basics of hydrodynamics -...

Basics of hydrodynamics

K141 HYAE Basics of hydrodynamics 2

Characteristics of cross section

ø D

O

S

B

b

y S O

pipe diameter D [m] channel depth y, h [m]

channel width - at bottom b [m],

- at water level B [m]

mean depth [m] BSys

flow area, cross sectional area S [m2]

wetted perimeter O [m]

hydraulic radius [m]

- circular pipeline with diameter D:

OSR

- wide channel B > (2030)y S ≈ By, O ≈ B R ≈ y

44

2 D

D

D

O

SR


Trajectory and streamline (at particular time)

streamline

trajectory

elementary stream fibre elementary stream tube

elementary discharge

substantial particle

(primary element)

u

at point M – envelope curve of immediate

velocity vectors

- real path of particle at time

dS

M

stream fibre - elementary volume of liquid defined by

pack of streamlines

whole flow – body of all flow fibres

udSdQ

point velocity dt

d u

s

ds


discharge (mass discharge)

SS

udSdQdt

dVQ

S – flow area

to streamlines (axis)

flow

mean velocity

S

udS S

1

S

Qv

umax

v

pipe

channel S


Kinds and forms of flow

• unsteady

• steady non-uniform S ≠ const., v ≠ const. uniform S = const., v = const.

• with free level – flow limited by solid walls, free level on surface, motion caused by own weight of liquid

• pressure flow – flow limited by solid walls from all sides, motion caused by difference of pressures

• jets – limited by liquid or gas surroundings, motion by own weight or by delayed action (inertia)

• laminar

• turbulent

tQQ

Q const.


Laminar and turbulent flow • laminar – particles of liquid move at parallel paths

• turbulent – motion of particles of liquid: irregular and inordinate, fluctuations of velocity vector in time and space, mixing inside flow

Criterion – Reynolds number

L – characteristic length:

diameter D for pipelines,

hydraulic radius R for other profiles

ReD < 2320 laminar (ReR !)

vLRe


S S S

1

2

dL

Continuity equation

tdLd

L

QQ

tdQ

td

t

LdS

td

t

LdStdLd

L

QQtdQ

LdS

tdLd

t

StdLd

L

Q

0

t

S

L

Q

general continuity equation for flow of compressible liquid

at definite cross section under unsteady flow

- expresses the law of perdurability of matter


Forms of continuity equation

unsteady flow of incompressible liquid

= const.

steady flow of incompressible liquid

0

t

S

L

Q

QSvSv 2211

S1 S2

v1 Q v2

0

t

S

L

Q

0t

S

0

L

Q

Q = const.


Euler hydrodynamic equation (ideal liquid)

0duudzgdp

dzcosds,dt

dsu

amF

amcosgdsSSpSdpp

Application of the 2nd Newton’s kinetic law:

Euler hydrodynamic equation

balance of forces:

dt

dudsScosgdsSSpSdpp

dt

dudsSam


const.g2

u

g

pz

2

Bernoulli equation for ideal liquid

under steady flow

.const2u

zgp

0u

duuzdzg

pdp

2

Integration of Euler hydrodynamic equation

Bernoulli equation BE (ideal liquid)

considering the mean cross-sectional velocity

Econst.2g

v

ρg

pz

2


work performed by flow on EV: dsSpA

kinetic energy of EV: 2

udsS

2

umE

22

k

potential energy of EV: zgdsSzgmEp

total mechanical energy of EV: JEEAE pk.mech

Total mechanical energy Emech. per unit of gravity :

m.constg2

u

g

pz

dsSg

EAEh

2kp

E

force F

volume of EV

Principle of conversation of mechanical energy:

.constE .mech

Derivation of BE from the balance of mechanical energy

of elementary volume EV


z geodetic head,

potential energy head of position [m]

pressure head,

potential energy head of pressure [m]

velocity head,

kinetic energy head, dynamic head [m]

g

p

2g

v2

2g

v

ρg

pz

2g

v

ρg

pz

2

222

2

111

E

Components of BE for ideal liquid


• point velocity u v tube:

• in technical calculations - mean velocities v

a) Coriolis number - coefficient of kinetic energy

g2

v2

depends on - shape of cross section

- form of velocity profile:

circular pipelines and regular channels = 1,05 1,2,

laminar flow = 2,

current technical calculations of pipelines ≈ 1,0

Bernoulli equation BE (real liquid)

a) Coriolis number

b) hydraulic resistances


b) hydraulic resistances

• motion of real (viscous) liquid hydraulic resistances

– internal friction in liquid

– friction of liquid around solid walls

– deformation of velocity and pressure field in singularities

(reduction and enlargement of flow, bends, closures ...)

• part of energy is consumed losses Z

energy decreases in the flow direction

line of energy decreases

non-uniform

velocity field


Form of Bernoulli equation for real liquid

Zg

v

g

ph

g

v

g

ph

22

2

222

2

111

Z – loss head (losses)

,

2g

vfZ

2

energy decreases in the flow direction

line of energy decreases

dL

dZiE

hydraulic slope

(gradient, friction slope)


Application of Bernoulli equation (for Z = 0)

Pitot tube Suction effect of flow

u

zg

u

2

2

g

p

1g

p

2

g

p

g

u

g

p

2

2

1

2

g

uz

g

pp

2

2

12

zgu 2

energy l.

p0

g

p

1

2g

v2

2

2gv2

1

gp 2

Hs

p.l.

r.l.

balance of relative pressures:

2g

v

gρ

p

2g

v

gρ

p 2

22

A

2

2

11

A

1

sB2 gHρp 0gρ

pH

B

2s


from mechanics of primary element:

umH

12i

i

2u

1u

uuQFd

udQFd

udQFd

dt

uddtQFd

dtQdm,dt

uddmadmFd

12

iiii

vvQF

FF,vu,FFd

for the whole flow:

Momentum equation in flow of liquid

momentum of primary element


FFi

12 vvQF

AR FF

outletv

entrancev

2

1

velocity

A21 FGFFF

FR

1

2

F

F

v

v

G F

1

1

2

2

A

x

y

determined volume of liquid

- external forces:

F1 = p1S1 ... pressure force in entrance profile

F2 = p2S2 ... pressure force in outlet profile

FA ... force of solid wall acting on liquid inside

FR ... force of liquid acting on solid wall

basics of hydrodynamics -...

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