U.
S.
DEPA
RTMENT
OF
COMMERCE
NATIONAL
BUREAU
OF
STANDARDS
RESEARCH PAPER
RP1151
Part
of
Journal of
Research
of
the
}.{ational
Bureau
of
Standards,
Volum.e
21,
Decem.ber
1938
LAWS
OF
TURBULENT
FLOW
IN
OPEN
CHANNELS
By
Garbis
H.
Keulegan
ABSTRACT
The
theoretical
investigations
of
Prandtl
and
Karman,
and
the
experimental
work
of
Nikuradse,
have
led
to
rational
formulas
for
velocity
distribution
and
hydraulic
resistance for
turbulent
flow
in
circular
pipes.
With
certain
assumptions
regarding
the
effects of
secondary
currents
and
of
the
free surface,
and
with
the
adoption
of
the
hydraulic
radius
as
the
characteristic
length,
similar
rational
formulas
are
deduced
for
open
channels.
The
validity
and
the
applications
of
these
formulas
are
illustrated
by
a
study
of
Bazin's
experiments.
In
this
study
the
equivalent
sand
roughn
esses of
the
channels
used
by
Bazin
are
determined
.
The
criterion
for
determining
the
conditions
under
which a
channel
with
wooden-
plank
surfaces is
to
be
considered
hydrodynamically
wavy
or
hydrodynamically
rough
is also
evaluated.
The
rational
formulas
with
constants
determined
from
Bazin's
experiments
are
expressed
in
the
form
of power laws.
It
is
shown
that
Manning's
empirical
formula
is a good
approximation
to
the
rational
formula
for
rough
channels
when
the
relative
roughness
is
lar
ge.
CONTENTS
1.
Introduction
__________ - _ - - -
___
- _ - _ - - - - - - - - - _ - _________ -
___
- - -_
II.
Development
of
the
theory
____
__ __
________
___
_____ _____ _______ _
1.
Karman
law
of
velocity
distribution
_______
_
___
___
________ _
(a)
General
law
of velocity
distribution
near
a solid
walL
(b)
General
equation
of velocity
distribution
for
smooth
walls ___________ _____ _______ _____________
___
_
(c)
General
equation
of velocity
distribution
for
wavy
walls ____ _____
__
______________________
___
_
__
_
(d)
General
equation
of velocity
distribution
for
rough
walls _________________ _____ ________
__
_____ __ _
2.
Derivation
of resistance
formula
s _________________________ _
(a)
Resistance
formulas
for
circular
and
infinitely wide
rectangular
pipes
__________ ________________
___
_
(1)
Smooth
walls ___________________________ _
(2)
Rough
walls ___________________
__
______ _
(b)
Resistance
formulas
for
channels
with
polygonal
cross
sections _______
__
_________________________
___
_
(1)
Trapezoidal
channels
__________________
__
_
(2)
Channels
of
other
shapes
________________ _
(3)
Error
from
neglect of
the
correction
terms
__ _
(c)
Equivalent
sand
roughness ______________________ _
III.
Applica
t ion
to
Bazin's
experimental
data
__
___
______ _______
__
__
__
_
1.
Description
of
Bazin's
channel
installation
___
_____ _______ _ _
2.
Equivalent
sand
roughness
for
some of
Bazin's
channels
_
___
_
3. Surfaces of wood
as
examples
of waviness
_____
________ ____ _
4. Effect of
shape
of cross
section
on
mean
velocity
__
____
___
__ _
5.
Distribution
of velocity
in
rough
channels
____
__
__ __
____
___
_
6.
Maximum
velocity
in
rough
rectangUlar
channels
_______ _
__
_
IV.
Manning's
formula
for
rough
channels-power-law
formula
s ____
___
_
V.
References ________ _
___
_
__
__
__________ __________ _____________ _ _
707
Page
708
709
709
709
711
712
713
715
715
716
717
717
718
720
720
722
723
723
724
727
730
732
734
737
741
708 Journal of Research
of
the
National Bureau of Standards
[Vol
.
11
1.
INTRODUCTION
The
theoretical investigations of
the
phenomena of
turbulent
flow
by
Prandtl
and
by
Karman
have
prepared
the
way for
rational
inter-
pretations
of experimental results.
In
these investigations two
distinct
relationships of basic
importance
are
expressed.
First,
in
a
unidirectional
turbulent
flow
the
apparent
shear
at
any
point
depends
on
the
square of
the
velocity
gradient
at
that
point. Second,
the
factor
of proportionality
in
the
relation of
the
apparent
shear
to
the
square
of
the
velocity
gradient
is
determinate
once
the
similarity
of turbuJence is assumed. As a
result
of these two relationships,
not
only
the
velocity
distribution,
but
also
the
hydraulic resistance factor
for circular pipes,
has
been expressed
in
a rational form;
th
at is, in a
form derived
by
analytical
methods
involving only two
constants
to
be
determined
by
experiments. One of these
constants
is a uni-
versal cons
tant,
which
IS
a characteristic of turbulence.
The
other
constant
is a characteristic of
the
surface of
the
pipe, which will be
called
the
"surface
characteristic"
in
this
paper.
In
his classic
experiments
on
circular pipes,
Nikuradse
[1,
2]
1
determined
the
universal
constant
of
turbulence
and
the
surface characteristics for
smooth
pipes
and
for pipes
with
one
type
of rough surface. These
genera] results
are
well
known
and
are presented
in
detail
in
recent
works [3].
This
paper
is
an
attempt
to
apply
these
same
principles
to
the
problem of
turbulent
flow
in
open channels, mainly for
the
purpose
of developing formulas for resistance
or
for
mean
flow
in
forms similar
to
those
obtained
for circular pipes.
Two
assumptions
are
made
in
developing
the
formulas.
The
first
assumption
IS
that
Karman's
universal law of velocity
distribution
near
a solid
boundary
is of
general applicability.
The
second assumption is one
in
regard
to
the
effects of secondary
currents
and
of
the
free surface.
It
is assumed
that
the
average resllIt of these effects over
the
cross section of a
channel is,
in
general, a small
quantity
which
can
be merged
in
the
surface characteristics entering
the
flow formulas.
Furthermore,
it
is found
that
when
the
hydraulic
radius is
adopted
as
the
characteristic
length
of a channel cross section,
the
resulting formulas become prac-
tically
independent
of
the
shape
of
the
channel except for a correc-
tion
of geometrical origin which
can
likewise be merged
with
the
surface characteristic.
Thus
the
formulas for flow
in
open
channels
are identical
in
form
with
those for flow
in
circular pipes,
the
differ-
ences being
in
the
values of
the
surface characteristics.
The
necessary experimental
data
used
in
this
paper
are
taken
exclusively from
Bazin's
pioneer research
[4]
on
flow
in
open channels,
which
constitutes
an
outstanding
monument
to early scientific work
in
this
field. These experiments were carried
o'-!
k .9n . a large scale
near
Dijon,
France,
in
the
years
1855 to 1860.
As a
matter
of convenience,
the
following definitions
are
adopted.
A
"pipe"
is a
conduit
for
carrying
water
in
which
the
flowing
stream
is entirely bounded
by
solid surfaces. A
"channel"
is a
conduit
for .
carrying
water
in
which
the
flowing
stream
is
in
part
bounded
by
an
air
surface;
that
is,
the
stream
has
a free surface.
The
word
"channel"
is
thus
used as a
synonym
for
the
expression
"open
channel."
1 Figur
es
in brackets indicate
the
literature references
at
the end of
th
is paper.
I
.J
KeuleganJ
Turbulent Flow
in
Op
en Channels 709
II.
DEVELOPMENT
OF
THE
THEORY
1.
THE
KARMAN
LAW
OF
VELOCITY
D
ISTRIBUT
ION
The
discussion here of
the
theory of velocity distribution relates
to unidirectional flow in the neighborhood of
a plane wall of large
extent. When
it
is necessary to consider experiments for verifica-
tion of
the
theoretical results, reference will be made
to
studios of
flow
in circular pipes, since numerous and reliable
data
are available
for this case. These references are permissible because the flow
in
circular pipes is also unidirectional.
Again,
in
these discussions the thin lamin
ar
layer
at
the wall will
be ignored, since the portion
that
it
contributes to the total
flow
is
negligible.
(a)
GENERAL
LAW
OF
VELOCITY
DISTRIBUTION
NEAR
A
SOLID
WALL
Prandtl
[5]
has
given the following expression for the
turbulent
shear stress
at
any
point in a fluid moving
past
a solid wall (see
fig.
1):
,-
du
-V7/p=Zdy
(1)
where
7
=t
he shearing stress
at
the point,
p=the
density of the fluid,
u=the
velocity
at
the point,
y=the
distance of the point from the wall,
and
l=the
so-called mixing len
gth
of
momentum
exchange.
For
the purpose of finding an approximate law of velocity distribu-
tion
in
the
neighborhood of the wall, we write
eq
1
in
the
form
,-
du
,-
-V
70/
p=
Z
dy
-V
70/
7
,
where
70
is the
shear
in
the fluid
at
the wall.
The
quantity
in
the
left member I
of eq 2
has
the dimensions
of
a
velocity, and, because
of
its signifi-
rl7
cance,
has
been called
by
Prandtl
I
II
the shear velocity, denoted
by
the
symbol
u*.
To
mal"e the concept
>,
I
u
TURBULENT
CORE
of shear velocity more concrete,
it
I I
may
be pointed
out
that
it
can be I I tii
__
1_~
LAMINAR~~~
expressed
in
the following simple
*~;,;~wm/,fhY,i/7#,0W/l/##///7
/
manner in terms
of
more customarily
To
used quantities:
u*=.JRig,
(3)
FIGURE
I.-Diagram
of
velocity distri-
bution
in
a stream flowing past a solid
wall
to
illustrate
notot1
:on
.
both
for flow
in
circular pipes and
for uniform flow
in
wide channels, where
R=the
hydraulic radius,
i=the
hydraulic gradient,
and
g=the
acceleration of gravity.
Introducing the notation for the shear velocity, we write eq 2 as
~
70)ll
t
u
-l
- .
*-
dy
T
(4)
1
710 Journal oj Research
oj
the
National Bureau
oj
Standards
[Vol.!1
As
a consequence of his principle of similarity of
turbu
lence,
Karman
derived
the
following expression for the mixing length l
at
any
point y in terms of
the
velocity gradient
at
y:
u'
l=K-;; ,
u
(5)
where
the
primes indicate differentiation with respect to y, and K
is a universal constant characterizing
the
turbulence
[5].
This
latter
expression for l, however, requires a modification for the following
reason.
The
two experimental determinations of l from the inde-
pendent
expressions, eq 4
and
5,
using
the
velocity measurements
in smooth or rough circular pipes,
may
be made to agree with each
other
for small values of y
by
selecting K properly.
But,
then, for the
same value of
K the two values of l show gradually increasing differ-
ences for increasing
y, the largest difference occurring for Y=Ym,
Ym
being the wall distance corresponding
to
r=O
[1].
Obviously, the
agreement between the two determinations would be improved if now
we select
in
the
place of eq
5,
(5a)
where kl
and
k2
are constants.
Since
in
the
typ
es
of
flows
we shall be considering, the component
of
acceleration normal to the direction of mean flow would be negligible,
r=ro(l-Y/Ym),
eq 4
then
may
be written also as
(4a)
When l from eq 5 is substituted in eq 4 there results, for values of Y
approaching zero,
which
can
be integrated, yielding
u 1
-=-
In
(Y/Yo),
u*
K
(6)
(7)
where
Yo
is a constant of integration.
The
other
constant resulting
from the first integration of eq 6 is
put
equal to zero as
the
consequence
of
the
limiting value of
u'
at
the wall. This is
Karman's
law of
velocity distribution
in
the neighborhood of a solid wall.
The
deriva-
tion is
mad
e for small values of y. Experience, on
the
other hand,
shows
that
eq 7 is sufficiently accurate also for large values of y, even
when
Y is as large as
Ym.
In
the second approximate solution, the
differential equation corresponding
to
eq 6 would contain
Y/Ym.
The
form of the resulting equation is necessarily complex and hardly suit-
able for the usual computations.
Equat
io
n 7, being of a simpler
form
and
also sufficiently accurate, will serve as the basis for the
elementary mathematical development of
thiR
paper.
J{eulegan]
Turbulent Flow
in
Open
Channels
711
Theoretical considerations indi
cate
that
K
sh
ould
be
independent
of
the
nature
of
the
wall surface.
This
has
been verified
in
the
cases of
flo
w in circular pipes
and
between parallel walls
[6]
. On
the
other
h
and
,
the
constant
of i
nteg
ration
Yo,
a length
that
we
may
te
rm
the
"characterist
ic l
ength
of
turbulence,"
va
ries
with
th
e shear velocity,
and
the
roughness of
the
wa
ll.
The
next
step is to consider
th
e evalu
at
ion of
Yo.
Let 0 be
the
minimum distance from
the
wall at which eq 7 holds. (See fig. 1.)
That
is, 0 is
the
thickness of
the
boundary
layer in which
th
e viscous
str
esses either predominate
or
are of
the
same order as
the
apparent
st
resses due
to
momen
tum
exchan~e.
It
can
be
shown easily
that
the
rat
e of energy dissipation
EiD
this
layer
per
unit
of surface is
(8)
where
u~
is
the
velocity for y =
o.
From
eq 7:
and
eliminating
Ua
from
the
last
t
wo
equations:
(9)
Unfortunately,
in
the
present
state
of
our
knowledge of
turbulence
E
and
0
cannot
be comp
uted,
so
that
eq 9 does
not
serve as a
means
of
computing
Yo.
In
th
e absence of
an
adequate
theory,
the
natural
procedure is
to
resort to experience, using
th
e
method
of dimensional reasoning as a
gu
ide. Based on
th
e
hydrodyn
amical effects
that
they
produce, solid
boundaries are usually classified as smoo
th,
wavy,
or
rough.
For
th
ese
thr
ee
types of walls,
Yo
may
be
expressed
in
three
distinct
forms
which
will now be developed.
(b)
GENERAL
EQUATION
OF
VELOCITY
DISTRIBUTION
FOR
SMOOTH
WALLS
If
the
surface of
the
wall is smooth,
Yo
will depend solely
on
u*
and
v,
and
dimensional reasoning
the
n furnishes
the
relation
where
m=a
constant,
and
You*
-1I-=m,
v=the
kinematic viscosity.
Substitution
of this value of
Yo
in
eq
7 yields
-=a
s+-
In
-
,or
u 1
(yu
*)
u*
K
11
u i_
+2.30
I
(yu
*)
---as
--
og - ,
u*
K
11
(10)
(11)
1