Fluid Mechanics – I
Figures Shown in the
Class
68_ch07_332-382.qxd
9/23/08
10:46 AM
Page 334
CHAPTER 7 - Similitude and Dimensional Analysis
334
Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
∆ p!
∆ p!
D, ρ , µ – constant
V, ρ, µ– constant
V
D
(a)
(b)
∆ p!
∆ p!
D, ρ , V– constant
D, V, µ – constant
Figure 54
ρ
(c)
µ
(d)
F I G U R E 7.1 Illustrative plots showing how the pressure drop in a pipe may be affected
by several different factors.
in terms of basic dimensions such as mass, M, length
force, F, L, and T as basic dimensions, since from New
CHAPTER 7 - Similitude and Dimensional Analysis
F ! MLT
D ∆p!
_____
ρV 2
ρ
VD
____
µ
Figure 55
1
F I G U R E 7
data using dimensionle
As noted in Chapter 1, we will use T to represent the basic dimension of ti
CHAPTER 7 - Similitude and Dimensional Analysis
Figure 56
CHAPTER 8 – Viscous Flow in Closed Conduits
8.1
General Characteristics of
p2 ≠ p1
(1)
p1 = p2
(1)
Q
Q
(2)
(a)
F I G U R E
(2)
(b)
8.2
(a) Pipe flow. (b) Open-channel flow.
For all flows involved in this chapter, we assume that the pipe is com
Figure 57
fluid being transported as is shown in Fig. 8.2a. Thus, we will not consider a
which rainwater flows without completely filling the pipe, as is shown in F
called open-channel flow, are treated in Chapter 10. The difference between o
aml,
most immediately becomes blurred and spreads across the entire pipe in a random fashion. These
three characteristics, denoted as laminar, transitional, and turbulent flow, respectively, are illustrated in Fig. 8.3b.
The curves shown in Fig. 8.4 represent the x component of the velocity as a function of
time at a point A in the flow. The random fluctuations of the turbulent flow 1with the associated
particle mixing2 are what disperse the dye throughout the pipe and cause the blurred appearance
illustrated in Fig. 8.3b. For laminar flow in a pipe there is only one component of velocity,
CHAPTER 8 – Viscous Flow in Closed Conduits
Turbulent
Dye
Pipe
D
Q = VA
Dye streak
Transitional
Smooth, well-rounded
entrance
Laminar
(a)
F I G U R E
Figure 58
8.3
(b)
(a) Experiment to illustrate type of flow. (b) Typical dye streaks.
CHAPTER 8 – Viscous Flow in Closed Conduits
er 8 ■ Viscous Flow in Pipes
uA
Turbulent
Q
A
x
Transitional
Laminar
t
F I G U R E
Figure 59
8.4
Time dependence of fluid velocity at a point.
V ! uiˆ. For turbulent flow the predominant component of velocity is also
4
106
quite complex. However, once the fluid reaches the end of the entrance region, section 122 of Fig.
8.5, the flow is simpler to describe because the velocity is a function of only the distance from
the pipe centerline, r, and independent of x. This is true until the character of the pipe changes
in some way, such as a change in diameter, or the fluid flows through a bend, valve, or some
other component at section 132. The flow between 122 and 132 is termed fully developed flow. Beyond the interruption of the fully developed flow [at section 142], the flow gradually begins its
CHAPTER 8 – Viscous Flow in Closed Conduits
Fully developed
flow
Entrance region
flow
Boundary layer
Inviscid core
D
r
x
(1)
(2)
(3)
!e
(6)
(5)
x6 – x5
Fully developed
flow
F I G
Figure
60
system.
U R E
8.5
(4)
x5 – x4
Developing
flow
Entrance region, developing flow, and fully developed flow in a pipe
those
nt flow.
a direct result of momentum transfer among the randomly moving molecules 1a m
nomenon2. The shear stress in turbulent flow is largely a result of momentum tran
randomly moving, finite-sized fluid particles 1a macroscopic phenomenon2. The n
the physical properties of the shear stress are quite different for laminar flow tha
CHAPTER
8 – Viscous Flow in Closed Conduits
flow.
p
Fully developed
flow: ∂ p/∂ x = constant
Entrance flow
Entrance
pressure
drop
∆p
x3 – x2 = !
x1 = 0
Figure 61 F I G U R E 8.6
x2 = !e
x3
Pressure distribution along a horizontal pipe.
x
JWCL068_ch08_383-460.qxd
9/23/08
10:53 AM
Page 413
CHAPTER 8 – Viscous Flow in Closed Conduits
8.4
413
Dimensional Analysis of Pipe Flow
0.1
0.09
Wholly turbulent flow
0.08
0.05
0.04
0.07
0.06
0.03
0.05
0.02
0.015
0.04
f
0.01
0.008
0.006
0.03
0.004
0.025
0.002
0.02
0.001
0.0008
0.0006
Laminar
flow
0.0004
0.015
0.0002
Smooth
Transition range
0.0001
0.00005
0.01
0.009
Figure 62
0.008
2(103)
103
4
2(104)
6 8
104
4
2(105)
6 8
4
105
6
2(106)
8
106
4
2(107)
6 8
4
6 8
0.00001
107
ρ VD
Re = _____
µ
F I G U R E 8.20 Friction factor as a function of Reynolds number and relative roughness for round pipes—the Moody
chart. (Data from Ref. 7 with permission.)
∋
__
D

OFFRE COLOPLASTE 2015 - espace medical webstore

Solution