03 basics of hydrodynamics

Basics of hydrodynamics

K141 HYAE Basics of hydrodynamics 2

Characteristics of cross section

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 us

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 !)

vL

Re


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 Newtons 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

g

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

g

pz

2g

v

g

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

1

g

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

v22

2gv21

gp 2

Hs

p.l.

r.l.

balance of relative pressures:

2g

v

g

p

2g

v

g

p 222

A

2

2

11

A

1

sB2 gHp 0g

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

03 basics of hydrodynamics

Documents

flow of compressible

forms of flow

axis flow

flow body

free level flow

bsys flow area

flow criterion reynolds

g2ugpz2bernoulli equation