Hydrodynamic

Introduction

1

Pictures from and based on the books :

Ship Dynamics for Mariners (IC Clark, The Nautical Institute)

Ship resistance & flow (SNAME 2010)

Viscous Fluid Flow (Franck White)

Fluid characteristic

Definition of a fluid : A continuous, amorphous substance whose molecules move freely past one another and that has the tendency to assume the shape of its container; a liquid or gas.

Properties :

• Isotropy : same characteristics whatever the considered directiondirection

• Mobility : it will take the shape of a tank

• Viscosity : is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress

• Compressibility : the density depends on the temperature and the pressure (for water, we consider it’s independent of the pressure

2

Forces on a fluid

• Gravity : volume force

• Pressure : force per surface

• Friction : interaction between particles and surface

• Inertia : proportional to acceleration

• Capillarity• Capillarity

• Surface tension

• Chemical forces

• Magneto hydrodynamic force

3

In general,

smaller than

the other 4.

In static

Only the first 2 forces have to be considered:

So, it become : p + ρ g h = 0

The difference of pressure between two points dependsThe difference of pressure between two points depends

only on the vertical distance between the points :

Pb-Pa = ρ g Z

Unit of the pressure : Pascal (Pa)

1 Pa = 1 N/m²

4

Statical pressure on a boat

The pressure forces are perpendicular to the plate.

The statical pressure is quite easy to calculate.

5

Dynamic pressure• The static pressure is a kind of

potential energy per unit volume.

• If we make a small hole, because of

this pressure, there will be a jet.

Potential energy will be changed

into kinetic energyinto kinetic energy

• It give the dynamic pressure : ρ g h =

½ ρ V²

6

BernoulliDaniel Bernoulli (Groningen, 8

February 1700 – Basel, 8 March

1782) was a Dutch-Swiss

mathematician and was one of the

many prominent mathematicians

in the Bernoulli family. He is in the Bernoulli family. He is

particularly remembered for his

applications of mathematics to

mechanics, especially fluid

mechanics, and for his pioneering

work in probability and statistics.

(from wikipedia)

7

BernoulliBernoulli’s theorem shows the conservation of energy.

8

It can be written :

ρ g h + ½ ρ V² + p = constant

Bernoulli

Let’s consider this pipe.

Liquid

incompressible, so

same volumetric flow

rate : A1V1=A2V2

9

ρ g h1 + ½ ρ V1² + p1 = ρ g h2 + ½ ρ V2² + p2

Because same

pressure

1

222

21

1 −∆=AA

hgV

Bernoulli : the pitot tube

The pitot tube measure

the pressure (static

and dynamic) with

one opening and the

static pressure with

the other one.the other one.

10

ρ)(2 st pp

V−=

Bernoulli around a shipAround the hull, the

flow is modified as in

the previous tube.

There are 3 zones (high,

low and high

pressure), wavepressure), wave

Two stagnation point :

pressure =1/2 ρ V²

11

Too simple.

No friction considered…

Surface tension

• Due to molecular forces

• Try to reduce the surface for the

volume (that’s why the drops are

spherical).

• In still water, force to « open » the sea• In still water, force to « open » the sea

= force to « close » no effect.

• In rough water, the spray : it costs

energy

12

Viscosity• Due to intermolecular attractive forces

• When we move the upper plate, there is a resistance force.

• Viscosity is define as the ratio

• So the frictional force : F = η A V / S

SV

AFso

rateStrain

stressShear

/

/=η

13

Viscosity• The classical formulation is (for 2D) :

• Behaviour of the fluids :

• Fortunately, water is a

y

u

∂∂= µτ

newtonian fluid

• Unit of µ Ns/m² or kg/(ms)

14

Laminar flow

All the particle trajectories

are parallel.

The energy is transfered by

the viscosity. the viscosity.

Resistance proportionnal to

the speed of the flow.

15

Turbulent flow

If the speed increases or if the surface length becomes too big, it

will be instable.

Turbulent

Particles move in all

16

Particles move in all

direction and the

Kinetic energy is directly

transfered.

Resistance proportionnal to the

square of the flow speed.

Turbulent flowAt the beginning,

laminar.

After, turbulent.

It occur in the boundarylayer (zone in whichviscosity isviscosity isconsidered).

17

At the end of the plank, the wake.

Bernoulli’s law doesn’t apply as energy is being dissipated in

turbulence. The streamline doesn’t fully converge increase

of resistance : form drag.

Reynolds

Osborne Reynolds (23 August

1842 – 21 February 1912) was

a prominent innovator in the

understanding of fluid

dynamics. Separately, his

studies of heat transfer studies of heat transfer

between solids and fluids

brought improvements in

boiler and condenser design.

18

Reynolds number

O. Reynolds worked on the

transition of laminar to

turbulent flow in pipe.

He concluded that the transition υη

ρ VDVD ==ReHe concluded that the transition

is function of the ratio inertia

force / viscosity force.

19

υημ is the dynamic viscosity

ν is the kinematic viscosity

Reynolds number for a shipThe length to consider is no more the diameter. We consider the

hull length.

The critical Re (for

transition) is from

0.4x106 to 106 (even

107), function of the

20

107), function of the

hull and the

roughness.

For sea water, μ=1.87x10-3 at 0°C and 0.97x10-3 Ns/m² at 25°C.

So, with μ=1.4x10-3 and ρ=1025 kg/m³, Re=112x107

Transition point is around 0.2 % of the length.

Foil

A flow on a profile produces a

lift and a drag forces.

Great to make the aircraft flying but also for the ships :

21

Great to make the aircraft flying but also for the ships :

Foil

• The force is created by the asymmetrical flow.

• It’s the combination of a symmetrical flow (no lift)

• And the circulation

• Difference of pressure proportionnal to V²

22

Foil

• The drag costs energy and the lift is what we want.

• If the profile is symmetrical and no angle of attack no lift

• If the profile is asymmetrical or angle of attack lift

23

Foil• Along the profile, the separation occurs at the end of the

profile.

• So, there is a wake.

• If the angle of attack is to big, the seperation point will be

more in the beginning of the profile stall

24

Foil• What does the lift depend on?

25

So :

- The angle of the rudder

should be limited.

- The rudder area of a fast

boat will be smaller.

- The force will increase

linearily with the area.

Cavitation• 2 problems : the lifting force can not increase (the difference

of pressure is limited).

• The bubbles appear but collapse when the pressure

decrease damage

• It can be a problem for propellers

26

Cavitation

• If the difference of pressure is to big, the water will « boil »

(changes state from liquid to water vapour)

• The vapour pressure should counteract the surface tension.

27

Resistance : the separate components

Hull still water resistance

Frictional or ResiduaryFrictional or skin resistance

Form drag

Residuaryresistance

Wavemaking

Eddy making

AppendagesAir

resistance

28

Resistance : skin friction

• Skin friction and residuary resistance are not linked.

• Skin friction is function only of the speed, the viscosity, the

wetted area and the length of the hull.

• So, it depends on Re and the wetted area…

• Tests were done with plates in towing tanks (so no residuary• Tests were done with plates in towing tanks (so no residuary

resistance) and curve fitting has been doen.

29

Resistance : skin friction

• We often work with coefficient of resistance.

• It’s a way to have adimensional value.

SV

RC f

f21 ρ

=

• So, following the ITTC conference of 57 :

30

SV 2

21 ρ

( )210 2Relog

075.0

−=fC

Eddy making resistance• If the change of flow direction is too severe (>20°), it will fail

to follow the contour

Separation and creation of eddy making resistance

Increase of resistance and, here, problem for steering

31

Eddy making resistance

Separation occurs later when turbulent boundary layer (water

« fills » more easily the available space)

For eddy making resistance, it is better to have turbulent flow

It can also appear in the fore part, if the waterline is too convex.It can also appear in the fore part, if the waterline is too convex.

32

Eddy making resistanceWhen a ship is relatively slow moving for its length : 2 main components

of resistance :

- Friction

- Eddy making resistance if bad shape

Spherical to reduce the wetted area

Aft part very narrow to redure eddy making resistance

Large bow because slow speed, so no wave

Cods head and mackerel tail (1585)

To increase the deadweight, adding

at the midship section33

Lord KelvinWilliam Thomson, 1st Baron Kelvin,

(26 June 1824 – 17 December 1907)

was a mathematical physicist and

engineer.

At the University of Glasgow he did

important work in the mathematical important work in the mathematical

analysis of electricity and

formulation of the first and second

Laws of Thermodynamics, and did

much to unify the emerging

discipline of physics in its modern

form. Lord Kelvin is widely known

for realising that there was a lower

limit to temperature, absolute zero. 34

Kelvin wave pattern of a moving

disturbance• Group velocity of a wave is

the velocity with which the overall shape of the wave's amplitudes

• The phase velocity of a • The phase velocity of a wave is the rate at which the phase of the wave propagates in space

• Here, group velocity=0.5 phase velocity

35

Kelvin wave pattern of a moving

disturbance

• Speed of the wave phase Cw =V sin Q

• Speed of the wave group Cg =0.5 V sin Q

36

Kelvin wave pattern of a moving

disturbance : two kinds of waves

• Transverse waves: same phase velocity, perpendicular to the

motion

• Divergent waves : slower phase speeds, angle which

decreases for waves of lower phase speed. (includes a whole

spectrum of waves)spectrum of waves)

37

Kelvin waves : submarine

• During the 2nd world war, the waves created by the periscope

made them visible…

38

Kelvin waves for a shipOn a ship, creation of such wave system on points where we

have change of pressure gradient. On a ship : 2 points

• High pressure centre at about 5% aft of the bow where the

streamlines start to converge causing pressure to reduce

downstream, so the waves originates as crests

• Low pressure at ~5% forward the stem, divergence of

streamline, pressure increases troughsstreamline, pressure increases troughs

39

Interference

• Because wavelengths depend on the speed and 2 systems of waves

are created Interference between the waves

• Speed of wave:π

λ2

gV =

• Half l:

• Number of half l:

• Because 180° difference of phase: odd N : constructive interference

even N : destructive 40

g

V 2

5.0πλ =

2

9.0

5.0

9.0

V

LgLN PPPP

πλ==

Trend in wave making

41

Wave resistance• Fr = 0.38 is the limit for displacement ship

• (Friction resistance has to be added)

• Above that, the bow wave increases.

• To reduce the wave resistance, the waterline should be as

smooth as possible.

• But contradiction with the goal of merchant ship which is to

increase the deadweight concave shape

• Contradiction with seakeeping performance (concave ships

have more buoyancy reserve). 42

Bulbous bow

• Goal of the bulbous bow: to create a wave, which will makedestructive interference

• Problem : it is done for certain speeds. At different speed, wemay have constructive interference

• Other advantage: add forward buoyancy waterplane maybe finer

43

Appendage resistance

• Rudder, stabilisers fins, propeller, etc increase the resistance

• Not placed for towing tank test

(too many variable)

• They have their own Fr and Re

44

Air resistance

• In air resistance, we consider frictional and

eddy making resistance

• In calm conditions : ~4%• In calm conditions : ~4%

• When wind, it can increase considerably

45

Form drag

• Frictional resistance is considered equal to the resistance of a

flat plate with the same wetted area

• But, if we make test at low Froude (so wave making resistance• But, if we make test at low Froude (so wave making resistance

can be considered as negligible), the total resistance is not

the frictional resistance. There is an additional residuary

resistance : the form drag

• Form drag is siginificant for wider boat

46

Form drag

• It is due to the boundary layer which is thicker when the

beam to length ratio increase.

• Bernoulli flow is forced to undergo a greater acceleration,

which make the boundary layer thickness.

• The stern pressure is lower, so the wake is bigger. • The stern pressure is lower, so the wake is bigger.

47

Towing tank

• Why?

• CFD is not yet very accurate to estimate the power of a boat.

• Statistical laws are limited

• Is it possible to use the results?• Is it possible to use the results?

• Yes, with some conditions…

48

Towing tank

3 kinds of forces are involved :• Inertial force (ma)

• Gravitationnal force (mg)

proportional to r U² l²

proportional to g l³r•

• Viscous force

49

proportional to µ U l

If the ratio of these forces are

the same, the flow will be

similar

Towing tank

gl

U

lg

lU

Gravity

Inertia 2

3

22

==ρ

ρ

µρ

µρ Ul

Ul

lU

Viscous

Inertia ==22

gl

UFr =⇒

νµρ UlUl ==⇒ Re

50

µµUlViscous

Viscous

Inertia

Gravity

Inertia

Viscous

Gravity1−

=

νµ

So, it means that if the Re and

the Fr numbers are the

same, the flows will be

similar.

Same Fr and Re numbers

λ=M

S

l

l

λS

S

MSM

SMSM

U

gl

glUU

gl

U

gl

UFrFr ==⇒=⇒=

The scale

51

λSSM glglgl

23ReRe

λννν

ννS

S

M

S

MSM

S

SS

M

MMSM l

l

U

UlUlU ==⇒=⇒=

Great, we can have similar flows…

We just need to respect the two relations above.

No problem, let’s replace the water by a liquid with another

viscosity, there is just 2 100 000 l to put and if we change the scale, we will replace it again, it’s easy

Towing tank test

• Following Froude, friction and residuary

coefficient are independent.

( ) ( ) )(ReRe, FrCCFrC RFT +=

• So, if we can obtain the friction resistance, we

can calculate the total resistance with respect

of Froude number.

52

Towing tank test

• Froude’s method :

• Perform the resistance tests with the model.

• So, we have R

( ) ( ) )(ReRe, FrCCFrC RFT +=

• So, we have RTM

• We know that :

• And that for the model and the

ship

53

MMM

TMTM SV

RC

221 ρ=

RFT CCC +=

Towing tank test (2)

• Following ITTC 57 :

(we can calculate it for the model and the ship

• Calculate CRM

• Thanks to Froude similitude :

( )210 2log

075.0

−=

RnCF

FMTMRM CCC −=

• Thanks to Froude similitude :

• We make the same procedure by the other

side…

54

RSRM CC =

Towing tank test (3)

• Following ITTC 57 :

for the ship

• Calculate CTS

(the last term is the roughness allowance :0.0004)

( )210 2log

075.0

−=

S

FSRn

C

FFSRSTS CCCC ∆++=

(the last term is the roughness allowance :0.0004)

• Finally :

• So, we can calculate the power :

55

SSSTSTS SVCR ×××= 2ρ

STSE VRP ×=

Towing tank test (ITTC-78)

• Some differences with the 57th method. The

decomposition is in a viscous resistance, which includes

the form effect on friction and pressure and wave

resistance.

• Assumption is : ( ) ( ) ( ) ( )FCCkFnC ++=+ Re1Re• Assumption is :

• Compute CFM

• Calculate the form factor k

56

MMM

TMTM SV

RC

221 ρ=

( ) ( ) ( ) ( )nwFT FCCkFnC ++=+ Re1Re 0

Towing tank test (ITTC-78)(2)

• Calculate CWM

• Remark : the wave resistance is smaller than the

residuary resistance for Froude method

• Compute the roughness allowance (according to

Bowden): 1 Bowden):

Where kMAA is the roughness in microns according to the

MAA method. ITTC recommend 150 microns.

• Determine the air resistance coefficient:

Where AT is the frontal area of the ship above the waterline57

S

AC T

AA ×= 001.0

33

1

1064.0105 −×

−

×=∆L

kC MAA

f

Towing tank test (ITTC-78)(3)

• Calculate the total resistance coefficient CWM

• Calculate the total resistance coefficient as before.

( ) AAFWSFSTS CCCCkC +∆++×+= 1

• Calculate the effective power as before also.

58

Form factor

• It includes the ratio of the viscous resistance and the

resistance of the equivalent flat plate.

• So, it includes the form effect.

• Empirical formula (Watanabe):

• Another way is to calculate it at low Re (<0.15) (Cw=0)

but small forces, so problems on measurement

59

T

B

B

L

Ck B

26.25095.0

+−=

Form factor

• Method of Prohaska : assumption: wave resistance

coefficient is proportional to the 4th power of the Fr.

• So:

• Or:

( ) 411 FnkCkC FT ++=

( )T Fnkk

C 4

1 ++=

• If the assumption in the wave resistance is correct:

60

( )FF

T

Ckk

C 11 ++=

Planing

• When a ship goes faster and exceed Fr = 0.38 (with enough

power and adequate shape, it can go faster than its wave.

• A lifting force appears.

• So, displaced water decreases and resistance is smaller.

61

Planing

• A part of the flow goes forward : spray

• Hard chine is better.

• The weight has to be lower

• Planing hull is common for pleasure craft

(in some case, not enough buoyancy)62

Shallow water

• The Bernoulli pressure distribution distorts the waterline.

• It will be more pronounced if the depth is small.

• Between the river bottom and the hull, water is accelerated,

creating a depression reduction of the under keel

clearance, called the squat.clearance, called the squat.

• It depends on :

– Static pressure, so it will increase in proportion of V²/g

– The sectional area of the water flow ( blockage factor)

– The block coefficient (the flow will be more restricted in case of high

Cb

63

Squat

• Squat is NOT an augmentation of the draft.

• It is the total reduction in under keel clearance.

• (water level also goes down)

64

Squat

• Blockage factor :

• So,

• Squat can be like :

( )015.0 WWD

dBS

+×=

S=Ship’s immersed midship sectional area

Sectional area of the unobstructed canal

• Squat can be like :

65

( ) mB

n CKSKg

VKS 32

2

1 ××=∆

Squat in narrow channels

• Following A. D. Watt :BCS

g

VsSquat ××=∆ 2

2

2.2

• With

• And V the speed in m/s (and g=9.81 m/s²)

66

SC

S

AA

AS

−=2

Squat in narrow channels

• Following Dr C. B. Barrass:

• With

Bk CS

VsSquat ××=∆ 81.0

08.2

20SA

S =• With

• And Vk the speed in knots

• Attention: these formulas are available in a narrow channel

67

S

S

A

AS =

Comparison of the method

• Speed : 8 kts

• Sectional area Ac = 0.5 (40+60) x 12 = 60 m²

• Sectional area As = 8 x 20 m²

• Block coefficient : 0.8

• Following Watt : SSquat 8.0160514.08

2.2 ××××=∆• Following Watt :

• Following Barrass :

68

mSSquat

SSquat

1.1

8.0160600

160

81.9

514.082.2

=∆

×−

×××=∆

mSSquat

SSquat

04.1

8.0600

160

20

881.008.2

=∆

×

×=∆

Squat in open shallow water

• The previous formulas were available for narrow channel, but

in shallow water, the squat phenomenon is also present.

• Dr I. Dand proposed a formula :

Bk CD

dVSSquat ×××=∆ 2

95

1

• With : Vk speed in knots, d deep water draft, D water depth

and CB the block coefficient

69

Bk CD

VSSquat ×××=∆95

Squat in open shallow water

• Barrass proposed an empirical formula. His philosophy was to

consider a width of influence, function of the beam of the

ship.

• Width of influence :

• Open water blockage factor S

• Open water squat

70

)(04.7

85.0 mC

BF

B

B =

Effect of squat on trim and list

• Distorsion of waterline may change the fore and aft position

of the center of buyoancy.

• If a vessel’s centre of buyancy is forward of midship (the bow

is fuller than the stem) head trimming momentis fuller than the stem) head trimming moment

• Faster flow on the fore part, so more « succion » head

trimming moment

• Acceleration on the propeller stern trimming moment

71

Squat over a shoal

• If the water depth is small: constant squat…

• But if the vessel sails over a shoal ?

72

Squat and heel

• What about the heel?

73

Other effects of squat

• Frictional resistance is increased, wave making resistance also

the ship slows down

• This increase of resistance loads more the propeller more

slip and the propeller revolution tend to decrease

• Proximity of the seabed greater vibration• Proximity of the seabed greater vibration

• Increase of turbulence and vibration under the stern if soft

sediment, water can be discoloured.

• Higher bow wave

• Response to helm action slower

• Motions (rolling, pitching) tend to be dampened by the

cushioning effect of the seabed

74

Wave making resistance in shallow

water

• Waves depend on the water depth (when water depth is

reduced to less than ~40% of the wavelength, it’s influenced

by the seabed).

• Phase and group speed decreases

• First, the waves with longer wavelength are modified : higher• First, the waves with longer wavelength are modified : higher

and longer

75

l (deep water)=

l (12m water)=

mg

V64

2 2

=π

mD

g

V101

2tanh

2 2

=

+λππ

Waves in shallow water

• Waves are longer when depth decreases

• So, angle of 19.28° is no more available

• It will increase when the speed increases and the depth

decreases.

76

Waves in shallow water

• Limit speed : kind of wall in front of the ship: as sound wall

• After this limit, resistance decreases

77

Waves in shallow water

• This effect was discovered accidentally in British canals, around

1844 when barges were towed by horses.

• A horse took fright and ran with the barge.

• The prominent bow wave suddenly disappeared and the speed

was much more bigger. was much more bigger.

• It was because :

– Canals were artificially built with a depth around 1 m (critical speed 3 m/s)

– Barges: long and narrow

– Barges towed from ashore, so no squat by the propeller

78

Fluid dynamic

Fluids have different kind of properties :

• Kinematic properties : linear and angular velocities, vorticity,

acceleration and strain rate. Properties of the flow more than

the fluid itself

• Transport properties : viscosity, thermal conductivity and • Transport properties : viscosity, thermal conductivity and

mass diffusivity

• Thermodynamic properties : pressure, density, temperature,

enthalpy, entropy, specific heat, Prandtl number, bulk

modulus, coefficient of thermal expansion

• Other miscellaneous properties : surface tension, vapor

pressure, eddy-diffusion coefficients, …

79

2 formulations : Lagrangian

• Consider a rocket lifting off

• You control it from the ground. You will see the

different parts separating from the main part and

you can follow the different trajectories

• Lagrangian description

• Very useful for the solid mechanic80

2 formulations : Eulerian

• Now, you want to follow the flow out of the nozzle. From the

ground, you’ll see a complicated unsteady flow.

• But if you examine them from the rocket, you will observe a

nearly steady flow… nearly steady flow…

• Eulerian description

• You can choose coordinate with a good orientation, making

the flow appear more steady

• You don’t study all the particle but the velocity field 81

Differential

• Fundamental laws are Lagrangian in nature (formulated for

particles).

• Variation of a function Q

dtQ

dzQ

dyQ

dxQ

dQ∂∂+

∂∂+

∂∂+

∂∂=

• We follow an infinitesimal particle, so : dx=udt, dy=vdt, dz=wdt

• So:

• dQ/dt is called substancial derivative, particle derivative or

material derivative 82

dtt

dzz

dyy

dxx

dQ∂

+∂

+∂

+∂

=

z

Qw

y

Qv

x

Qu

t

Q

dt

dQ

∂∂+

∂∂+

∂∂+

∂∂=

Material derivative

• The way to write it :

• In the vectorial form:

• With

Dt

DQ

( )QVt

Q

Dt

DQ ∇⋅+∂∂=

∂∂∂• With

• If the speed is = 0:

83

zk

yj

xi

∂∂+

∂∂+

∂∂=∇

t

Q

Dt

DQ

∂∂=

Deformation

4 types of motion or deformation:

• Translation

• Rotation

• Extensional strain or dilataion

• Shear strain

Rotation

Dilatation

Shear strain

• Shear strain

We work with rate, i.e. a time

derivative

84

Translation

Shear strain

Angular rotation

• Angular rotation= avergage

counterclockwise rotation of

the side AB (-dβ) and BC (-

dα).

• So : ( )βα ddd z −=Ω 1• So :

• Following the schema:

85

( )βα ddd z −=Ω2

dtx

v

dxdtx

udx

dxdtx

v

ddt ∂

∂=

∂∂+

∂∂

= −

→

1

0tanlimα dt

x

v

dxdtx

udx

dxdtx

v

ddt ∂

∂=

∂∂+

∂∂

= −

→

1

0tanlimβand

Vorticity

• Instead of working with

• And is called vorticity

• In term of vector :

dt

dΩ= 2ωωωω

VVVVVVVV ×∇== curlωωωω• In term of vector :

• One property :

• If ωωωω=0, the fluid is irrotational

86

VVVVVVVV ×∇== curlωωωω

0==•∇= VVVVcurldivdiv ωωωωωωωω

Shear-strain rate

• 2 lines are initially perpendicular.

• This angle decreases: measured by shear-strain rate.

• By the same way as vorticity, we can change the formulation

+=dt

d

dt

dd xy

βαε2

1

• By the same way as vorticity, we can change the formulation

of αand β

• So :

• The last element are dilatation :

87

∂∂+

∂∂=

∂∂+

∂∂=

∂∂+

∂∂=

y

w

z

vand

z

u

x

w

y

u

x

vyzxzxy 2

1

2

1;

2

1 εεε

( )dt

x

u

dx

dxxdxdtudxd xx ∂

∂=−∂∂+= /ε

Shear-strain rate

• Shear-strain tensor is symmetric :

• It may be visualized as a array :

jiij εε =

= yzyyyx

xzxyxx

ij

εεεεεεεεε

ε

• Another property : it exist 1 (and only 1) set of axes for whichthe shear-strain rate vanish :

• These axis are called principal axes

88

zzzyzx εεε

=

3

2

1

00

00

00

εε

εε ij

Coefficient of viscosity

• Lets consider, as prevously, 2 walls.

• The upper wall moves at a speed V

• The shear stress (the stress to move the wall at the

constant speed) is constant

• The speed has only 1 component : u(y)

xyτ

• The speed has only 1 component : u(y)

• So, only 1 strain rate:

• After experiment, one remarks that

• For Newtonian fluid, linear relationship, so :

89

dy

duxy 2

1=ε

( )xyxy f ετ =

dy

duxy µτ =

Boundary conditions

Five types of boundaries :

1. A solid surface

2. A free liquid surface

3. A liquide-vapor interface3. A liquide-vapor interface

4. A liquid-liquid interface

5. An inlet or exit section

90

Boundary conditions (2)

1. Solid surface: in the wall :

(no slip conditions)

If the wall is permeable Vnormal = 0

2. Free liquid surface : open surface exposed to an atmosphere

solidfluid

solidfluid

TT

VV

=

=

2. Free liquid surface : open surface exposed to an atmosphere

of either gas or vapor. 2 cases can be considered : ideal

surface (exerts only a pressure on the liquid boundary) or

complicated case (pressure but also shear, heat flux, mass

flux, etc)

Conditions : the fluid particles must remain attached (kinematic

conditions) and pressure of liquid and gas must balance

91

Boundary conditions (3)

• Liquid-vapor or liquid-liquid interface

In free surface: heat transfer and shear stress negligible.

In liquid-vapor or liquid-liquid : the fluids are strongly coupled so

kinematic, stress and energy constraints.

;; TTVV === ττ

• Inlet and exit boundary conditions

Often, we try to limit the analysis to a finite region through

which the flow passes. So, we need the properties at all

boundaries. Specifications of distributions of V, T and o

92

212121 ;; TTVV === ττ

Fundamental equations

• Equations known for more than 100 years.

• But impossible to solve with the mathematical techniques

because the boundary conditions become randomly time-

dependent. dependent.

• They have been developped by Navier and Stokes and comes

from 3 laws of conservations:

1. Conservations of mass (continuity)

2. Conservation of momentum (Newton’s second law)

3. Conservation of energy (first law of thermodynamics)

93

Conservation of mass

• The mass should be constant :

• In Eulerian terms :

• Variation of volume = dilatation rate x Volume :

.constm == Vρ

( )Dt

D

Dt

D

Dt

D

Dt

Dm ρρρ VV

V +=== 0

• Variation of volume = dilatation rate x Volume :

• We know that :

94

Dt

Dzzyyxx

V

V

1=++ εεε

VdivV

z

w

y

v

x

uzzyyxx

⋅∇==∂∂+

∂∂+

∂∂=++ εεε

Conservation of mass (2)

• So, if we combine the equations, we finally

obtain :

( ) 00 =+∂=+ VdivordivVD ρρρρ

• If the density is constant (incompressible flow)

95

( ) 00 =+∂∂=+ Vdiv

tordivV

Dt

D ρρρρ

0=divV

When we consider a fluid volume,

if the volume decreases or

increases.

Density has to increase or

decrease (to keep the mass)

Conservation of momentum

• Newton’s second law : F=ma

• We work with density (more convenient):

• Body force : gravity (we ignore magnetohydrodynamics force)

surfacebody fffDt

DV +==ρ

• Body force : gravity (we ignore magnetohydrodynamics force)

• Surface forces are those applied by external

stresses on the sides. Tensor like strain rate

96

gfbody ρ=

Conservation of momentum (2)

• Total force on each direction :

• If the element is in equilibrium, forces balanced.

• But if acceleration, front- and back-face will be different by

differential amount:

...=

++=

y

zxyxxxx

dF

dxdydxdzdydzdF τττ

∂τdifferential amount:

• If we just take the net force :

• So, per volume unit : 97

dxxxx

backxxfrontxx ∂∂+= τττ ,,

dxdydzz

dxdzdyy

dydzdxx

dF zxyxxxx

∂∂+

∂∂

+

∂∂= τττ

zyxf xzxyxx

x ∂∂+

∂∂

+∂

∂= τττ

Navier-Stokes equations• Finally :

• We can transform it for Newtonian fluid :

ijgDt

DV τρρ •∇+=

divVx

u

x

up ij

i

j

j

iijij λδµδτ +

∂∂

+∂∂+−=

• Navier-Stokes equations :

98

xx ij ∂∂Kronecker

operator

100

010

001 Second

coefficient of

viscosity

Lamé constant or

coefficient of bulk

viscosity

+

∂∂

+∂∂

∂∂+∇−= divV

x

u

x

u

xjpg

Dt

DVij

i

j

j

i λδµρρ

Thermodynamic properties

• The first thermodynamic law :

• With: dE : the change in total energy of the system

dQ : the heat added to the system

dW : the work done on system

dWdQdE +=

dW : the work done on system

• Finally, after the same kind of transformation than before :

With

99

( )j

iij x

uTkdiv

Dt

Dp

Dt

Dh

∂∂+∇+= 'τρ

divVx

u

x

uij

i

j

j

iij λδµτ +

∂∂

+∂∂='

Fluid mechanics equations

• Finally :

pgDt

DVij −∇•∇+= 'τρρ

( ) 0=+∂∂

Vdivt

ρρ

• Three variables : V, p and T

• Four variables (assumed known from auxiliaryrelation) : r,m,h and k

100

( )j

iij x

uTkdiv

Dt

Dp

Dt

Dh

∂∂+∇+= 'τρ