of fluid.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 11 / 27
Pressure forces
FP ∝ pL2
−→ Force equals pressure times areap is the pressure.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 12 / 27
Elastic Fluid Forces
Fe ∝ ǫνEνL2
where:
1 ǫν is the compression ratio
2 Eν is the volume elasticity ( or compressibility)
3 ǫν Eν = elasticity modulus K
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 13 / 27
Surface Forces
FS ∝ σL
where,σ is the surface tension.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 14 / 27
Froude number Fn
1 The ratio between inertia and gravity:
Inertia force
Gravity force=
Fi
Fg∝
ρU2L2
ρgL3=
U2
gL
2 Dynamic similarity requirement between model and full scale:
U2M
gLM=
U2F
gLF
∴
UM√gLM
=UF√gLF
= Fn
3 Equality in Fn in model and full scale will ensure that gravity
forces are correctly scaled.
4 Surface waves are gravity driven −→ equality in Fn will ensure
that wave resistance is correctly scaled.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 15 / 27
Reynolds number Re
1 The ratio between inertia and gravity:
Inertia force
Viscous forces=
Fi
Fv∝
ρU2L2
µUL=
UL
ν= Re
2 ν is the kinematic viscosity,
ν =µ
ρ
3 Equality in Re will ensure that viscous forces are correctly scaled
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 16 / 27
Kinematic Similarity
To obtain both Fn and RN for a ship model in a scale of 1 : 10 wouldrequire νm = 3.5 × 10−8
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 17 / 27
Mach number
1 The ratio between inertia and and elastic fluid forces:
Inertia force
Viscous forces=
Fi
Fe∝
ρU2L2
ǫνEνL2
2 By requiring ǫν to be equal in model and full scale means:
(
ρU2L2
ǫνEνL2
)
M
=
(
ρU2L2
ǫνEνL2
)
F
UM√
Eν,M
ρ
=UF
√
Eν,F
ρ
= Mn
3
√
Eν
ρis the speed of sound
4 Fluid elasticity is very small in water, so usually the mach numbersimilarity is not required.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 18 / 27
Weber number Wn
1 the ratio between inertia and surface tension forces is:
Inertia force
Surface tension forces=
Fi
Fs∝
ρU2L2
σL=
ρU2L
σ
(
ρU2L2
σL=
ρU2L
σ
)
M
=
(
ρU2L2
σL=
ρU2L
σ
)
F
σ = 0.073 at 200C
UM√
σM
ρL M
=UF
√
σF
ρL F
= Wn
2 When Wn > 180, a further increase in Wn does not influence the fluidforces.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 19 / 27
Scaling ratios used in testing of ships and offshore
structures
Symbol Dimensionless number Force ratio Definition
Re Reynolds number Intertia/Viscous ULν
Fn Froude number Intertia/gravity U√
gl
Mn Mach number Inertia/Elasticty U√
Eνρ
Wn Weber number Inertia/surface tension U√
σ
ρL
St Strouhal number - fνDU
Kc Keulegan-Carpenter number Drag/Inertia UATD
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 20 / 27
Froude scaling
UM√gLM
=UF√gLF
−→ UF = UM
√
LF
LM= UM
√λ
Using the geometrical similarity requirement: λ = LF /LM
If you remember this, most of the other scaling relations can be easilyderived just from the physical units.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 21 / 27
Froude scaling table
Physical parameter Unit Multiplication factor
Length [m] λ
Structural mass [kg ] λ3 × ρF/ρMForce (N] λ3 × ρF/ρM
Moment [Nm] λ4 × ρF/ρMAcceleration [m/s2] aF = aM
Time [s]√λ
Pressure [Pa = N/m2 ] λ× ρF/ρM
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 22 / 27
Hydroelasticity
1 Additional requirements to the elastic model
1 Correctly scaled global stiffness.2 Structural damping must be similar to full scale.3 The mass distribution must be similar.
2 Typical applications1 Springing and whipping of ships2 Dynamic behaviour of marine risers and mooring lines.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 23 / 27
Scaling of elasticity
Deflection δ ∝ FL3
EI,
Hydrodynamic force: F ∝ CρU2L2
Geometric requirements: δFLF
= δMLM
−→ δF = λδMRequirements for structural rigidity:
(
U2L4
EI
)
F
=
(
U2L4
EI
)
M
−→ (EI )F = (EI )M λ5
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 24 / 27
Scaling of elasticity continued
1 Geometrically similar model implies: IF = IMλ4
2 Must change the elasticity of material: EF = EMλ
3 Elastic propellers must be made geometrically similar, using a verysoft material: EM = EF /λ
4 Elastic hull models are made geometrically similar only on the outside.Thus E is not scaled and IM = IF × λ−5
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 25 / 27
Cavitation
1 Dynamic similarity requires that cavitation is modelled.
2 Cavitation is correctly modelled by equality in cavitation number
σ =(ρgh + p0)− pv
0.5ρU2
3 To obtain equality in cavitation number, atmospheric pressure p0might be scaled.
4 pv is vapour pressure and ρgh is hydrostatic pressure.
5 Different definitions of the velocity U is used.
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 26 / 27
General modelling and scaling laws
1 Dimensionless numbers
2 Similarity requirements
3 Derivation of dimensionless used in model testing
4 Froude scaling
5 Hydroelasticity
6 Cavitation number
Philip Wilson (Southampton) UFPA Ship Hydrodynamics September 15, 2015 27 / 27