10.1-Classifying Flow, fluid mech
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Classifying Flow
10.1 Classifying Flow
This section describes how to classify flow in a conduit by considering (a) whether the flow is laminar or
turbulent, and (b) whether the flow is developing or fully developed. Classifying flow is essential for selecting
the proper equation for calculating head loss.
Laminar Flow and Turbulent Flow
Flow in a conduit is classified as being either laminar or turbulent, depending on the magnitude of the Reynolds
number. The original research involved visualizing flow in a glass tube as shown in Fig. 10.1
a
. Reynolds 1 in
the 1880s injected dye into the center of the tube and observed the following:
·
When the velocity was low, the streak of dye flowed down the tube with little expansion, as shown in Fig.
10.1
b
. However, if the water in the tank was disturbed, the streak would shift about in the tube.
·
If velocity was increased, at some point in the tube, the dye would all at once mix with the water as shown
in Fig. 10.1
c
.
·
When the dye exhibited rapid mixing (Fig. 10.1
c
), illumination with an electric spark revealed eddies in the
mixed fluid as shown in Fig. 10.1
d
.
Figure 10.1
Reynolds' experiment.
(a) Apparatus.
(b) Laminar flow of dye in tube.
(c) Turbulent flow of dye in tube.
(d) Eddies in turbulent flow.
The flow regimes shown in Fig. 10.1 are laminar flow (Fig. 10.1
b
) and turbulent flow (Figs. 10.1
c
and 10.1
d
).
Reynolds showed that the onset of turbulence was related to a πgroup that is now called the Reynolds number
(Re = ρ
VD
/) in honor of Reynolds' pioneering work. Reynolds discovered that if the fluid in the upstream
reservoir was not completely still or if the pipe had some vibrations, then the change from laminar to turbulent
flow occurred at Re ~ 2100. However, if conditions were ideal, it was possible to reach a much higher Reynolds
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Classifying Flow
number before the flow became turbulent. Reynolds also found that, when going from high velocity to low
velocity, the change back to laminar flow occurred at Re ~ 2000. Based on Reynolds' experiments, engineers use
guidelines to establish whether or not flow in a conduit will be laminar or turbulent. The guidelines used in this
text are as follows:
(10.1)
In Eq. (10.1), the middle range (2000 ≤ Re ≤ 3000) corresponds to a the type of flow that is unpredictable
because it can changes back and forth between laminar and turbulent states. Recognize that precise values of
Reynolds number versus flow regime do not exist. Thus, the guidelines given in Eq. (10.1) are approximate and
other references may give slightly different values. For example, some references use Re = 2300 as the criteria
for turbulence.
There are several equations for calculating Reynolds number in a pipe.
(10.2)
These equations are derived by using the definition of Re, the definition of kinematic viscosity from Eq. (2.8),
and the flow rate equations from Eqs. (5.8 and 5.9).
Developing Flow and Fully Developed Flow
Flow in a conduit is classified as being developing flow or fully developed flow. For example, consider laminar
fluid entering a pipe from a reservoir as shown in Fig. 10.2. As the fluid moves down the pipe, the velocity
distribution changes in the streamwise direction as viscous effects cause the plugtype profile to gradually
change into a parabolic profile. This region of changing velocity profile is called
developing flow
. After the
parabolic distribution is achieved, the flow profile remains unchanged in the streamwise direction, and flow is
called
fully developed flow
.
The distance required for flow to develop is called the
entrance length
(
L
e
) This length depends on the shear
stress that acts on the pipe wall. For laminar flow, the wall shearstress distribution is shown in Fig. 10.2. Near
the pipe entrance, the radial velocity gradient (change in velocity with distance from the wall) is high, so the
shear stress is large. As the velocity profile progresses to a parabolic shape, the velocity gradient and the wall
shear stress decrease until a constant value is achieved. The entry length is defined as the distance at which the
shear stress reaches to within 2% of the fully developed value. Correlations for entry length are
(10.3a)
(10.3b)
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Figure 10.2
In developing flow, the wall shear stress is changing. In fully developed flow, the wall
shear stress is constant.
Eq. ((10.3)) is valid for flow entering a circular pipe from a reservoir under quiescent conditions. Other
upstream components such as valves, elbows, and pumps produce complex flow fields that require different
lengths to achieve fully developing flow.
In summary, flow in a conduit is classified into four categories: laminar developing, laminar fully developed,
turbulent developing, or turbulent fully developed. The key to classification is to calculate the Reynolds number
as shown by Example 10.1.
EXAMPLE 10.1 CLASSIFYIG FLOW I CODUITS
Consider fluid flowing in a round tube of length 1 m and diameter 5 mm. Classify the flow as laminar
or turbulent and calculate the entrance length for (a) air (50°C) with a speed of 12 m/s and (b) water
(15°C) with a mass flow rate of
.
PROBLEM DEFINITION
Situation:
Fluid is flowing in a round tube (two cases given).
Find:
1. Whether each flow is laminar or turbulent.
2. Entrance length (in meters) for each case.
Properties:
1.
Air (50°C), Table A.3, ν = 1.79 × 10
5
m
2
/s.
2.
Water (15°C), Table A.5, = 1.14 × 10
3
N · s/m
2
.
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Sketch:
Assumptions:
1. The pipe is connected to a reservoir.
2. The entrance is smooth and tapered.
PLAN
1. Calculate the Reynolds number using Eq. (10.2).
2. Establish whether the flow is laminar or turbulent using Eq. (10.1).
3. Calculate the entrance length using Eq. ((10.3)).
SOLUTION
(a) Air
Since Re > 3000, the
(b) Water
Since Re < 2000, the
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