03 basics of hydrodynamics
DESCRIPTION
derivation of flow governing equation and bernoulli's equationTRANSCRIPT
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Basics of hydrodynamics
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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
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K141 HYAE Basics of hydrodynamics 3
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
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K141 HYAE Basics of hydrodynamics 4
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
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K141 HYAE Basics of hydrodynamics 5
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.
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K141 HYAE Basics of hydrodynamics 6
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
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K141 HYAE Basics of hydrodynamics 7
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
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K141 HYAE Basics of hydrodynamics 8
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.
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K141 HYAE Basics of hydrodynamics 9
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
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K141 HYAE Basics of hydrodynamics 10
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
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K141 HYAE Basics of hydrodynamics 11
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
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K141 HYAE Basics of hydrodynamics 12
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
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K141 HYAE Basics of hydrodynamics 13
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
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K141 HYAE Basics of hydrodynamics 14
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
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K141 HYAE Basics of hydrodynamics 15
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)
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K141 HYAE Basics of hydrodynamics 16
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
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K141 HYAE Basics of hydrodynamics 17
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
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K141 HYAE Basics of hydrodynamics 18
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