Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
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172 4 Perfect Flow
uvv
r
(13)
so that the mass flow rate relative to the deflector must be changed accord-
ingly by
AvQm
rrr
3
2
3
2
1
UU
(14)
AvQm
rrr
3
1
3
1
2
UU
(15)
The Eqs. (1), (2), (5) and (6) will be solved for
x
f and
y
f , with Eqs. (14)
and (15), using the relative (moving) frame locked onto the deflector by
replacing
v by
r
v . We can thus obtain, by a heuristic argument, which the
forces are written by
¸
¹
·
¨
©
§
cos
3
1
1
2
AuvF
x
U
(16)
sin
3
1
2
AuvF
y
U
(17)
D
D
E
cos311
sin31
tantan
11
x
y
F
F
(18)
and
cos610
3
1
222
AuvFFF
yx
U
(19)
The kinetic energy
s
W given to the deflector is thus expressed as follows
¸
¹
·
¨
©
§
cos
3
1
1
2
uAuvuFW
x
s
U
(20)
(iii) For a series of deflectors, the actual force on a deflector is zero before
and after the liquid jet strikes the deflector. The situations considered are
analogous to an impulse turbine, such as a Pelton wheel (refer for further
exercises in Section 4.2). In order to examine the problem, we will idealize
the situations that the jet (the discharge from a nozzle) is deflected with an
angle of
D
on average, where the jet strikes the deflector and is deflected
with the relative velocity
r
v to the deflector, as discussed in the previous
D
D
D
D
Problems 173
problem (ii), i.e. uvv
r
according to Eq. (13). That said, the mass flow
rate continuously striking the deflector, one after another, is defined in Eqs.
(3) and (4) on average. Therefore, we solve for
x
f and
y
f with Eqs. (1),
(2), (5) and (6) by replacing
v
with
r
v , but we can consider the discharge
Q with Eqs. (3) and (4). Resultantly, then, we have a force acting upon a
deflector on average
)()( ˞cos
3
1
1 AuvvF
x
U
(21)
˞sin
3
1
AuvvF
y
)(
U
(22)
D
D
E
cos31
sin31
tantan
1
x
y
F
F
(23)
and
DU
cos610
3
1
22
vAuvFFF
yx
(24)
The kinetic energy
d
W given to run the deflectors with the absolute speed
u in the direction of the discharge (the direction of liquid jet) is thus given
by
¸
¹
·
¨
©
§
DU
cos
3
1
1uAuvvuFW
xd
(25)
where
Z
r
u and
r
is the radius to which each deflector rotates with the
angular velocity
Z
, which is analogous to the Pelton wheel.
Problems
4.1-1 In the potential flow, examine the flow field when one source of
strength q and one sink of strength q are spaced equidistant ar
from the origin in the
x
-axis in an uniform flow
f
U (as shown in
Fig. 4.25). The complex potential
)(zW will be given by following
formula
–1
–1
174 4 Perfect Flow
»
¼
º
«
¬
ª
»
¼
º
«
¬
ª
»
¼
º
«
¬
ª
¿
¾
½
¯
®
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
°
¿
°
¾
½
°
¯
°
®
ff
f
f
21
21
11
2
2
2
2
22
ln
2
ln
2
tantan
2
ln
4
T
S
T
SSS
S
S
\
I
qq
yUir
q
r
q
xU
ax
y
ax
yq
yUi
yax
yaxq
xU
izW
)(
Fig. 4.25
One potential flow with source and sink
Ans.
>@
(oval) ovoid Rankine The
4.1-2 Siphons are used to discharge liquid from a reservoir as a simple and
inexpensive way of pumping, as illustrated in Fig. 4.26. Find the
elevation
3
z
and the static pressure at the highest point ٤
2
, and dis-
cuss the function of the siphon, denoting that the discharge flow rate
is Q and the level of point
٤
2
is known as
2
z (given) and the pipe
has an uniform cross sectional area A . Neglect the effect of friction
and assume that the area of the reservoir tank is large enough, com-
pared to the siphon cross-sectional area, to disregard the kinetic en-
ergy at suction
٤
1
compared to the other kinetic energies. The sur-
rounding pressure is the atmospheric
a
p
and the density of the
liquid is
U
.
Problems 175
Fig. 4.26 Siphon
Ans.
»
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
«
¬
ª
t
pressureVapor
2
2
32
2
2
3
sat
PP
zzP
AQ
z
g
g
U
4.1-3 In open channels, such as broad shallow rivers and irrigation canals,
the flow rate can be measured by a simple mean, the so-called broad-
crested weir as illustrated in Fig. 4.27. The weir is usually raised
from the channel bed by a concrete crest with a height of
0
z , letting
the width of the crest to be b. The upstream crest should be well
rounded to minimize a loss due to flow separation. When
0
z is suffi-
cient to choke the flow and the crest is long enough so that the over
flow streams are in parallel to the crest surface.
For the choked flow, a critical flow condition exists, and condition
y may be
c
yy 23
, in which the flow is tranquil. Assuming that
the kinetic energy at point
٤
1
is neglected, i.e. 0
1
v , and Bernoulli
equation from
٤
1
to ٤
2
can be applied, determine the velocity
2
v at
point
٤
2
, and then estimate the flow rate Q by considering y as the
measuring parameter.
176 4 Perfect Flow
Fig. 4.27 Broad-crested wire
Ans.
»
»
¼
º
«
«
¬
ª
2
3
2
3
2
3
2
2 byQyyv
c
gg ,
4.1-4 A nozzle is discharging a liquid of density
U
into the atmospheric
pressure
a
p , as illustrated in Fig. 4.28. Find the pressure at point ٤
1
and calculate the force
1x
F
of the fluid acting on the nozzle in parts
٤
1
–٤
2
. The liquid can be treated as a high velocity jet to give dis-
charge to a Pelton wheel or in fire fighting. The force, in this prob-
lem, is that fire fighters must hold the nozzle.
Fig. 4.28 Nozzle
Ans.
»
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
¸
¸
¸
¹
·
¨
¨
¨
©
§
2
2
1
1
2
1
2
1
2
2
2
1
1
1
2
11
2
A
A
A
m
F
AA
m
p
x
U
U
4.1-5 The liquid jet is discharged from a nozzle of the cross sectional area
2
A
with the discharge mass flow rate of .m
Calculate the force
F
Problems 177
of the liquid jet acting on the deflector as a function of the inclined
angle
D
, as illustrated in Fig. 4.29, and compare the divided mass
from rate of
1
m
and
2
m
. Discuss the difference between the force
1x
F (the force acting on the nozzle, Problem 4.1-4), and the force
x
F
(the
x
component of force acting on the deflector). Assume the liq-
uid is a perfect fluid and the column of the jet is kept constant with-
out the vena contracta.
Fig. 4.29 Deflector
Ans.
»
»
¼
º
«
«
¬
ª
DDD
U
cos1
2
cos1
2
sin
21
2
2
2
m
m
m
m
A
m
F
x
,,
4.1-6 In treating a drag force acting on a flat plate, there would not be any
drag force on the plate if the fluid flowing on the plate is a perfect
fluid. This is due to the reason that there is no viscosity, so that the
flow over the plate is kept constant at the approaching velocity
f
U ;
namely, the flow slips (the slip condition) on the plate and there
would not be any momentum change of the fluid, thus the drag force
x
F is null, as depicted in Fig. 4.30(a). However, in a more realistic
situation, fluid has a viscosity and the velocity at the surface of the
plate is kept at zero; a no-slip condition. Consequently, the velocity
on the plate has a distribution to the perpendicular direction ( y -
direction) against the flow direction (
x
-direction), which is called
the boundary layer. Due to the boundary layer, there would be a
momentum change in the control volume (1-2-3-4), including the
boundary layer, as illustrated in Fig. 4.30(b). Determine the overall
178 4 Perfect Flow
drag force, and the drag coefficient of the plate, assuming that the
velocity profile at the trailing edge is linear within the boundary
layer, whose thickness is
h . Consider the flow is steady with the
density
U
and tackle the problem by applying Eq. (4.1.44) for the
control volume, as indicated in Fig. 4.30(b).
Fig. 4.30 Drag force on a plate
Ans.
»
»
¼
º
«
«
¬
ª
f
hC
hU
FD
D
x
3
1
6
2
,
U
*Problem 4.1-8 is helpful to think of this problem
*More details in the boundary layer theory
4.1-7 Draw a Kàrmàn-Trefftz airfoil (wing) and a Mises airfoil (wing), as
respectively given by Eqs. (14) and (15) in Exercise 4.1.7, with ap-
propriate constants. Illustrating with a computer will be helpful.
4.1-8 Give an idea to measure a drag force on an airfoil, the velocity pro-
file in downstream is measured as
yuu
, as illustrated in Fig.
4.31.
Problems 179
Fig. 4.31 Drag on an airfoil
Ans.
»
»
¼
º
«
«
¬
ª
³
ooo
f
dSuUuFD
x
4321
᧩
U
4.1-9 With the same manner as the Blasius’ first theorem, obtain the mo-
ment
0
M from the Blasius’ second theorem by substituting Eq. (27)
into Eq. (24) in Exercise 4.1.6.
Ans.
>
@
BeUM
i
D
SU
f
2Im
0
4.1-10 A sprinkler, as illustrated in Fig. 4.32, discharges water (the density
is
U
) at the volumetric flow rate of Q from each nozzle, whose
opening area is A . Determine the rotational velocity (the angular
velocity) of the sprinkler and the torque to hold the sprinkler sta-
tionary. Ignore mechanical loss and air resistance.
180 4 Perfect Flow
Fig. 4.32 Sprinkler
Ans.
»
»
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
«
«
¬
ª
?
¸
¸
¹
·
¨
¨
©
§
?
21
2
0
2
2
2
1
21
2222
1
11
1
21
22
2
1
1
1
0
rr
A
Q
T
A
Q
rr
rr
r
A
Q
uwvr
A
Q
uwv
QQQ
vrQvrQT
u
u
u
u
r
UZ
ZZ
UU
,
4.2 General Theories of Turbomachinery
Turbomachines are the devices in which mechanical energy is transferred
either to, or from, a continuously flowing fluid by means of the dynamic
action of rotating propellers or vanes. Generally speaking, two main cate-
gories of turbomachine are identified. A turbopump (including fans and
compressors) adds energy to a system to increase the fluid pressure (or
head) with the result of an enthalpy increase in the fluid. A turbine (includ-
ing wind, hydraulic, steam and gas turbines) extracts energy from a system
to produce power by expanding fluid to a lower pressure (or head) with the
result of an enthalpy decrease in the fluid. More often, the energy trans-
ferred to those of turbomachines is mechanical work that is converted to or
form electric power. In Fig. 4.33(a) a single-suction pump; (b) a single-
stage reaction turbine, are schematically displayed.
4.2 General Theories of Turbomachinery 181
Fig. 4.33 Radial-flow turbopump and turbine nomenclature
The chief objective of this section is to study the general theories of
turbomachinery based on the fluid mechanics and thermodynamics. The
majority of content in this section is devoted to pumps that are typical ex-
ample of turbomachineries. A fundamental treatment of compressible
flows is also studied in applications of fans and gas turbines. Dimensional