Fluid Mechanics Module-1 Basic concepts in FM

Fluid Mechanics Module-1 Basic concepts in FM

1. Reynolds Experiment
•Reynolds Number
•Laminar flow: Fluid moves in smooth
streamlines
•Turbulent flow: Violent mixing, fluid
velocity at a point varies randomly with time
•Transition to turbulence in a 2 in. pipe is at
V=2 ft/s, so most pipe flows are turbulent









2
lowfTurbulent4000
flowTransition40002000
flowLaminar2000
Re
Vh
Vh
VD
f
f


Laminar Turbulent

2. Shear Stress in Pipes
•Steady, uniform flow in a pipe: momentum
flux is zero and pressure distribution across
pipe is hydrostatic, equilibrium exists
between pressure, gravity and shear forces
D
L
hhh
ds
dhD
z
p
ds
dD
sD
ds
dz
sAsA
ds
dp
sDWAs
ds
dp
ppAF
f
s








0
21
0
0
0
0
4
4
)]([
4
)(0
)(sin)(0





•Since h is constant across the cross-section of the
pipe (hydrostatic), and –dh/ds>0, then the shear
stress will be zero at the center (r = 0) and increase
linearly to a maximum at the wall.
•Head loss is due to the shear stress.
•Applicable to either laminar or turbulent flow
•Now we need a relationship for the shear
stress in terms of the Re and pipe roughness

3. Darcy-Weisbach Equation
)(Re,
)(Re,
;Re;
,,:variablesRepeating
),(
),,,,(
2
0
2
0
2
0
321
214
0
D
e
FV
D
e
F
V
VD
e
DV
F
eDVF
















 V  D e
ML
-1
T
-2
ML
-
3
LT
-1
ML
-1
T
-1
L L
)(Re,8
2
)(Re,8
2
)(Re,
4
4
2
2
2
0
D
e
Ff
g
V
D
L
fh
D
e
F
g
V
D
L
D
e
FV
D
L
D
L
h
f
f














Darcy-Weisbach Eq. Friction factor

4. Laminar Flow in Pipes
•Laminar flow -- Newton’s law of viscosity is valid:






















2
0
2
0
2
0
2
1
4
44
2
2
2
r
r
ds
dhr
V
ds
dhr
CC
ds
dhr
V
dr
ds
dhr
dV
ds
dhr
dr
dV
dr
dV
dy
dV
ds
dhr
dy
dV












•Velocity distribution in a pipe (laminar flow)
is parabolic with maximum at center.

















2
0
max
1
r
r
VV

5. Discharge in Laminar Flow
ds
dhD
ds
dhr
Q
rr
ds
dh
rdrrr
ds
dh
VdAQ
rr
ds
dh
V
r
r











128
8
2
)(
4
)2()(
4
)(
4
4
4
0
0
22
0
2
0
22
0
22
0
0
0




 

ds
dhD
V
A
Q
V


32
2



6. Head Loss in Laminar Flow
2
21
12
2
12
2
2
2
32
)(
32
32
32
32
D
VL
h
hhh
ss
D
Vhh
ds
D
Vdh
D
V
ds
dh
ds
dhD
V
f
f
















Re
64
2
2/)(
Re
64
2/))((64
2/
2/32
32
2
2
2
2
2
2
2





f
V
D
L
fh
V
D
L
V
D
L
DV
V
V
D
VL
D
VL
h
f
f












7. Pipe Entrance
•Developing flow
–Includes boundary layer and core,
–viscous effects grow inward from the
wall
•Fully developed flow
–Shape of velocity profile is same at all
points along pipe




flowTurbulent 4.4Re
flowLaminar Re06.0
1/6
D
L
e
eL
Entrance length L
e
Fully developed
flow region
Region of linear
pressure drop
Entrance
pressure drop
Pressure
x

8. Entrance Loss in a Pipe
•In addition to frictional losses, there are minor
losses due to
–Entrances or exits
–Expansions or contractions
–Bends, elbows, tees, and other fittings
–Valves
•Losses generally determined by experiment
and then corellated with pipe flow
characteristics
•Loss coefficients are generally given as the
ratio of head loss to velocity head
•K – loss coefficent
–K ~ 0.1 for well-rounded inlet (high Re)
–K ~ 1.0 abrupt pipe outlet
–K ~ 0.5 abrupt pipe inlet
Abrupt inlet, K ~ 0.5
g
V
Kh
g
V
h
K
L
L
2
or
2
2
2


9. Elbow Loss in a Pipe
•A piping system may have many minor
losses which are all correlated to V
2
/2g
•Sum them up to a total system loss for pipes
of the same diameter
•Where,







m
m
m
mfL
K
D
L
f
g
V
hhh
2
2
lossheadTotal
L
h
lossheadFrictional
fh
mh
m
fittingfor lossheadMinor
mK
m fittingfor tcoefficienlossheadMinor