Equations of Continuity and Motion (5/6)
14.1 Frictionless Flow
Objectives
- Derive 3D equations of continuity and motion
- Derive Navier-Stokes equation for Newtonian fluid - Study solutions for simplified cases of laminar flow
- Derive Bernoulli equation for irrotational motion and frictionless flow - Study solutions for vortex motions
1) Irrotational motion of incompressible fluid in a real fluid
- Bernoulli eq. can be obtained by integrating Navier-Stokes equation for entire flow field.
- Viscous force term is dropped due to irrotational and incompressible fluid.
2) Frictionless flow of compressible fluid
- Bernoulli eq. can be obtained by integrating Navier-Stokes equation along a streamline.
- N-S eq. for ideal compressible fluid (m = 0) is used without assuming irrotational motion.
- Special case of the Navier-Stokes Equation
For inviscid flow
→ Assume no frictional (viscous) effects but compressible fluid flows
→ Bernoulli eq. can be obtained by integrating Navier-Stokes equation along a streamline.
Eq. (11.5): N-S eq. for ideal compressible fluid (m = 0)
g p
2q
m ( )
3 q
m q q ) q
t
=
)
p q
g q q
t
=
(14.1)Substituting (11.7) into (14.1) leads to
Multiply (element of streamline length) and integrate along the streamline
)
p q
g h q q
t
=
g = g h
(14.2)
dr = idx jdy kdz dr
) )
1
q
g h dr p dr dr q q dr C t
t
=
(14.3)dr d
=
For a scalar function of space,
where
) )
dp q
gh dr q q dr C t
t
=
I
(14.4)
) ) )
I = q q dr = dr q q = q dr q
II
By the way,
) ) )
II dr dx dy dz
x y z
= =
dr ) q q dx q dy q dz dq
x y z
= =
2
I = q dq = d q
)
2 22 2
q q
q q dr d
= =
Thus, Eq. (14.4) becomes
2
)
2
dp q q
gh dr C t
t
=
(14.5)For steady motion,
q
0;C t ) C
t
=
Divide by g
For incompressible fluids, = const.
2 .
dp q
gh const
=
along a streamline (14.6)2
2 .
p q
gh const
=
2
2
p q
h C
g =
along a streamline (14.7)- Eq. (14.7) is identical to Eq. (13.17).
- However, constant C is varying from one streamline to another in a rotational flow, Eq. (14.7); it is invariant throughout the fluid for
irrotational flow, Eq. (13.17).
Summary of Bernoulli equation forms
• Bernoulli equations for steady, incompressible flow 1) For irrotational flow
constant throughout the flow field
2
2
p q
H h
g
= =
(14.8)2) For frictionless flow (rotational)
2
2
p q
H h
g
= =
constant along a streamline (14.9)3) For 1-D frictionless flow (rotational)
2
2
p V
H h Ke
g
= =
constant along finite pipe (14.10)4) For steady flow with friction → include head loss hL
2 2
1 1 2 2
1 2
2 2
Lp q p q
h h h
g g
=
(14.11)(1) Efflux from a short tube
• Zone of viscous action (boundary layer): frictional effects cannot be neglected.
• Flow in the reservoir and central core of the tube: primary forces are pressure and gravity forces. → irrotational flow
• Apply Bernoulli eq. along the centerline streamline between (0) and (1)
p0 = hydrostatic pressure = d0 , p1 = patm → = 0
q0 = 0 (neglect velocity at the large reservoir)
→ Torricelli’s result
0 0 1 1
0 1
2 2
p q p q
z z
g g
=
1gage
p
0 1
z = z
2 1
2
0q d
g =
1
2
0q = gd
(14.12)If we neglect thickness of the zone of viscous influence
2
4 1
Q =
D q• Selective withdrawal: Colder water is withdrawn into the intake channel with a velocity q1 (uniform over the height b1 ) in order to provide cool
condenser water for thermal (nuclear) power plant.
During summer months, large reservoirs and lakes become thermally stratified.
→ At thermocline, temperature changes rapidly with depth.
Heated water discharge from power plant
(14.13)
p0 = hydrostatic pressure = ( – D)(d0 – a0)
2 2
0 0 1 1
0 1
2 2
p q p q
a b
g g
=
0
0
q
)
1 0 1
p = d D h b
(14.14)
)
2 1
0 0
2
q h d a
g
= D D
)
1 2
1 2 0 0
q g h d a
D
= D
(14.15)1 2
q = g h D
→ Torricelli’s result (14.16) Selective withdrawal
→ Measure velocity from stagnation or impact pressure
(14.17) (14.18)
2 2
0 0
0
2 2
s s
s
p q p q
h h
g g
=
0 s
,
h = h q
s= 0
2
0 0
2
q p
sp g h
= = D
0)
0
2 p
sp
q
=
(A)By the way,
1 s 2 0 m
p = p D = h p = p D h
)
0
s m
p p = D h
Combine (A) and (B)
)
0
2 h
mq
D
=
(B)
