Chapter 5
VISCOUS FLOW:
PIPERS AND CHANNELS
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Chapter 3: the fluid was considered frictionless, or in some cases
losses were assumed or computed without probing into their
underlying causes.
This chapter deals with real fluids, i.e., situations in which
irreversibilities are important.
Viscosity is the fluid property that causes shear stresses in a moving
fluid; it is also one means by which irreversibilities or losses are
developed. In turbulent flows random fluid motions, superposed on
the average, create apparent shear stresses that are more important
than those due to viscous shear.
This chapter:
 concept of the Reynolds number developed
 characteristics that distinguish laminar from turbulent flow are
presented, and the categorization of flows into internal versus
external is established
 concentrates on internal-flow cases.
 steady, laminar, incompressible flows are first developed, since
the losses can be computed analytically.
5.1 LAMINAR AND TURBULENT FLOWS;
INTERNAL AND EXTERNAL FLOWS
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Laminar flow: fluid moves in layers, or laminas, one layer gliding
smoothly over an adjacent layer with only a molecular interchange of
momentum.
 Any tendencies toward instability and turbulence are damped out
by viscous shear forces that resist relative motion of adjacent
fluid layers.
Turbulent flow has very erratic motion of fluid particles, with a violent
transverse interchange of momentum.
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The nature of the flow, i.e., whether laminar or turbulent, and its
relative position along a scale indicating the relative importance of
turbulent to laminar tendencies are indicated by the Reynolds
number.
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This section: the concept of the Reynolds number
Sec. 3.5: an equation of motion was developed with the assumption
that the fluid is frictionless, i.e., that the viscosity is zero. More
general equations that include viscosity have been developed by
including shear stresses.
These equations (see Sec. 5.2) are complicated nonlinear partial
differential equations for which no general solution has been
obtained.
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Two flow cases are said to be dynamically similar when
1)
They are geometrically similar, i.e., corresponding linear
dimensions have a constant ratio.
2)
The corresponding streamlines are geometrically similar, or
pressures at corresponding points have a constant ratio.
In considering two geometrically similar flow situations, Reynolds
deduced that they would be dynamically similar if the general
differential equations describing their flow were identical. By
changing the units of mass, length, and time in one set of
equations and determining the condition that must be satisfied to
make them identical to the original equations, Reynolds found that
the dimensionless group ulρ/μ must be the same for both cases.
The quantity u is a characteristic velocity, l a characteristic length,
ρ the mass density, and μ the viscosity. This group, or parameter,
is now called the Reynolds number R,
(5.1.1)
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To determine the significance of the dimensionless group, Reynolds
conducted his experiments on flow of water through glass tubes (Fig.
5.1)
 A glass tube was mounted horizontally with one end in a tank
and a valve on the opposite end. A smooth bellmouth entrance
was attached to the upstream end, with a dye jet so arranged
that a fine stream of dye could be ejected at any point in front of
the bellmouth. Reynolds took the average velocity V as
characteristic velocity and the diameter of tube D as
characteristic length, so that R = VDρ/μ.
Figure 5.1
Reynolds apparatus
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For small flows the dye stream moved as a straight line through the
tube, showing that the flow was laminar. As the flow rate increased,
the Reynolds number increased, since D, ρ, μ, were constant and V
was directly proportional to the rate of flow.
With increasing discharge a condition was reached at which the dye
stream wavered and then suddenly broke up and was diffused
throughout the tube. The flow had changed to turbulent flow with its
violent interchange of momentum that had completely disrupted the
orderly movement of laminar flow.
By careful manipulation Reynolds was able to obtain a value R =
12,000 before turbulence set in.
A later investigator, using Reynolds' original equipment, obtained a
value of 40,000 by allowing the water to stand in the tank for several
days before the experiment and by taking precautions to avoid
vibrations of the water or equipment.
These numbers, referred to as the Reynolds upper critical numbers,
have no practical significance in that the ordinary pipe installation
has irregularities that cause turbulent flow at a much smaller value of
the Reynolds number.
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Starting with turbulent flow in the glass tube, Reynolds found that it
always becomes laminar when the velocity is reduced to make R less
than 2000. This is the Reynolds lower critical number for pipe flow and
is of practical importance.
With the usual piping installation, the flow will change from laminar to
turbulent in the range of Reynolds numbers from 2000 to 4000. For the
purpose of this treatment it is assumed that the change occurs at R =
2000.
In laminar flow the losses are directly proportional to the average
velocity, while in turbulent flow the losses are proportional to the
velocity to a power varying from 1.7 to 2.0.
There are many Reynolds numbers in use today in addition to the one
for straight round tubes. For example, the motion of a sphere through a
fluid can be characterized by uDρ/μ, in which U is the velocity of sphere,
D is the diameter of sphere, and ρ and μ are the fluid density and
viscosity.
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The nature of a given flow of an incompressible fluid is characterized
by its Reynolds number.
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For large values of R:
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One or all of the terms in the numerator are large compared with the
denominator  large expanse of fluid, high velocity, great density,
extremely small viscosity, or combinations of these extremes.
The numerator terms are related to inertial forces, or to forces set up by
acceleration or deceleration of the fluid; the denominator term is the
cause of viscous shear forces  the Reynolds number parameter can
also be considered as a ratio of inertial to viscous forces.
Large R indicates a highly turbulent flow with losses proportional to the
square of the velocity. The turbulence may be fine-scale, composed of
a great many small eddies that rapidly convert mechanical energy into
irreversibilities through viscous action; or it may be large-scale, like the
huge vortices and swirls in a river or gusts in the atmosphere. The
large eddies generate smaller eddies, which in turn create fine-scale
turbulence. Turbulent flow may be thought of as a smooth, possibly
uniform flow, with a secondary flow superposed on it. In general, the
intensity of turbulence increases as the Reynolds number increases.
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For intermediate values of R both viscous and inertial effects are
important, and changes in viscosity change the velocity distribution and
the resistance to flow.
For the same R, two geometrically similar closed-conduit systems (one,
say, twice the size of the other) will have the same ratio of losses to
velocity head. The Reynolds number provides a means of using
experimental results with one fluid to predict results in a similar case
with another fluid.
Internal and External Flows
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Another method of categorizing flows is by examining the extent of
the flow field.
 Internal flow involves flow in a bounded region, as the name
implies.
 External flow involves fluid in an unbounded region in which the
focus of attention is on the flow pattern over a body immersed in
the fluid.
The motion of a real fluid is influenced significantly by the presence
of the boundary.
 Fluid particles at the wall remain at rest in contact with the wall. In
the flow field a strong velocity gradient exists in the vicinity of the
wall, a region referred to as the boundary layer.
 A retarding shear force is applied to the fluid at the wall, the
boundary layer being a region of significant shear stresses.
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This chapter deals with flows constrained by walls in which the
boundary effect is likely to extend through the entire flow.
The boundary influence is easily visualized at the entrance to a pipe
from a reservoir (Fig. 5.2).
Figure 5.2 Entrance zone of pipeline.
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At section A-A, near a well-rounded entrance, the velocity profile is
almost uniform over the cross section. The action of the wall shearing
stress is to slow down the fluid near the wall. As a consequence of
continuity, the velocity must increase in the central region. Beyond a
transitional length L' the velocity profile is fixed since the boundary
influence has extended to the pipe centerline. The transition length is a
function of the Reynolds number; for laminar flow Langhaar [2]
developed the theoretical formula
(5.1.2)
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In turbulent flow the boundary layer grows more rapidly and the
transition length is considerably shorter than given by Eq. (5.1.2).
In external flows, with an object in an unbounded fluid, the frictional
effects are confined to the boundary layer next to the body (examples:
golf ball in flight, an airfoil, and a boat)
The fully developed velocity profile, presented in Fig. 5.2 for an internal
flow, is unlikely to exist in external flows. Typically interest is focused
on drag forces on the object or the lift characteristics developed on the
body by the particular flow pattern.
5.2 NAVIER-STOKES EQUATIONS
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The equations of motion for a real fluid can be developed from
consideration of the forces acting on a small element of the fluid,
including the shear stresses generated by fluid motion and viscosity.
The derivation of these equations, called the Navier-Stokes equations,
is beyond the scope of this treatment. They ire listed, however, for the
sake of completeness, and many of the developments of this chapter
could be made directly from them.
First, Newton's law of viscosity, Eq. (1.1.1), for one-dimensional
laminar flow can be generalized to three-dimensional flow (Stokes'
law of viscosity)
(5.2.1)
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The first subscript of the shear stress is the normal direction to the
face over which the stress component is acting. The second subscript
is the direction of the stress component.
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Chap. 3: in developing the Euler and energy equations, z was taken
as the vertical coordinate, so that z was a measure of potential
energy per unit weight.
In dealing with problems in Chaps. 5 and 8 it is convenient to allow
the x, y, z system of right-angular coordinates to take on any arbitrary
orientation.
Since gravity, the only body force considered, always acts vertically
downward, h is taken as a coordinate which is positive vertically
upward; then dh/dx is the cosine of the angle between the x axis and
the h axis, and similarly for the y and z axes.
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When the Navier-Stokes equations are limited to incompressible
fluids, they become
(5.2.2)
(5.2.3)
(5.2.4)
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in which v is the kinematic viscosity, assumed to be constant, d/dt is
differentiation with respect to the motion
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and
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For a nonviscous fluid, the Navier-Stokes equations reduce to the
Euler equations of motion in three dimensions, (8.2.3) to (8.2.5).
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For one-dimensional flow of a real fluid in the l direction (Fig. 5.3)
with h vertically upward and y normal to l (v = 0, w = 0, du/dl = 0), the
Navier-Stokes equations reduce to
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For steady flow
Figure 4.4
Model tests
showing the
influence of a
bulbous bow
on bow wave