Chapter_2.pdf

Chapter 2

PIPE FLOW

2.1 Generalities

On both historical and pedagogical grounds, fully developed flow in

a smooth pipe of circular cross section is an ideal point of entry for

technical fluid mechanics. The reason is that a crucial quantity, the

friction at the wall, is proportional to the axial pressure gradient,

which is usually easily measured. More than a century ago, exper-

iments by Hagen, Poiseuille, Couette and others used this property

to confirm the hypothesis of Newton that the viscosity of ordinary

fluids, particularly water and air, is a real physical quantity that

depends on the state of the fluid but not on the particular motion.

These early experimenters encountered turbulence in larger facilities

at higher speeds, and the issue quickly became the need for a better

qualitative and quantitative appreciation of turbulence. In fact, it

was an investigation of transition in pipe flow by Reynolds that led

to the discovery of the fundamental dimensionless number that bears

his name.

Study of turbulent flow in smooth round pipes led about 1930

to development of the mixing-length model, which in some situa-

tions still represents the best available approach to the problem of

turbulent flow near a wall. An important mechanism, just beginning

to be understood, involves the effect of the no-slip condition at the

CHAPTER 2.

PIPE FLOW

wall on turbulent fluctuations. Pipe flow is the vehicle of choice for

exploring the effect of wall roughness and the effect of drag-reducing

polymers on turbulent flow. Pipe flow is also a useful vehicle for in-

vestigating heat transfer, with numerous practical applications. Sec-

ondary flow in non-circular pipes exposes the non-Newtonian nature

of the Reynolds stresses, unfortunately without exposing any plau-

sible constitutive relations. Other variations on pipe flow, such as

flow with curvature or flow with an abrupt change in cross section,

reveal strong effects on mixing processes. Similarity laws originally

developed for pipe flow provide a point of contact with the boundary

layer and the wall jet.

The main experimental disadvantage of pipe flow is difficulty of

access for instrumentation, particularly optical instrumentation. A

major source of experimental scatter in fundamental work is failure

to provide sufficient length for the flow to become fully developed.

2.1.1

Equations of motion

The Reynolds equations of mean motion were derived in SECTION

1.3.4 for the cylindrical polar coordinates sketched in FIGURE 2.1.

These equations are easily specialized for the case of steady flow in

a round pipe of radius

. To make the notation consistent with

that for plane flow, take the axial, radial, and azimuthal coordinates

as (

x,r,θ

) and the corresponding velocity components as (

u,v,w

Take the mean flow to be fully developed, which is to say rectilinear.

Thus

= 0, and

= 0. All derivatives of mean quantities

with respect to

are zero, as are all derivatives with respect to

except

∂

p/∂x

, this term being the engine that drives the flow. The

continuity equation is moot. For simplicity, the overbar indicating

a mean value will be suppressed hereafter except in the Reynolds

stresses. The three momentum equations are reduced to

0 =

−

∂p

∂x

(

−

′

)

;

(2.1)

0 =

−

∂p

∂r

(

′

−

r ρ

′

)

;

(2.2)

2.1. GENERALITIES

Figure 2.1:    Cylindrical polar coordinate system and

notation for pipe flow.

0 =

−

′

(2.3)

The fluctuations

′

vanish at the wall. The three equations

(2.1)–(2.3) are rigorously correct, and must be satisfied by the mean

flow under the specified conditions. The equations are obviously not

complete, since there are three equations for six unknown quanti-

ties. Two of the Reynolds stresses,

−

′

and

−

′

, fail to ap-

pear. Of these, the absence of the mean product

′

is expected

if the turbulent motion is random. Given a sampled value for the

axial fluctuation

′

, it is reasonable to suppose, in view of the axial

symmetry of the problem, that positive and negative values for the

azimuthal fluctuation

′

are equally probable. That this argument

can be dangerous will be demonstrated in SECTION 2.5.5, where it

leads to a wrong conclusion for the viscous sublayer near a wall. In

the case of

−

′

, the failure of the Reynolds equations even to con-

tain this streamwise normal stress is embarrassing. It appears that

nothing can be learned about this stress from the laws of mechanics

in Reynolds-averaged form. At the same time, it is this Reynolds

stress

−

′

that is most easily and most commonly measured.

CHAPTER 2.

PIPE FLOW

Consider the three equations of motion (2.1)–(2.3). The third

equation (2.3) has the integral

′

= constant, and the constant

is zero whether evaluated on the axis or at the wall. The second

equation (2.2) is more instructive. Formal integration from

with the boundary condition

at the wall, gives

′

−

∫

(

′

−

′

)

r .

(2.4)

When the lower limit is taken at the pipe axis,

= 0, the integral

diverges unless

′

on the axis. The equality is intuitively

obvious. If two traverses are made along a diameter of the pipe

to measure in one case the radial velocity fluctuation

′

along the

traverse direction and in the other case the azimuthal velocity fluc-

tuation

′

normal to it, the two measurements must be statistically

equivalent on the axis.

As the lower limit

in equation (2.4) approaches the upper

limit

, the definite integral is eventually small compared with the

term

′

on the left. In some vicinity of the wall, therefore, a useful

approximation suggests itself;

′

= constant

(2.5)

This approximation will be examined more closely in SECTION X

Finally, the quantity in parentheses in the first momentum

equation (2.1) is evidently the total shearing stress

, here defined

with a change in sign in the derivative and in

′

because

−

is a more natural independent variable for an observer viewing the

flow from the wall;

−

(

−

′

)

(2.6)

At the wall of the pipe, where all velocity fluctuations vanish, the

corresponding value is

−

(

)

(2.7)

Unclear reference, possibly 5.2.1

2.1. GENERALITIES

By assumption, the terms in parentheses in equations (2.1) and

(2.2) do not depend on

, so that

∂

∂x

∂

∂r∂x

= 0

(2.8)

It follows that

∂p/∂x

is a constant, independent of

and

, although

itself depends on

and also on

if the flow is turbulent, according

to equation (2.4). Equation (2.1) in the form

0 =

−

∂p

∂x

−

rτ

(2.9)

can therefore be integrated from

, with the boundary condition

when

, to obtain

Rτ

−

rτ

−

(

−

)

∂p

∂x

(2.10)

On putting

= 0, this becomes

−

∂p

∂x

(2.11)

where

= 2

is the diameter. The last expression is also easily

obtained from a global momentum balance on a length of the pipe,

given that

∂p/∂x

is constant. Elimination of

∂p/∂x

between equa-

tions (2.10) and (2.11) yields finally the linear stress profile shown in

FIGURE 2.2,

(2.12)

So far, the flow has usually not been specified to be either

laminar or turbulent. In either case, equation (2.11) provides an

accurate and unambiguous method for determining

for fully de-

veloped flow, and this property is the main reason that pipe flow is

discussed here before all other flows involving walls. In a real pipe,

there will be problems with entrance flow and development length,

to be discussed in SECTION 2.2.1. The fact that the stress

de-

fined by equation (2.6) is linear in

also provides an opportunity for

proof and calibration of instruments, such as hot-wire anemometers,

CHAPTER 2.

PIPE FLOW

Figure 2.2:    The parabolic mean-velocity profile and

the linear shearing-stress profile for steady laminar

flow in a circular pipe.

commonly used for measurement of the Reynolds shearing stress.

See, for example, NEWMAN and LEARY (1950), KJELLSTR

and HEDBERG (1970), and PATEL (1974). A more cogent use of

equation (2.12), as one criterion for equilibrium at second order of a

developing turbulent flow, will be taken up in SECTION 2.5.1.

2.1.2

Laminar flow

If the flow is laminar,

is independent of

, according to equation

(2.2), so that

∂p/∂x

becomes d

. The velocity profile is the

integral of equation (2.12), given

−

, that takes the value

(

for centerline) at

= 0 and satisfies the boundary condition

= 0 at

. In dimensionless form, this profile is the parabola

shown in FIGURE 2.2,

= 1

−

(2.13)

2.1. GENERALITIES

where the centerline velocity

is related to the other parameters of

the problem by

= 2

(2.14)

This solution (2.13) of the Navier-Stokes equations for laminar pipe

flow, like solutions of the Stokes approximation for low Reynolds

numbers, represents a balance between pressure forces and viscous

forces. However, the transport terms vanish here because of the

special geometry, not because the Reynolds number is necessarily

small.

Finally, the volume flow

in the pipe can be calculated, and

a mean velocity

defined, from

∫

πru

πR

(2.15)

(the tilde, here and elsewhere, is intended as a mnemonic for an

integral mean value). Given equation (2.13) for

, it follows on inte-

gration that

(2.16)

Several of the earliest experiments with pipe flow and with

flow between concentric rotating cylinders in the 19th century were

undertaken primarily for a reason that may now seem almost unnat-

ural. The equations of NAVIER (1823) and STOKES (1849) inde-

pendently incorporate a hypothesis first proposed by NEWTON in

his

Principia Mathematica

, published in 1687. According to STAN-

TON and PANNELL (1914), the proper attribution to Newton was

first pointed out by Sir George Greenhill, who would presumably

have been at home in both the subject and the Latin language. The

relevant passage introduces Section IX, “The circular motion of flu-

ids,” in Book II, “The motion of bodies.” In the elegant prose of the

revised translation by Cajori (1934):

Hypothesis: The resistance arising from the want of lu-

bricity in the parts of a fluid is, other things being equal,

CHAPTER 2.

PIPE FLOW

proportional to the velocity with which the parts of the

fluid are separated from one another.

This hypothesis is explicit in the tensor relation

def

in the introduction. The scalar constant of proportionality, the vis-

cosity, is assumed to be an intrinsic or state property of the fluid (it

may, for example, vary significantly with temperature), independent

of the motion. In the case of pipe flow, the technique was and is

used to show the existence of such a fluid property by showing that

the quantity

, expressed theoretically for the case of fully developed

laminar pipe flow by a combination of equations (2.11), (2.14), and

(2.16),

−

(2.17)

is experimentally independent of particular choices for

and

, since

these must be precisely compensated for by variations in d

. As

a practical matter, it is usually not the mean velocity

that is mea-

sured, but the volume flux

defined by equation (2.15). Equation

(2.17) is therefore better expressed as

−

πD

128

(2.18)

to emphasize that in capillary-tube viscometry the diameter

needs

to be very accurately known. Many common fluids, including air and

water, possess the property of viscosity in the sense just defined and

are therefore referred to as Newtonian fluids. A second important is-

sue, the validity of the no-slip boundary condition at the wall, has for

practical purposes been resolved experimentally in favor of no slip.

Residual doubts about this condition for the case of non-wetting

combinations, such as mercury on glass, or water on tetrafluoroethy-

lene (teflon), have been mostly quieted by BINGHAM and THOMP-

SON (1928) and by BROCKMAN (1956), respectively. Exceptions

to Newtonian behavior are known, and these present formidable dif-

ficulties in formulating a constitutive relationship between the stress

and rate-of-strain tensors. The relatively unstructured literature of

rheology and of turbulence modeling testifies, in a familiar idiom,

that Newton is a hard act to follow.

2.1. GENERALITIES

The Reynolds number for both laminar and turbulent pipe flow

is commonly defined in terms of the mean velocity

and the pipe

diameter

;

(2.19)

Usage varies in defining a dimensionless friction coefficient (see

SECTION X).

In mechanical and aeronautical engineering, where

the boundary layer and its global momentum balance are in the fore-

ground, the usual form for the dimensionless friction, and the one

that I will adopt here, is

(2.20)

This quantity is also denoted by

and called the Fanning factor

by mechanical engineers. For laminar flow, equations (2.14), (2.16),

(2.19), and (2.20) imply

(2.21)

To the extent that pipe flow can be viewed as a boundary layer

on the inside of a cylindrical body, it might be more consistent to

use

and

instead of

and

in the definition (2.19) for the

Reynolds number, and

instead of

in the definition (2.20) for

the friction coefficient. Probably the usage described here has sur-

vived because no value for

is available for most of the existing

pipe data. In practice, the definition of dimensionless coefficients to

characterize pipe flow over a large range of Reynolds numbers has

been preempted by the turbulent case, for which the wall friction is

only weakly dependent on the viscosity, and the dynamic pressure is

the important reference quantity.

In principle, both

and

are easily measured for fully de-

veloped pipe flow, the former in terms of

∂p/∂x

and the latter by a

variety of methods. These include direct evaluation from the integral

definition (2.15), if a velocity profile is available; or measurement of

volume flow rate, by weighing or by use of a calibrated volume if

Possibly section 2.4.1

CHAPTER 2.

PIPE FLOW

the fluid is a liquid; or by blowdown techniques if the fluid is a gas.

Other methods include use of a calibrated venturi, orifice plate, or

other type of flow meter; or, for best regulation, use of a constant-

displacement pump or even a piston-cylinder displacement mecha-

nism.

2.1.3

An extremum principle

It was pointed out by LIN (1952), in a short paper whose roots

lie in work by HELMHOLTZ (1868) and KORTEWEG (1883) on

arbitrarily slow steady viscous motions, that the parabolic laminar

profile in a round pipe can be obtained from an extremum principle.

Consider all possible axisymmetric rectilinear motions

(

) satisfying

the no-slip condition at the wall, and minimize the total rate of

energy dissipation,

Φ =

∫

πr

(

)

r ,

(2.22)

subject to the constraint of a constant volume flux,

∫

πru

πR

= constant

(2.23)

The notation

Φ means a volume integral of the local rate of dissi-

pation over the pipe cross section and over unit length in the flow

direction.

The problem just formulated is an example of what COURANT

and HILBERT (1953, Vol. 1, Chapter IV) call the simplest problem

in the variational calculus. This is to find the function

(

) that

minimizes the integral

Φ =

∫

(

r, u, u

′

r ,

(2.24)

2.1. GENERALITIES

subject to the constraint

∫

(

r, u, u

′

= constant

(2.25)

The prime here indicates differentiation with respect to

. The Euler

equation for the problem is

(

′′

∂

∂u

′

∂u

′

∂

∂u∂u

′

∂

∂r∂u

′

−

∂

∂u

)

(

λG

) = 0

(2.26)

where

is a Lagrange multiplier. In the present case, with

= 2

πμr

(

′

)

, G

= 2

πru ,

(2.27)

equation (2.26) becomes

(

)

−

λr

= 0

(2.28)

The indefinite integral of this equation is

(

λr

)

(2.29)

where

and

are constants of integration. A logarithmic term

appears in equation (2.29), but not in equation (2.13), because the

earlier derivation already required

(i.e., d

) to be linear in

The logarithmic term is needed if the no-slip boundary condition is

to be satisfied for the more general case of axial flow in an annulus.

For flow in a pipe, the boundary conditions

= 0 and

= 0 at

require

= 0 and

= 2

μu

−

λR

4. Equation

(2.14) is recovered,

= 1

−

(2.30)

with now

−

μu

(2.31)

CHAPTER 2.

PIPE FLOW

Use of the three equations (2.31), (2.14), and (2.11) implies

= 2

(2.32)

as does a direct comparison of equation (2.28) with the laminar ver-

sion of the momentum equation (2.1).

For the parabolic profile in a pipe, therefore, the rate of dissi-

pation is an extremum. That the extremum is a minimum is easily

shown by adding to the parabolic profile an arbitrary axisymmetric

perturbation that vanishes at the wall and does not contribute to

the volume flux. The result (2.32), derived here for a very special

motion of an incompressible viscous fluid, should be read in the same

sitting as SOMMERFELD’s argument (1950, pp. 89–92) that for an

incompressible inviscid fluid the pressure

can be interpreted as a

Lagrange multiplier representing the constraint of incompressibility.

Finally, the pressure gradient and the rate of dissipation can

be related directly for fully developed pipe flow, whether laminar or

turbulent. The general form for energy loss from the mean flow per

unit time and per unit volume is

(cite introduction)

grad

, so

that equation (2.22) can be written

Φ =

∫

πrτ

r .

(2.33)

Integration by parts, with

= 0 at

= 0 and

= 0 at

, gives

Φ =

−

∫

rτ

r .

(2.34)

Use of equation (2.12) for

, (2.15) for the volume flow

, and (2.11)

for

then gives

Φ =

−

∂p

∂x

(2.35)

The point of this exercise in the calculus of variations for lam-

inar pipe flow is that a similar principle may hold for turbulent flow.

If so, it would not be surprising if the resulting mean velocity profile

turned out to be logarithmic.

2.2. DEVELOPMENT LENGTH

Figure 2.3:    A schematic diagram of flow in the

laminar development region of a circular pipe with

uniform entrance flow.

2.2 Development length

In any attempt to define the properties of a hypothetical fully de-

veloped or equilibrium flow experimentally, a useful first step is to

observe the rate of approach to equilibrium from a disturbed ini-

tial condition. In the case of pipe flow, the process most commonly

observed is the evolution of the flow downstream from the pipe en-

trance. The response to other disturbances, such as a step change in

diameter, can also provide estimates of a characteristic time scale or

spatial scale for approach to equilibrium.

2.2.1

Laminar flow

The parabolic profile requires some distance to develop in the en-

trance region of a real pipe, especially at large Reynolds numbers,

and the need for adequate development length is not always recog-

nized by experimenters. Let this development length be estimated

approximately in terms of the inward growth of internal laminar

boundary layers that start at the pipe entrance at

= 0, as shown

in FIGURE 2.3. Note that the artist was apparently not told of the

doubling of the velocity on the pipe axis during laminar develop-

CHAPTER 2.

PIPE FLOW

ment. Global parameters available for making the estimate dimen-

sionless are the kinematic viscosity

, the pipe diameter

, and the

mean velocity

. That the latter quantity is independent of

in the

development region can be shown by reinstating the axisymmetric

continuity equation in the form

∂u

∂x

∂rv

∂r

= 0

(2.36)

and calculating the derivative of

from the definition (2.15); thus

πR

∫

πr

∂u

∂x

−

∫

∂rv

∂r

= 0

(2.37)

since

vanishes at both limits.

A qualitative estimate of development length in terms of dif-

fusion time and transport time can be obtained by using the device

of a moving observer applied to a growing internal boundary layer.

Assume that the entrance is cut square with the axis so that the

origin for

is well defined. Take the flow to be uniform at the pipe

entrance; i.e.,

= 0. Vorticity generated at the wall by

the axial pressure gradient diffuses inward through a distance

in a

time

∼

/ν

(see the Rayleigh problem in the introduction)

An observer following the mean flow travels a distance

in a time

. At equal times,

x/δ

∼

uδ/ν

. If the development length

is defined as the value of

when

2, then

is proportional to

DRe

and

x/X

is proportional to

x/DRe

Since the first papers by BOUSSINESQ (1890a,b,c; 1891a,b),

work on the problem of laminar flow development in a round pipe

has become almost a cottage industry. Of more than forty exper-

imental, analytical, or numerical papers on this topic, about half

aim at estimates of a particular constant

belonging to the art

of capillary-tube viscometry (see the next SECTION 2.2.2). The

remainder view the problem as an exercise in classical fluid mechan-

ics. The first analysis using boundary-layer theory was carried out

by SCHILLER (1922). This paper was a natural application of the

integral method of Karman and Pohlhausen, published a year ear-

lier. The model combined parabolic profiles near the wall with a

2.2. DEVELOPMENT LENGTH

X/(R&)D

FIGURE

3.25.

SOme

analytical

results

for

laminar

flow

development

smooth

pipe.

The

dependent

variable

uc/u

has

asymptotic

limit

Figure 2.4:    Some analytical results for laminar flow

development in a smooth pipe.  The dependent variable

has an asymptotic limit of 2.

flat profile near the centerline. The predicted evolution of the axial

velocity on the pipe axis is one of the curves shown in FIGURE 2.4.

The analysis ignores the fact that a boundary-layer approach is com-

plicated by effects of lateral curvature and by interaction between

the axial pressure gradient in the inviscid core flow and the grow-

ing displacement thickness of the boundary layer, each affecting the

other. Moreover, the boundary-layer approximation fails before the

flow development is complete, and an asymptotic analysis is even-

tually required. The growing power of computers now allows the

laminar problem to be treated by numerical integration of the full

Navier-Stokes equations for quite large Reynolds numbers.

The other data in FIGURE 2.4 are chosen from the work of

DORSEY (1926)

ULLER (1936) figure 4

ATKINSON and GOLDSTEIN (1938)

LANGHAAR (1942) figure 1

ASTHANA (1951)

TATSUMI (1952)

SIEGEL (1953)

CHAPTER 2.

PIPE FLOW

RIVAS and SHAPIRO (1956)

BOGUE (1959)

TOMITA (1961)

CAMPBELL and SLATTERY (1963)

COLLINS and SCHOWALTER (1963)

LUNDGREN et al. (1964)

SPARROW et al. (1964) figure 3

CHRISTIANSEN and LEMMON (1965) figures 2, 3

HORNBECK (1965) figures 3, 6

McCOMAS and ECKERT (1965)

VRENTAS et al. (1966) figures 3–8

McCOMAS (1967)

FRIEDMANN et al. (1968) figures 1–4

LEW and FUNG (1968) tables

SCHMIDT and ZELDIN (1969)

FARGIE and MARTIN (1971) figure 4

CHEN (1973) figure 1

KESTIN et al. (1973) figure 8

KANDA and OSHIMA (1986) figures 5–8

The main imperfection of these contributions lies in the fre-

quent assumption of a flat velocity profile at the station

= 0, with

an unphysical upstream flow that is often left undefined. However,

there is general agreement that the independent variable

x/DRe

a natural and appropriate one in the laminar development region,

whether a boundary-layer model is used or not.

A few experiments include the work of

BOND (1921)

RIEMAN (1928)

ZUCROW (1929)

KLINE and SHAPIRO (1953)

SHAPIRO et al. (1954)

KREITH and EISENSTADT (1957)

WELTMANN and KELLER (1957)

RESHOTKO (1958) figure 9

PFENNINGER (1961) figure 12

2.2. DEVELOPMENT LENGTH

McCOMAS and ECKERT (1965) figure 3

ATKINSON et al. (1967) figure 4

DAVIS and FOX (1967) figure 11

EMORY and CHEN (1968)

BERMAN and SANTOS (1969) figures 3–7

BURKE and BERMAN (1969) figures 3–6, 8

FARGIE and MARTIN (1971) figure 4

WYGNANSKI and CHAMPAGNE (1973)

MESETH (1974)

MOHANTY and ASTHANA (1979)

(

Why does nobody plot

? Mention honeycombs. No calcula-

tions for square-cut entrance with bubble.)

From all of this work, the evidence is persuasive that develop-

ment is complete for practical purposes when

≈

(2.38)

provided that the Reynolds number

is not less than about 100.

For a pipe of given diameter and length, the largest Reynolds number

for which a parabolic profile can be established is about 14

L/D

. For

a given Reynolds number, the smallest

L/D

is about

Re/

14. Con-

sequently, to identify the record holder for highest Reynolds number

with fully developed laminar flow, it is necessary only to look for

large values of

L/D

and to test the state of the exit flow. By this

criterion, the record is about

uD/ν

= 13

000 and is held by LEITE

(1958).

With sufficient care, it is possible to maintain laminar flow

in relatively short pipes up to very large Reynolds numbers, 50,000

or more. However, the parabolic profile is never fully developed in

such cases, and the decisive element for transition becomes the dis-

turbance level outside the boundary layer in the early development

region. In any contest to establish the highest achievable laminar

Reynolds number in short pipes, the results are a measure of the de-

gree of care taken to avoid such disturbances, rather than of any di-

rectly useful physical quantity. For example, blowdown methods can

reduce disturbance levels far below the best obtainable in flows driven

CHAPTER 2.

PIPE FLOW

by an upstream pump or compressor, and a gas flow can be quieted

by use of a sonic orifice to isolate the test section from a downstream

pump. As far as I know, the record here is held by PFENNINGER

and MEYER (1953), who used a long conical contraction fitted with

13 screens, as well as elaborate vibration isolation, to obtain a flow

free of turbulence at a Reynolds number

uD/ν

= 88

000. The corre-

sponding number

ux/ν

was about 50

, an order of magnitude

higher than the number that can be obtained on flat plates in con-

ventional wind tunnels. The boundary layer in Pfenninger’s pipe

occupied about 60 percent of the area at the pipe exit. To obtain a

fully developed parabolic profile in air under these conditions would

need a length of about 140 meters, together with heroic measures

to minimize disturbances in the flow and to account for changes in

density. Even if this could be done, I see no particular profit from

an attempt.

Finally, several experimenters have found that it is not safe to

assume that a laminar pipe flow in the laboratory will automatically

be axisymmetric. LEITE (1958), RESHOTKO (1958), HOULIHAN

(1969), WYGNANSKI and CHAMPAGNE (1973), and BREUER

(1985), all working on difficult questions of stability and transition,

and all working with air in pipes having a diameter of a few centime-

ters, describe problems in obtaining axisymmetric fully-developed

flow. Some of their results are illustrated in FIGURE 2.5. They

comment in particular on the need for close control of thermal in-

homogeneities, pipe alignment, and axial symmetry of conditions far

upstream. Most other experimenters have had a better experience

or have not tested their flows for symmetry.

2.2.2

Capillary-tube viscometry

Numerous devices are commercially available for measuring the vis-

cosity of fluids used in laboratory experiments or manufacturing pro-

cesses. They include cone-plate, rotating-cylinder, falling-ball, and

capillary-tube viscometers. The choice is usually governed by the

range of viscosity involved, by the volume of the sample committed

to the measurement, and occasionally by special conditions such as

2.2. DEVELOPMENT LENGTH

Figure 2.5:    Several examples from the experimental

literature showing non-axisymmetric laminar flow in

circular pipes.  The most likely cause is secondary flow

due to thermal inhomogeneity in a gravitational field or

asymmetry of the upstream channel.

CHAPTER 2.

PIPE FLOW

a need in process work to immerse the device in the bulk fluid. Typ-

ically, such viscometers are calibrated by the manufacturer or the

user, using fluids of known viscosity, to take into account various

effects of finite geometry.

The question of how a known viscosity comes to be known is

far from trivial. This question is the business of standards labora-

tories. For example, SWINDELLS, COE, and GODFREY (1952)

describe a painstaking program in capillary-tube viscometry, carried

on from 1931 to 1941 and from 1947 to 1952 at the U.S. National

Bureau of Standards (now the U.S. National Institute of Standards

and Technology), whose singular result was to establish the value

= 0

010019

000003 poise for pure water at 20

◦

. This value is

still the cornerstone of viscosity tables for water that are constructed

from more extensive but less accurate measurements. The geometry

in capillary viscometry at the standards level is apparently stan-

dardized, requiring a square-cut entrance and exit, although other

geometric details, such as the tube outer diameter, are usually left

open. The need for close control of several variables is self-evident. In

order to obtain four significant figures in the viscosity, according to

equation (2.18), the absolute temperature, the volume flow rate, the

tube length, the pressure difference, and especially the tube diameter

must all be known to better accuracy, with a further allowance for

errors introduced by differentiation of experimental data. It may be

necessary, for example, to represent the cross section of a real tube

by an ellipse rather than by a circle, with a corresponding adjust-

ment in the theory. A long tube of small diameter provides an easily

measurable pressure difference, although this pressure difference is

necessarily global rather than local if pressure taps cannot be pro-

vided along the length of the tube. At the same time, the diameter

must not be so small that it cannot be measured with the necessary

accuracy. The length is also limited in practice by the need to main-

tain the flow system at constant temperature in a bath of practical

size without having to coil the tube and accept a much more complex

theory.

The expected pressure distribution in a finite tube is shown

schematically in FIGURE 2.6. Certain general relationships that

2.2. DEVELOPMENT LENGTH

Figure 2.6:    A schematic representation of the

pressure distribution along a tube of length

with

uniform entrance flow at

= 0.  The flow may be laminar

or turbulent and the cross section need not be circular.

emerge from the figure do not depend on the shape of the tube cross

section or the shape of the tube entrance, as long as these are fixed.

Suppose for convenience that entrance and exit are cut square, so

that

is well defined. The quantities

and

are the static pres-

sures in the entrance and exit reservoirs. The quantity

′

is the

static entrance pressure associated with a fictitious fully developed

flow over the full length of the tube. Anomalous pressure effects at

both entrance and exit are included in principle, although exit effects

are not treated explicitly. It is not necessary to distinguish between

laminar and turbulent flow, or even mixed flow at Reynolds numbers

in the transition regime.

The strategy associated with FIGURE 2.6 is one of experi-

mental differentiation. Suppose that the Reynolds number is held

constant as the length of the tube is varied, and suppose that the

shortest tube is long enough to achieve fully developed flow. Then

a unit increment in the length of the tube will give rise to a unit

CHAPTER 2.

PIPE FLOW

increment in total pressure drop. Begin with the identity

−

= (

−

′

) + (

′

−

)

(2.39)

For the general case, define an ideal friction coefficient for fully de-

veloped flow,

(

′

−

)

Lρ

(2.40)

where

Q/A

is again a mean velocity over the cross section

For the special case of a circular tube, the definition (2.40) reduces

through equation (2.11) to equation (2.20). Define also a global

friction coefficient,

(

−

)

Lρ

(2.41)

where the circumflex over

denotes a mean value over the entire

length of the tube, reservoir to reservoir. Equation (2.39) then takes

the form

(

−

′

)

(2.42)

For fully developed flow at the exit, the first term on the right is a

constant that is characteristic of the complete development process

for particular entrance conditions and Reynolds number. Without

the factor of 2 in the denominator, this constant is usually denoted

in the literature of capillary-tube viscometry;

(

−

′

)

(2.43)

so that

(2.44)

Except possibly for a square-cut entrance, the development parame-

ter

, defined graphically in figure 2.6, will depend on the entrance

geometry and flow conditions for a particular tube. A plot of

L/D

against

L/D

is a straight line whose slope

and intercept

2 are

characteristic for the Reynolds number in question. In particular,

∂

(

L/D

)

∂

(

L/D

)

(2.45)

2.2. DEVELOPMENT LENGTH

A simple implementation of this differentiation scheme was

proposed and used by COUETTE (1890), but has since been used

only rarely, and then mostly by professionals such as Swindells

et al

The technique is to construct two capillary viscometers by connect-

ing two tubes in series with a third reservoir inserted between them.

The two tubes are assumed to be identical in diameter and all other

geometrical details except length. When in equilibrium, they have

the same flow rate. The two parameters

and

are then readily

evaluated from a single experiment.

Discuss

from

(

)

. Cite bibliography.

Equation (2.44) is usually cast in a more direct and more literal

form for use in viscometry with a round tube. From equation (2.39),

with

′

−

= 4

Lτ

= 8

uL/R

, and with

Q/πR

, this form is

(

−

)

ρQ

πμL

ρQ

(2.46)

For constant

and

and variable

, a dimensional plot of (

−

)

against

yields a straight line whose slope is proportional to

and

whose intercept is proportional to

. Such a plot is a practical

realization of FIGURE 2.6.

A different form, appropriate for a single tube with changing

flow rate, is obtained on multiplying by

and making the important

assumption that

is independent of Reynolds number;

(

−

)

ρQ

πμL

(2.47)

Now, for constant

and

and variable

, a plot of (

−

)

against

yields a straight line whose slope is proportional to

and

whose intercept is proportional to

. This formulation is the one used

by Swindells et al. Such a plot is sometimes called a Knibbs plot,

after KNIBBS (1895). The assumption that the entrance parameter

defined by equation (2.44) is independent of Reynolds number in

some range of

needs verification for both smooth and square-cut

entrances, since the parameters

, and

define two lengths,

and

ν/

, together with their ratio

CHAPTER 2.

PIPE FLOW

Finally, still for a single tube, multiplication of equation (2.47)

ρQ

yields

(

−

) =

ρmQ

+ 8

πμLQ .

(2.48)

The conclusion that the overall pressure drop should be the sum of

a term in

(or

) and a term in

(or

) was noted by HAGEN

(but not by Poiseuille) and is probably the reason that the term

containing

is often referred to, not quite correctly, as a kinetic-

energy correction.

For the case of a circular tube, it is also possible to calculate

−

′

in FIGURE 2.6 from first principles, given a suitable model

of the flow in the development region. By definition,

(

−

′

)

(

−

) + (

−

)

−

(

′

−

)

(2.49)

Visualize an ideal entrance, for which the velocity profile at an initial

station

= 0 is uniform, with velocity

and pressure

that do not

depend on

. Assume frictionless acceleration from rest upstream of

this station. The Bernoulli equation then determines the first term

on the right in equation (2.49);

−

(2.50)

The middle term can be partially evaluated for laminar flow.

For steady axisymmetric laminar flow in the development region,

the equations of motion

(see introduction)

can be written, with

no boundary-layer approximation, as

∂ru

∂x

∂rv

∂r

= 0 ;

(2.51)

(

∂ruu

∂x

∂ruv

∂r

)

−

∂p

∂x

μr

(

∂

∂x

∂

∂r

∂u

∂r

)

(2.52)

When both equations are integrated over a cross section of a tube of

constant diameter, the result can be reduced to a simple momentum