THE SECONDARY FLOW IN RECTANGULAR DUCTS
by
RAPHAEL MOISSIS
B. Sc. Tech., The Victoria University of Manchester
(1956)
SU3MITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF
SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1957
Author
Department ptf
Signature redacted
Certified by....
dby.
dhairman, Dor
1
.
Engineering, May 20, 1957
.
nature nf
.
Ri
ec i.a-n
Signature redacted
A1
/1/
Thesis Suoervisor
Signature redacted
ental Committee on G
duate Students
Ttc
OCT 25 1957
L IBRA
Graduate House
M. I. T.
Camoridge 39, Mass.
May 20, 1957
,Mr. Leicester F. Hamilton
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge 39, Massachusetts
Dear Sir:
I hereby submit in partial fulfillment of the requirements for the degree of Master of Science this thesis
entitled "The Secondary Flow in Rectangular Ducts."
Very truly yours,
Signature redacted
Raphael Moissis
ABSTRACT
The Secondary Flow in Rectangular Ducts
by
Raphael Moissis
Submitted to the Department of Mechanical Engineering on
May 20, 1957 in partial fulfillment of the requirements for
the degree of Master of Science.
14
An analysis is presented by which it is shown that
secondary flows cannot exist in fully developed laminar flow
through a straight tube of any cross-sectional shape.
The proof is derived for the case of a rectangular duct,
but it can also be applied to a duct of any cross-sectional
shape.
Thesis Supervisor:
Title:
Warren M. Rohsenow
Professor of Mechanical Engineering
TALE OF CONTENTS
Acknowledgement
Introduction and Summary
A. INTRODUCTION TO SECONDARY FLOW
1. Definition of Secondary Flow
2. Kinds of Secondary Flow
(M) The Secondary Flow about an
Oscillating Cylinder
(ii) The Secondary Flow in a 3end
(iii) Secondary Flows in Straight Ducts
of Non-Circular Section
3. SECONDARY FLOWS IN NON-CIRCULAR DUCTS
1. Evidence Suggesting the Existence
of these Flows
2. Explanation of Secondary Flows
3. Conclusion
Page
4
5
7
7
7
8
9
12
15
15
24
26
C. SECONDARY FLOWS IN LAMINAR FLOW THROUGH A DUCT 29
1. Solution Using the Perturbation Theory
29
34
2. More Exact Proof
D. CONCLUSIONS AND REMARKS
40
APPENDIX
Measurements of the Velocity Distribution
in a Square Duct
46
Nomenclature
References
5
ACKNOWLEDGEMENT
This note is the last part of my thesis that I
have to write. It is the easiest part, yet perhaps the
most essential. Essential, because the help and encouragement that I have received from the members of
the M.I.T. staff, and my class and work mates, has contributed so much to my work, that it must be mentioned
in the presentation.
Professors Warren M. Rohsenow and Ascher H. Shapiro
have obliged me with their valuable advice and criticism.
But I feei particularly obliged to Mr. Lawrence C. Hoagland who suggested the problem, and listened patiently
to many rambling monologues while this work was taking
shape.
This research project is under the sponsorship of
the John B. Pierce Foundation of New Haven, Connecticut,
and I wish to take this opportunity to express my thanks
to the Foundation for their financial support.
The material presented in the Appendix has been
taken from Mr. Deane Kihara's 3achelor's thesis.
Miss Harriet Nicholson, who did the typing, and Miss
Marguerite Derosier, who wrote the equations, share between them the credit for the presentability of this work.
This note of acknowledgement would be incomplete
without mention of my parents and of Dido, whose interest
and encouragement contributed greatly toward the completion of this work.
R. M.
-5-
Introduction and Summary
The material presented in this work is part of a
general investigation of forced convection heat transfer
in odd shaped cylindrical tuoes.
The ultimate aim of
this investigation is to develop a generalized solution
for heat transfer and friction in the turbulent flow
regime.
The solution sought would provide a means for
correlating all the data on such surfaces and would make
it unnecessary to conduct experimental work to determine
the characteristics of each new surface geometry.
A thorough understanding of the flow conditions in
odd shaped tubes is essential for the development of any
further analyses.
A study of the literature on these flow
conditions revealed the fact that many writers oelieve
that there exist in fully developed turbulent flow in
non-circular conduits, secondary flows which act in a
plane normal to the through-flow direction and are superimposed on the primary flow.
These secondary flows are
considered to have a negligiole effect on the main flow
velocity distribution and hence on the heat transfer conditions, in the interior of the flow sections.
In the
corner regions, however, where the primary flow velocities
are low, the secondary flow effect on the heat transfer
conditions may be appreciable.
It was therefore considered of particular importance
to investigate these secondary flows.
A review of the
literature which is presented here indicated that there
is sufficient evidence that such flows do exist in fully
develooed turoulent flow through odd shaped tubes.
How-
ever, no adequate explanation for the ultimate cause of
these flows has been found.
Furthermore, there was no
sufficient evidence showing whether these flows are
essentially a result of turbulence, or whether they can
also exist in laminar flow.
For this reason an analysis was developed in order
to determine whether secondary flows can exist in fully
developed laminar flow through a rectangular duct.
different solutions are presented.
Two
The first is approx-
imate, but the second, which is a little more mathematically involved, is exact.
The conclusion drawn from
these analyses is that secondary flows cannot exist in
fully developed laminar flow through a rectangular duct,
or through a closed duct of any odd shape.
Therefore,
the solutions found in the literature for the velocity
distribution in laminar flow through non-circular ducts
(which are based on the assumption that secondary flows
do not come into play) are exact and can be used with
confidence as asymptotic cases in the remaining investigations on the turbulent flow.
A. INTRODUCTION TO SECONDARY FLOW
1. Definition of Secondary Flow.
Many basic fluid-motion studies are made with the
primary assumption that the motion is irrotational.
In practice, however, it often hapoens that the
stagnation pressure is not the same along all streamlines of the flow oattern, out varies slightly from one
streamline to another.
It is therefore concluded that
the flow is not completely irrotational but only approximately so.
The difference between the potential flow and this
approximately potential flow in the normal plane is called
secondary flow.
Consequently, the secondary flow must
contain vorticity.
This vorticity, in turn, constitutes
a motion of the fluid particles, which for reasons of
continuity, is not necessarily restricted to involve
motion of the fluid particles along the streamlines of
the ootential flow, but can define other streamlines.
Eaving thus defined secondary flow, we examine the
types of such flow that have been observed.
2. Kinds of Secondary Flow.
The types of secondary flow that have been observed
are usually classed in three categories:
(i)
Secondary Flow about an Oscillating Cylinder
(ii)
Secondary Flow in a Bend
(iii)
Secondary Flow in Straight Ducts 6f NonCircular Cross-Section
(i) The Secondary Flow about an Oscillating Cylinder
Calculations of the boundary layer on a body which
performs a reciprocating harmonic oscillation of small
amplitude in a fluid at rest were carried out by Tollmien
and Schlichting (ref 10).
The solution for the velocity
distribution contained a steady-state velocity component
which does not vanish at a large distance from the body,
i.e. outside the boundary.layer.
It was therefore con-
cluded that a steady, secondary motion is induced at a
large distance from the wall as a result of viscous forces.
Photographs of the flow pattern about a circular
cylinder which performs an oscillatory motion in a tank
filled with water were published by Schlichting and by
Andrade, and the conclusions drawn therefrom are in good
agreement with the theoretical pattern of streamlines,
which are shown in Figure 1.
The calculations together with the experiments mentioned above and others which were carried out by the
3ureau of Standards group several years later, give a
good explanation of the flow around bodies which are
placed in a field of standing accoustic waves and also
of Rundt's "dust figures".
This kind of secondary flow, which belongs to a
completely different field of interest than the other
two has thus been analyzed adequately, and needs no further mention in the present work.
(ii) The Secondary Flow in a Bend
We consider a fluid flow moving in a curved path.
Each particle of the fluid, as it moves in the curved
path, is acted upon by a centrifugal force.
Also, since
the fluid particles are touching each other, the motion
of one will influence the motion of the others within the
fluid region.
Under the action of this centrifugal force
each of the fluid particles is forced toward the outside
of the curved iath in the direction of a line drawn through
the path from the centre of the path.
Under the condition of non-uniform axial velocity
distribution in the flow in a curved pipe (for example),
the centrifugal forces acting on the fluid particles within
the cross-section of the pipe are clearly of different
magnitude.
These centrifugal forces acting in the plane
of the pipe perpendicular to its longitudinal axis then
create a motion of the fluid particles in this plane.
The
fluid particles with the higher axial velocity are acted
upon by a larger centrifugal force than the slower moving
particles.
The resulting motion will then have a non-
uniform velocity distribution within the cross-section of
-10the pipe, giving rise to a vorticity in this plane and
thus satisfying the definition of a secondary flow.
Thus, the main flow parallel to the longitudinal
axis of the pipe has superposed on it a secondary flow at
right angles to it, outwards at the centre of the pipe,
and inwards in the neighbourhood of the walls.
(See
figure 2.)
The flow at the bottom of a flat-bottomed circular
vessel when the fluid in the vessel is given a circular
motion is a second example of secondary flow due to "centrifugal forces".
The flow in the layer next to the
bottom is directed inwards owing to its smaller centrifugal force.
It is a matter of everyday observation that
small particles on the bottom of a vessel are carried to
the centre and piled up there.
The bottom flow which has
just been mentioned is the explanation of this phenomenon.
(See figure 3.)
An interesting analogy between the secondary flow in
bends and the motion of a gyroscope is given by Squire
and Winter (ref 12).
If a disc which is rotating about
its axis of symmetry, is turned about an axis in its plane,
it develops an angular velocity aoout a third axis perpendicular to the first two.
The analogy with the flow
through a bend is obtained by supposing the disc to lie
originally in the plane containing the stream direction.
-11-
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EXAMPLES OF SECONDARY FLOWS
Fig. 1 :
Pattern of streamlines of the steady secondary motion
in the neighbourhood of an oscillating circular
cylinder (after Schlichting)
Fig. 2 :
Secondary flow in a bent pipe of circular cross-section
(after Prandtl)
Fig. 3 :
Secondary flow at the bottom of a flat-bottomed circular
vessel (after Prandtl)
-t12The disc and the stream are then turned about an axis
normal to the stream direction.
The rotation of the disc
about its third axis corresponds to the secondary flow.
There have been a number of theoretical and experimental investigations on the flow through curved circular
A summary of these may be found
and non-circular pipes.
in reference 2.
The first
theoretical study on the flow
through a curved circular pipe was carried out by W. R.
Dean and that was followed by experiments and analyses
by various other men.
Extensive tests on the flow through
bent pipes of non-circular cross-section were carried out
by Nippert, Richter, Eichenberger, and others.
It may therefore be concluded that this kind of
secondary flow has also been extensively treated, and that
its cause--unequal centrifugal forces--is well understood.
(1ii) Secondary Flows in Straight Ducts of Non-Circular
Section
The existence of secondary flows in straight ducts
of non-circular section was first suggested by Nikuradse
in his two classical papers (refs 7, E).
Nikuradse meas-
ured the velocity distribution in what he assumed to be
"fully developed" turbulent flow through various non-circular ducts.
He attributed some peculiarities which he
found in the velocity profiles, to the existence of secondary flows in a plane normal to the through-flow direction.
-13The same peculiarities in the velocity profiles-which will be described in detail later--were observed by
Mayer, Eichenberger and very recently by Eckert and Irvine
at the University of Minnesota.
None of the investigators
could find other possible explanations to these peculiarities than Nikuradse's explanation, namely, the existence
of secondary flows.
These flow velocity profiles have
thus come to be considered as "evidence" for the existence
of secondary flows.
Some authors have tried to reason the origin and
cause of this type of secondary flow, but the ultimate
cause of the flows is not yet established.
The effects
of these flows on the velocities and subsequently on the
losses in the main flow, were always considered to be
correspondingly small.
This assumption, together with
the fact that the equations of motion become very complicated unless secondary flows are neglected, has led all
investigators to disregard secondary flows in their
analyses of the velocity or temperature distribution in
non-circular passages.
With the increasing industrial use of non-circular
ducts in heat exchangers and with the advent of nuclear
engineering and the resulting unconventional heat transfer
design problems, the problem of the exact understanding of
the nature of flow through non-circular passages becomes
more than just an academic question.
-14For example, in some applications, like gas turbines
or nuclear power plants, heat exchangers are needed which
ooerate at extremely high temperature levels.
Usually,
the limiting factor in the design of such exchangers is
the temperature which the exchanger-walls can safely
withstand.
It is to be expected that the heat transfer
from a fluid flowing through a non-circular passage and
cooling the duct walls will be lowest near the corners of
the passage.
Therefore, it is of very special interest
to know the velocity and heat transfer conditions in the
corner regions.
Since in such corner regions the through flow velocities are low, the corresponding secondary flow effects-if secondary flows do exist--may not be negligible.
It is therefore concluded that It is both of practical
and of academic importance to investigate more thoroughly
the problem.of secondary flows in straight non-circular
passages.
It seems necessary that the literature on the
subject suggesting the existence of these flows should be
critically examined.
The explanations found in the liter-
ature for the ultimate cause of these flows should also
be reviewed.
Finally, the material gathered from that
review should serve to point out the path toward a better
understanding of this kind of secondary flow.
-15-
3. SECONDARY FLOWS IN NON-CIRCULAR DUCTS
1. Evidence Su gesting the Existence of these Flows.
This section presents a review of the existing experimental data which support the theory that secondary flows
exist in fully developed flow through non-circular ducts.
It has already oeen mentioned that this theory emerges
as an explanation of some peculiarities in the velooity
These peculiarities and their explanation are
profiles.
now reviewed.
The first extensive investigations of the velocity
distribution in non-circular ducts were carried out by
Nikuradse at the University of GOttingen.
Velocity dis-
tributions were measured for flow of water in ducts of
triangular, and rectangular cross-sections.
The flow
considered was well in the turbulent region, and the
velocity field was measured in the exit plane of the
ducts.
The length of the ducts was 60 to 120 hydraulic
diameters, a length which was then considered to be quite
sufficient for the complete developing of the flow.
The
Reynolds number of the flow was varied between 20,000 and
30,000.
The results of Nikuradse's experiments were very
surprising.
Instead of a distribution with ever more
rounded isotachen (lines of equal velocity), he obtained
for triangular and rectangular channels the lines shown
in figures 4 and 5.
It is seen that in both cases the
velocities at the corners are comparatively very large,
the velocity near the corner being in fact larger than
near the central part of the wall.
No explanation for these results was given in
Nikuradse's original paper, but some time later L. Prandtl
was the first to explain the peculiar velocity profiles
(ref 9).
Prandtl suggested that the water flowing in all
channels of uniform but non-circular section, develops
"secondary motions" in such a manner that in a corner the
fluid along the middle of the angle flows into the corner,
and on both sides of the middle it flows out of the corner.
These flows enable the explanation of the observed velocity
profiles.
Momentum is ever communicated to the corners,
thus producing the great velocities there.
The form of
the secondary flows as suggested by Prandtl for the triangular and the rectangular channels,is shown schematically in figure 6.
The diagram for the rectangle shows
how the inward flow from the wall develops regions of
subnormal velocity at the ends of the long sides and also
in the middle of the short sides.
For the confirmation of these views, Nikuradse
carried out further experiments with three pipes of triangular section, two circular pipes with notches and one
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Figures 4 & 5 :
Figure 6
"Isotachen" for ducts of equilateral triangular,
and rectangular sections.
(after Nikuradse)
: Secondary flows in the above ducts. (Schematic)
-18nine of trapezoidal cross-section.
In all cases, the
velocity profiles were analogous to those shown in figures 4 and 5, thus confirming Nikuradse's previous observations and Prandtl's explanation.
In addition, ex-
periments carried out by Nikuradse in open channels of
rectangular section indicated that secondary flows come
into play in such channels.
The maximum velocity does
not occur near the free surface but at about one fifth
of the depth down, and the flow in the free surface is
not at all two-dimensional as miqht well be expected.
As a further and direct confirmation of the existence of secondary flows, Nikuradse carried out a set
of flow visualization experiments.
A milky fluid was
injected at various points in the cross-section of an
equilaterally triangular and of a circular duct in which
water was flowing in turbulent motion (Reynolds number
of 12,000 to 15,000).
Intensities of illumination in
the section were photographed by means of a Hartmannschen Mikrophotometer, and from the photometric measurements light profiles were constructed in a manner similar
to the construction of velocity profiles from oressure
measurements.
These light profiles were employed for
drawing lines of equal illumination (isophoten) in the
same way as velocity profiles are used to determine lines
of equal velocity (isotachen).
The isophoten obtained for the two sections considered are shown In figures 7 and 8.
The conclusion drawn
-'9-.
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Pig. 7
Fig. 8
"Isophoten" for ducts of equilateral triangular and circular
cross-sections (after Nikuradse)
-20therefrom is that secondary flows of the nature previously suggested do exist in the triangular duct but do not
exist in the circular pipe.
These visualization experi-
ments are, in the author's knowledge, the only attempts
ever to illustrate directly the existence of secondary
flows.
The next experiments of interest were carried out
by Edwin Mayer (ref 6).
Mayer's experiments considered
the velocity and pressure relations over a number of
entire cross-sections in a duct of constant area but with
a transition in cross-sectional shape from a circle into
a rectangle.
Measurements were made at sections along
the circular part of the section, along the transition,
and along the rectangular part.
It was observed that at
the cross-section at the end of the transition (i.e. at
the "entrance" to the rectangular duct), there was a sufficiently uniform velocity distribution over the entire
cross-section with a rapid drop toward the wall in a
manner analogous to the distribution in the circular
section.
At the' next section, however, a short distance
downstream in the rectangular duct, it was found that at
some positions, particles farther removed from the wall
had smaller velocities than those close to the wall.
This
is clearly the same observation that Nikuradse made, and
the explanation given is again the existence of secondary
flows.
In the section near the entrance to the rectangle,
-21the ohenomenon was only found in the direct vicinity of
the edges, but at subsequent sections downstream the
ohenomenon was observed in greater and greater regions,
and finally at the section where fully developed velocity
distribution was first found there were superimposed over
the entire primary flow, secondary flows that included a
very large region of the cross-section.
It may be noted
that the fluid in Mayer's experiments was air, and the
calming length was 80 and 90 hydraulic diameters.
Some mention of the existence of this type of secondary flows was also made at the Massachusetts Institute
of Technology by H. P. Eichenberger in 1952 (ref 4).
Eichenberger carried out a theoretical and experimental
investigation of the flow in bent pipes of rectangular
cross-section.
In addition to measuring velocity profiles
along and after the bend, he also measured stagnation
pressures in "fully developed" turbulent flow 28 inches
before the bend.
The profiles obtained were similar to
Nikuradse's and the conclusion dravn was that secondary
flows exist in the straight duct.
The measurements were taken at a section after a
calming length of 30 hydraulic diameters.
The subsequent
results of Eckert and Irvine indicate that for some Reynolds numbers, the flow is still developing at a much
greater distance from the entrance.
It is therefore very
doubtful whether Eichenberger's results can be considered
-22as evidence for the existence of secondary flows in
fully developed duct flows.
Perhaps the most thorough experimental and theoretical investigation on the flow characteristics of
triangular shaped passages was conducted by Eckert and
Irvine at the University of Minnesota (ref 3).
In this
investigation, flow visualization techniques, longitudinal pressure drop measurements, and probing of the
velocity field have all been used to investigate the flow
for two ducts over a Reynolds number range from completely
laminar to completely turbulent flow.
A number of intereating conclusions are drawn from
Eckert's and Irvine's experiments.
Firstly, measurements
of the velocity profile along a plane parallel to the
triangle-base and very close to the base indicated that
in that plane the velocities near the corner are larger
than in the centre.
This observation is clearly the same
as Nikuradse's and the explanation given for the phenomenon is, once again, secondary flow.
Secondly, in
Eckert's and Irvine's experiments, profiles which indicate
such a secondary flow were found only at the highest Reynolds number.
This appears to indicate that such sec-
ondary flow exists only for well developed turbulent flow.
Thirdly, flow visualization measurements using smoke indicated that over a large Reynolds number range both laminar
and turbulent flow exist side by side within the duct.
The
-W23flow near the corners remains laminar for Reynolds
numbers much higher than the Reynolds number at which
the flow through the middle of the cross-section becomes
turbulent.
This remarkable fact may possibly serve as a
basis for the explanation of the cause of secondary flows.
Finally, pressure gradient measurements indicated that at
some Reynolds numbers the flow is not completely developed
at a distance of 70 hydraulic diameters from the inlet.
This makes it doubtful whether fully developed conditions
were always obtained in the previously published measurements of friction factors in non-circular ducts.
Finally, measurements of the velocity distribution
for air flowing through a two inch square duct are being
carried out presently in the Heat Transfer Laboratory of
the Massachusetts Institute of Technology, as another part
of the general investigation of forced convection heat
transfer in non-circular tubes.
At the time of writing
this thesis, the above-mentioned experimental work was
at its initial stages.
Measurements were made only for a
few Reynolds numbers in the region of 40,000 to 50,000.
The calming length of the tube is apnroximately 75 hydraulic diameters.
The need was felt for a number of
minor improvements in the measuring device used, before
proceeding further with the experiment.
As far as the
present thesis is concerned, velocity profiles obtained
were of Value as they constituted one further piece of
-24evidence that secondary flows do exist in turbulent flow
through a straight square duct.
Unfortunately, the meas-
uring devices available at present were not sufficiently
sensitive to measure, with any accuracy, velocities in
the laminar regime.
Having reviewed the evidence suggesting the existence
of secondary flows, it is of interest to examine the explanation that has been given to this type of flow.
This
is given in the following article.
2. Explanation of Secondary Flows.
The previous article shows that a number of investigators have encountered secondary flows in their experiments with non-circular tubes.
It is therefore somewhat
surprising to find that no explanation has been attempted
for these flows (on the ground of today's better understanding of turbulence) other than L. Frandtl's original
explanation.
The original explanation can be summarized as follows:
Prandtl concluded from Nikuradse's velocity profiles that
there exists, in the case of turbulent fully developed
flow in non-circular sections, in addition to the to-andfro motion in the direction of maximum velocity drop, a
still stronger to-and-fro motion at right angles to it,
that is, in the direction of the isotachen (lines of constant velocity).
If this is the case, then it follows
that centrifugal forces thereby arise on the particles
-25which move up and down along the curvature of the isotachen.
These forces always point toward one direction,
namely, the convex sides of the isotachen and are stronger
the greater the curvature of the latter.
The result is
that the particles move outward along the line bisectors
of the corners of the cross section whereby the particles
located at the corners are forced, for reasons of continuity, to deviate first laterally and then inwardly.
Since
the particles flowing toward the corners possess a relatively large axial velocity, whereas, conversely, the
outward flowing particles as a result of the friction at
the wall have a smaller axial velocity, the unexpected
appearance of the velocity profiles and hence, the isotachen is thus explained.
The question as to the reason for the mixing to-andfro motion being of this nature is not answered by
Prandtl's consideration.
The question leads to the more
comprehensive question of the origin and inner mechanism
of turbulence in three dimensional flow about which
little was known at the time of Prandtl's explanation.
In spite of the considerable step toward a better understanding of the structure of turbulent shear flow which
has been taken since the time of the above explanation,
no attemot has been made to complete Prandtl's argument.
-26Conclus on.
The experimental data which has been reviewed in
article 1 of this section provides a sufficient evidence
that secondary flows do exist in turbulent fluid flow
through non-circular channels and ducts.
Even though a
few of the flows considered had probably not been completely developed, it can be stated with fair confidence
that at least some of the ducts considered--in particular,
the longer ones in Nikuradse's and in Eckert's experiments--were sufficiently long to satisfy the condition
of fully developed flow.
The photometric measurements
and the isophoten shown in figures 7 and 8 provide the
most direct and conclusive evidence.
With regard to the existence of secondary flows in
the laminar regime, the evidence is by no means conclusive.
The experimental methods which were employed to illustrate
the secondary flows in the high Reynolds number region,
cannot produce conclusive results in the case of laminar
flow.
Velocity measurements are difficult to make because
extremely accurate measuring devices are needed.
This is
especially true for the regions near the edges of the tube,
as in those regions the velocities are particularly low.
The only experiment that had available a measuring device
of good accuracy was that of Eckert and Irvine who developed
a micromanometer having sufficient sensitivity to measure
air velocities below two feet per second.
Even that in-
27strument, however, may not be sufficiently sensitive to
detect secondary flow effects, which, if existent in the
case of laminar flow, would be of extremely low absolute
magnitude.
The theoretical considerations of Prandtl provide
no answer to the question whether secondary flows exist
in the laminar region.
Prandtl's explanation of the sec-
ondary flow associates this flow with the phenomenon of
turbulence, but the explanation is incomplete, and the
ultimate cause of the flows is by no means established.
A possible explanation of the origin of secondary
flows is obtained by consideration of shearing stress at
the boundary.
Shearing stress is greater along the middle
of the boundary, while it is less at the corners.
This
may cause fluid to flow toward the centre of the tube
from the middle of the boundary, and from the inside of
the tube toward the wall at the corners.
If this is the
cause of secondary flows, then it is natural to expect
that these flows will exist also if the flow is laminar,
since the shearing stress consideration is true for laminar
as well as for turbulent flow.
It is clear from the above considerations that the
question whether secondary flows exist in the laminar reLion
has not been answered.
It is by no means clear
whether secondary flow is a phenomenon associated with
turbulence only, or whether it exists in laminar as well
as in turbulent fully developed duct flows.
All previous analyses of velocity distribution for
laminar flow have assumed that secondary flows are not
present.
These analyses are to serve as asymptotic cases
in the general analysis of forced convection heat transfer
in non-circular tubes.
given above, it
In view of the considerations
becomes an essential part of the general
investigation to examine the possibility of secondary
flows in the laminar case.
The analytical velocity dis-
tributions will be exact and suitable for use as asymptotic cases, only if secondary flows are proved to be
non-existent.
For this reason, an analytical investigation has
been carried out in order to establish or dismiss the
possibility of existence of secondary flows in the laminar region.
This analysis constitutes the essential
part of this thesis and is presented in the following
sections.
r_
C. SECONDARY FLOWS IN LAMINAR FLOW THROUGH A DUCT.
1. Solution Using the Perturbation Theory.
Consider a fluid flowing through a rectangular duct.
Let
a
and
b
be the duct dimensions and let U, V, W
be the components of velocity in the mutually perpendicular
x, y, z
The direction of fluid motion
directions.
is considered to be -the z-direction.
The equations describing the general motion are:
X
_2U __X.
Dt
e
.
39 + va)
/3
P (3
+V
I
V)
+
pv=Y_
_P .
a7L
Dt
pS3z
eU
4.i
4-
+ VaW)
W
We will consider an incompressible (G = 0) laminar
flow in the z-direction, and for the analysis the body
forces will be neglected, i.e. X z Y = Z 1 0.
The assumption is also made that the flow is fully
developed, and that consequently all derivatives of vel-
-30ocities in the z-direction are zero.
With these assump-
tions, the equations of laminar motion become:
U _
I - + { 2U +')U
+V
%P
ys 10D(.70x
+
v)W=-
?U1
and
dP
+ VP
= -
)dV
I
K7
i_
Pz
497%
t
+
U I0
(1)
2
\/v
+
_LV
I
(2)
Assuming that secondary flows do not exist, then
U = V = 0, and the first two equations in.(l) as well as
the equation of continuity (2), vanish.
The third equation
in (1) becomes:
e dr 2 3k I
where
has replaced
TA
'
Y.
4idP
3
(3)
since P varies with z only.
Also for fully developed flow
d
is constant.
A solution for the velocity distrioution presented by
Claiborne for a rectangular duct of section axbis: (ref 1)
X
Wz B[
where
IA - .Cn (2n+1) IT.
-1n+
a.
cos h
a
n b . cos(2n'L01V (4)
4- eJ:"+'YCS LX24 1 I
=
Constant
dP
-31We now consider the possibility that secondary flows
We may expect,
do exist in the laminar flow.
then, a vel-
ocity distribution of the form
xa-
u
Va
,
+
,and
c sha
2
. c S?
(5
)
U
2e
where u, v and w are of small order compared with W.
In
line with the above assumptions we also write:
P2 Cr .
(6)
where p is small compared with P, and is a function of
x, y
only.
Substituting (5) and (6) into the equations (1), and
in the case of the third equation of (1) if we subtract (3)
from it, we obtain:
(,
.
-.
FJ(
+
0(10)
48(n
-Fn 2.
(7)
92.t)r+
+
2~ J--Tm
io(
a.
0
UY-C,
+ ".dur
L
(9)
-- in
-32If we now comoare such terms as .44
-
with terms as
we see that the first one is of the order
and the second of the order of
of M?
--
r
* Thus
the first is negligible in comparison to the second, prois small compared to unity.
vided that
a reasonable assumption since
order.
U,
This seems
is by definition of small
With this assumption, equations (7), (E) and (9)
can be linearized.
In particular, equations (7), (8) and
(10) can be re-written as:
c
b*
/0*a x i
(11)
g'
(10)
+
and
Differentiating the first of (11) with respect to
the second with respect to
x:
33
+
%
Subtracting:
3C z +
V(12)
0)
y
and
-33Nowto ascertain that the equation of continuity is
always satisfied, define a streamfunction
#
from:
(13)
-
Substituting (13) into (12),
Or, the problem to be solved is now defined from the equationt
4(14)
With the following boundary conditions derived from the
u - v a 0
fact that at the walls
90 0o
at
=
at
X =4 4Land at
2
jC
(15)
and at
It is interesting to note that the problem is now
identical to the so-called fundamental biharmonic boundaryvalue problem of the theory of elasticity.
Equation (14)
and the necessary boundary conditions (15) are identical
to the equations describing the state of deflection
0
a thin rectangular plate, clamped at its four edges and
with no lateral loading on the plane of the plate.
It
can then be reasoned physically that the only possible
of
...34..
state of deflection that such a plate could take is
4
(x, y)
=
0.
It is clear that
0
satisfies equation
(x, y) w 0
(14) and all the boundary conditions.
3ut because of the
linearity of (14), the uniqueness theorem applies and
0 (x, y) a 0
(See for
is the only possible solution.
examDle pp. 86 and 272 in ref 11.)
Thus, subject to the linearizing assumption being
correct, it has been shown that secondary flow cannot
exist in laminar fully developed flow through a rectangle.
2. More Exact Proof.
The analysis developed in the previous section involves the assumption that the Navier-Stokes equations
may be linearized with no appreciable errors.
This
assumption is very probably acceptable for almost the
entire non-circular section, but it
may not be true for
the regions very near the corners.
Since these regions
are of particular interest, a more general and slightly
more involved analysis has been developed.
This analysis
treats the Navier-Stokes equations in their non-linear
form, and consequently the results of the analysis are
exact for fully developed laminar flow and are applicable
to the entire cross-section.
-35Consider again equations (7) and (6)
of the previous
analysis:
U.dLL
u
Multiply (7) by
L)
+.
7
+
and (&) by
+( Un
2V--I
Consider now the term
v
L
Vi.,
LLV . -= UL.
This can also be written as
Now from equation 6.74a in reference 5 we see that
V.4
and if we let
07.ii + i.V
u a f
and VLt~ K , we can then use the
above relation and write:
.uLVt=LLV. \ + W.. VA.
Comparing this with (18), we see that we can write:
(16)
-36Then using (19) and a similar expression for
equations (16) and (17) can be written as follows:
V-a
+ lfv.a
ax
CII
P(
LV&
(
)2]
e
which are the same as
u.9ta)
4 v.(*Vu.)-(Vu)]
.....
_a
-"'
__.
/)). .
+
U va)
=
+
Now, using the equation of continuity
we see that such terms as
+-L-4ivv
aj
?5at.+ pl
-L.9,&+ O
=0
(20ab, c)
are all zero, and therefore can be added to any part of
the equations without affecting their validity.
Thus we may write:
U
a"+
,ax
ax
(t)a
CU
37-
4 Va a"h
(..A).g.
*
2
2
Ud+
ax3
UV(L)
- (Var
-37Or
4v(LVLw) -(wJ']
U{v{)29))
9
_+
Addinz, and also using (20c) we obtain:
;-
It
+(+4
-
+
-
(21)
can be seen that
13
9E
12
is a divergence, since it is equal to V.F
=_
+__ +_
a
where
and also
F:(
-)
where V
Similarly,
is the velocity vector.
J.(ag+.2..-
)
V-1k
where
So equation (21) can be written as
V.U
"1,I
]. 9
and' in-
If we then multiply by an element of area
tegrate between the limits of the boundary of the rec-
6Atwe
tangle
have:
~
+ [~v~uvj4~
I~v(Vu)1d~d~.
(22)
3ut we have the divergence theorem, which, in a two dimensional case states that:
V.AdS
S
f.i
S
C
In our oroblem, we know that along the boundary the velocity vector is zero, i.e.
the boundary.
V: O
Thus, the line integrals of
along the boundary are zero.
P LO
so that
R
and
on
F
Then, using the divergence
theorem, equation (22) reduces to
0:
O+ O -
9
(Ve)'(V o)
The integral is taken over constant limits and is the integral over an area of a positive quantity.
The only
possible way that this integral can be zero is that the
quantity itself is zero.
-39Therefore
(VuA)t
-(va),
And as complex numbers are not acceptable, the only
possibility is that
u a v a 0.
It has thus been shown that in laminar fully developed flow through a rectangle, secondary flows cannot
exist.
This croof is general and is based on no other
assumpotion that
the assumption that fully developed flow
exists.
The analysis was carried out for a rectangle.
It
is clear, however, that it also applies to any crosssectional shace.
It can thus be concluded that secondary
flows cannot exist in fully developed laminar flows
through closed conduits.
D. CONCLUSIONS AND REMARKS
The conclusions that are drawn from the present work
can be summarized as follows.
There exist, in fully
developed turbulent flow through odd shaped tubes, secondary flows in a plane normal to the through-flow direction.
On the other hand, when the flow is fully developed
and laminar, no secondary flows can possibly exist.
There-
fore, turbulence is the cause of the secondary flows, and
secondary flows cannot exist in the absence of turbulence.
Two points seem to require some consideration.
The
exactness of the proof that secondary flows cannot exist
in the laminar regime, and the usefulness of this result
in practical considerations.
With regard to the exactness of the solution, the
assumptions that have been made must be critically examined.
It should be ascertained that after the assumptions
are made, the analysis still aoplies to a definite flow
pattern, and also that this flow pattern exists and is of
practical importance.
The following assumptions are made in the derivation
of the exact solution:
~41The Navier-Stokes equations are assumed to des-
(i)
cribe the fluid motion.
(ii)
The fluid flow is assumed to be laminar.
(iii)
The fluid is assumed to be incompressible.
(iv)
The flow is assumed to be fully developed.
Under ordinary circumstances, there is no reason to
doubt that the details of the laminar or turbulent motion
of ordinary fluids are described by the Navier-Stokes
equations.
This assumption has been queried for the
case of turbulent flow, but even in that case no inconsistencies have been found so far.
ture is well understood, it
In gases, whose struc-
is easy to show that depar-
tures can only occur at scales of fluid velocities outside ordinary experience.
Therefore, the first assump-
tion is perfectly reasonable and allowable, at least for
all points in the cross-section apart from the corner
points.
The corner points are "peculiar" and the author
believes that one should be cautious in drawing decisive
conclusions from application of mathematical equations
to corner points.
The second assumption is clearly permissible, as
flows with low Reynolds number occur in many practical
applications.
The assumption that the fluid is incompressible is
always permissible for liquids and for ideal gases when
the Mach number of the flow is small and the rate of heat
-42W
transfer from sources outside the fluid is low.
This
assumption then puts only a small restriction to the applicability of the analysis.
The last assumption, namely that the flow under consideration is fully developed, is one that is found in
all analytical work on -flow through non--ircular sections.
It
This assumption is made for reasons of simplicity.
has always been assumed, by analogy to flows through circular sections, that fully developed flow in non-circular
tubes does exist and is found at some distance downstream
of the entrance to the straight tube.
A fully developed
flow in a tube is defined as a flow such that the pressure
drop is constant along the length of the tube, and the
velocity profiles are identical at all sections.
This
means that all derivatives of velocities with respect to
the flow direction must be zero.
If the assumption of fully developed flow is made,
and the equations of motion are examined carefully, some
peculiar results seem to be deduced.
It appears that by
assuming the flow to be fully developed, one immediately
"drops" some of the boundary conditions and the mathematics may no more describe the problem exactly.
Also, it
is seen in section C that by assuming the flow to be
fully developed, the first two equations in equation (1)
become independent of W.
This is rather surprising since
W is the forcing factor in the flow and one expects it
to have an effect on all the other variables.
Furthermore, if one applies the Navier-Stokes equations to a point at a corner, the results obtained are
again most unusual and-suggest that fully developed flow
cannot exist in non-circular passages.
This last con-
sideration is indeed remarkable, but as it has been
pointed out earlier, one should be very cautious in accepting results which are obtained by application of
mathematical equations to a "peculiar" point.
However,
much more work is needed in order to confirm or disprove
the above consideration.
Another interesting consideration with regard to
the existence of fully developed flows is the following
which is derived by analogy with the theory of elasticity.
It is known that problems of fluid flow in non-circular
sections are described in a manner identical to the torsion problems for plates of the corresponding section.
It can be reasoned that deflections in a particular direction correspond to velocity gradients in that direction.
Similarly then, the usual assumption in the theory of
torsion that "plane sections remain plane after twisting"
corresponds to the assumption of fully developed flow in
fluid dynamics.
Sokolnikoff has shown in a very elegant
manner that the assumption that "plane sections remain
plane after twisting" is only acceptable and true for the
-44case of circular sections.
(See page 109 in ref 13)
If
the author was permitted to extend the elasticdty--fluid
dynamics analogy, he might deduce that the "fully develoved flow" assumption is also only acceptable and true
for the case of flow through circular sections.
All the above remarks may be promising but are still
at the stage of early consideration.
In spite of these
remarks which suggest that fully developed flows never
exist in non-circular tubes, the bulk of the experimental
data indicates that flow in non-circular tubes develops
just as does the flow in circular tubes.
Experimental
data which include such work as measurement of pressure
drop at different sections along the length, flow visualization methods, comparison of velocity profiles at
different sections, etc. indicate that fully developed
flows do exist in non-circular tubes, even though a
larger calming length is necessary.
It therefore ap-
pears that the assumption of fully developed flow is
reasonable, as the fully developed flow pattern is a
pattern that actually exists.
It
is thus concluded that all the assumptions that
have been made are reasonable, and the result can be considered as general.
The usefulness of the results of the present work
to future investigations on forced convection heat transfer is quite evident.
The results put a solid ground
-45under the existing analyses for the velocity distribution
in laminar flow, which had hitherto assumed, without
proof, that secondary flows are non-existent.
These
analyses can now be used with confidence as asymptotic
cases in the treatment of the turbulent flow problem.
Finally, it
has been shown that the secondary flow
phenomenon is one associated with turbulence only, and
its complete explanation must wait until a better understanding is acquired of the general structure of turbulence.
-
-46
APPENDIX
Measurements of the Velocity Distribution in a Square Duct
It has been mentioned on page 23 of the text that
measurements of the velocity distribution for air flowing through a square duct are being carried out presently
in the Heat Transfer Laboratory of the Massachusetts Institute of Technology.
This experimental investigation
has not yet been completed.
Some measurements of veloc-
ity distribution were obtained for a limited range of
Reynolds numbers.
The results of these measurements were
of interest to the present work in that they indicated
the existence of secondary flows in the case of turbulent flow through the square duct.
For this reason, a
brief description of the apparatus and the experimental
procedure, together with a typical plot of velocity contours, is presented in the following few paragraphs.
Under the influence of a blower located downstream,
air from the room was drawn through a smooth entrance
into the
square duct.
The duct was fabricated of alum-
inum and its important dimensions were 1.75 inch wide and
12 feet long.
The test section was located one foot from
the blower-end of the duct.
A 4.50 inch-long piece of
the upper plane of the duct was removed and replaced by
-47a block of plexiglass.
This block was held tightly on
the sides of the duct, so that the inner flow section
remained identical to the square section.
A hole was
drilled through the plexiglass and a total pressure probe
3y sliding the plexiglass
was inserted into the section.
block across the tube, movement of the probe across the
section was obtained.
The vertical and horizontal posi-
tion of the probe could be measured with good accuracy.
Measurements were recorded only for one quadrant of
the section.
However, a few measurements were taken on
corresponding points in the other quadrants to assure
that readings were symmetrical.
Measurements were taken
at fourteen traverses in one quadrant, the spacing
between the traverses close to the wall being 0.025 inches
and away from the wall 0.100 inches.
Velocity profiles
were plotted for each traverse, and these profiles were
employed to determine lines of equal velocity.
A typical plot of the velocity contours is presented
in Figure 9.
The contours are seen to be quite analogous
to those obtained by Nikuradse for the rectangular duct
which are shown in Figure 5.
It may therefore be con-
cluded that the contours presented in Figure 9 constitute
one further piece of evidence that secondary flows exist
in turbulent flow through a straight square duct.
VELOCITY CONTOUR
IN A SQUARE
DUCT /75.' . X .760.875S+
V1.o"~
..-.
Sgure
0.7
V-4
0.6-
43 1h
-
0.5
-
40
0.3
-
0
0.1
-
.
(A/,
-
0.8
465
-
N
47,900
in.
-49--.
NOMENCLATURE
x
coordinate distance
y
coordinate distance
z
coordinate distance - flow direction
U
component of velocity
V
component of velocity
W
velocity in flow direction
X
body force
Y
body force
Z
body force
P
pressure
p0
density
viscosity
6
compressioility
Q
substantial time derivative
V
the vector ooerator del :
V
the scalar operator
9
kinematic viscosity
u
component of secondary flow
v
comoonent of secondary flow
w
component of secondary flow
p
secondary pressure
0
streamfunction
V
i4.
.
V.V
-50-
LIST OF REFERENCES
1.
CLAIBORNE, H. C. "Heat Transfer in Non-Circular
Ducts - Part I" O.R.N.L.-985-1951.
2.
DETRA, R. W. "The Secondary Flow in Curved Pipes"
Dissertations from the E.T.H. Zurich, 1953.
3.
ECERT, E.R.G. and IRVINE, T.F. "Flow in Corners of
Passages with Non-Circular Cross-sections" Heat
Transfer and Fluid Mechanics Institute, Los Angeles
June 1955.
4.
"Shear Flow in Bends" M.I.T.
EICHEN3ERGER, F. P.
D.S. Thesis in M.E. 1952.
5.
FILDE3RAND, F. 3. "Advanced Calculus for Engineers"
Sixth Printing Prentice-Fall, Inc. Englewood Cliffs,
N. J.
6.
MAYER, E. "Efffect of Transition in Cross-Sectional
Shape on the Development of the Velocity and Pressure Distribution of Turbulent Flow in Pipes"
V.D.I. Forschungsheft 389, (1927) Translated as
N.A.C.A, T.M. 903.
7.
NIKURADSE, J. "Untersuchung uber die GeschwindigKeitsverteilun6 in turbulenten Str'xMungen" Thesis,
Gottingen (1926) VDI-Forschungsheft 381, Berlin (1926).
8.
NIKURADSE, J. "Turbulente Stromung in nicht Kreisformigen Rohren" Ing-Arch, 1.306 (1930).
9.
PRANDTL, L. "Turbulent Flow" Proc. 2nd International
Congress Appl. Mech. (1926), Zurich, 1927. Also
translated as N.A.C.A. T.M. 435.
10.
SCHLICHTING, H. "3oundary Layer Theory" McGraw-Bill
Book Co., Inc. New York.
11.
SOKOLNIKOFF, I. S.
"Mathematical Theory of Elasticity" McGraw-Hill 3ook Co., Inc., New York 1956.
12.
SQUIRE and WINTER "The Secondary Flow in a Cascade
of Airfoils in a non-uniform stream" J. of the Aeron.
Sciences Vol. 18 No. 4 (1951) p.271.