summary basic fluids

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


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


Applications


Types of similarity

What are the similarity requirements for model testing?

1 Geometrical similarity

2 Kinematic similarity

3 Dynamic similarity


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


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


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 .


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)


Gravitational Forces

Fg ∝ ρgL3

−→ Just mass × acceleration.g is the acceleration due to gravity.


Viscous forces

FV ∝ µdU

dtL2 ∝ µUL

where:µ is the dynamic viscosity and a function of temperature and type of fluid.


Pressure forces

FP ∝ pL2

−→ Force equals pressure times areap is the pressure.


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


Surface Forces

FS ∝ σL

where,σ is the surface tension.


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.


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


Kinematic Similarity

To obtain both Fn and RN for a ship model in a scale of 1 : 10 wouldrequire νm = 3.5 × 10−8


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.


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.


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


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.


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


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.


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


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


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.


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


summary basic fluids

Documents

similarity of forces

scale ratio

different forces

fg4 pressure forces

scale structures

inmodel scale

fp5 elastic forces

fv3 gravitational forces