BASIC HYDRODYNAMIC EQUATIONS FOR TURBULENT MOTION AND THEIR APPLICATION TO MOTION IN ROUND PIPES

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000100200003-6
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 18
• Document Creation Date: December 22, 2016
• Document Release Date: March 8, 2012
• Sequence Number: 3
• Publication Date: February 19, 1952
• Content Type: REPORT

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STAT
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BASIC HYI~roDYNA~ TC F~UATIoNS i'oi TURAUL!'NT TTioN ANr)
APPLICATION TO MOTION IN iOUND PIPES
:Stjepan Mohorovicic
(Zagreb, Yugoslavia).
Note: The following; is an article that appeared in the journal ZeitM
schrift fur technisehe Physik, paces 68.7L (192).7
Contents;
Introduction; New Hasic Equations :for Turbu1 nt Notion
I. Application to Turbulent I~2ow in Round. ipes in the Case Where the
Average Velocities are Stationary.
II. The Honstationary State, a Special Case.
STAT
It is well known that present day hydrodynamics ;kith the aid of Eulerian
and Navier-Stokes theory govern only "laminar" motions. As soon as, however,
the velocity exceeds in actuality a li.ntiting value, an entirely irregular turbu-
lent motion immediately ensues. It is assumed, nevertheless, ghat the basic
hydrody"namjc equations do not 1osi~ their validity in also this cese, although
tLe their,', here is undoubtedly in need of broadening. p're'viously several attempts
were made along these lines to expand he theory, and the most noteworthy attempts
htve been performed recently by J. Boussinesq 2) and 0. Reynolds 3).
Several years ago I had made an attempt, during an inve ~t' f
~ g ~~,g , ton of the
structure of wind - which is reported in more detail in another place, to treat
turbulent flow theoretically, and during it I had obtained rather good agreement
with the results of my experi.u,ental measurements . It is my intention to include
here generally "1r expansion of the theory, and to apply it to turbulent flow in
round pipes.
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Let us proceed from the familiar equations for an incompressible fluid;
U
-_ .. - ..-~?,
12
is composed of two parts, we shall divide ~accordin to Ieynolds the velocities
i
-1--
C
E
where we have introduced in addition th@ internal forces' ,J ,,,. which
act to retard the turbulent motion because of the large energy losses; epsilon
is the "virtual" internal friction or turbulence . wince the fluid motion
(-
3y
into two components
primary or secondary; that is, a part of tae velo~.ity drop causes the average
(2)
part of the velocity drop causes the turbulent motions, irrespective of whether
w' the components of turbulence (pulsation). k'iirther we shall assume that a
U
v
where we disignate by , v, w the averace (mean) velocities and byv',
spondin@ fractions of the velocity drop, we roust have
flow and the other part causes the turbulence. If 8 and mean the correw
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r r'
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the basic equations (11 2) then break up into the two systems:
a
'Y U
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Now we must first solve the first system (1j.1 2 ) and set the values found for
J,, v, w into the second system (s); only then can we calculate the turbulence
velocities,, v1, which in many cases will be superfluous. The very
cornplic2trd system (51,2
) makes us suspect that the pulsations (turbulence)
will be of a very cornpli.cated nature, which obrvat=i can also has conf.,iri~icd.
T. The utility of t e expansion develop(d here of tl:e theory T have already
ndicabed c arie~ in the investigation of the structure of wind (see footnote 14.);
1 will now apply this theory to the case of turbulent flow 1~ around pipes. In
this case we shall first assurnt? that the average velocities w are stationary;
the z-axis coincides ith the cylinder's axis; and the average velocities w are
distributed around the cylinder's axis sym:netrca1ly ' that is, the average
velocity w is only a function of the distance r from the z-axis. Since v
W / at Q, aw / az = p, X = Y Z = 0 (that is, we shall also be a~)le to
neglect gravity in the case where the pipe is in a horizontal position), we obtain
from (~1 } the farnilar basic relation:
Obviously we must make assumptions for g such that they not only lead us to
correct results but also correspond to the nature of our problem.
First Jet ua set;
where A, b, a also can be functions of the pipe 1 s radius.
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In order to solve our basic equation (6), let us set:
where K and 13 too are constants to be determined, and 9o should be a function
0
of r. From ($) and (6) we obtain:
where we have set
(8)
(9)
In order to be able to solve the differential equation (9) with regard to (?), we
and finally obtain from it
In the case whore m is any real number, then we shall have for
at once n ."2 + m
or na?2-m,
our problem. Therefore it follows then
and we see imrndiately that the negative values of n do not correspond to
KAa m KAb 2
r + ~,r +B (m? 2).
~'
m
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Since at the wall of the pipe (r R) no slip occurs, that is since we must
have w = 0 there, we finally get:
w M A . ?~-"? L::2 (1?n " xm) + j* ( f2 in 5 2)
E oz
For the total flow through a cross'section in 1 second we get;
and we ,form the average velocity:
C a ? Q/fI- n ; (1'7)
thus we o tain for tl:e velocity d:Lstributicn in the cross--section the
r b
expression: _ 'R .". rv') `r ~..~?
It will now be interesting to consider some special cases;
1. For A b 1 and a 0, our relations (1Sa) to (18) reduce to the
fa(liliar Poiseuille Law for laminar flow:
Therefore it follo~as that laminar flow represents an entirely special
case of natural flow. As is well lrnown, this law loses its validity at once
if the flow becomes turbulent. It T~1ows from (19) that for r 0 we get
3
immediately;
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on the other hand, measurements of turbule nt flows yield
1.16c up to 1.23'
For b = 0 and A' a ao, we have already a kind of turbulont flow; the
relations (15a) to (16) reduce to:
/GtJ
-:
0*
a0
h c.
n
o p
(.,i +2) #?
',
w)
2.
L
vvr
1
In this case we should not forget ? as we have already emphasized this - that
a also could be a function of R; that is,
0
From (21 ) it follows for r 0 immediately that
3
w m - C (m52)
o m
and in this case m is always to be chosen so that (21)) agrees with (20),
however, we must not forget that the relation (21) also must agree with the
1
results of measurertients. Let us make the following; table therefore, where
we shall consider only some integral values of m:
m 23 14 5 6 7 8 9 10
(m + 2)/m : 2 1.66? 1.5 1.)4 1.333 1.286 1.25 1,222 1.2
M 1 M
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(20)
(212)
(213)
(22)
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M : ll 12 13 100 X000 a~
(m + 2)/m : 1.182 1.167 1.154 1.02 1.000 1
Only for the case m = 2 is it valid for laminar flow, since then we have
n m M2 0; all other cases m,> 2 represent turbulent flcws Lor us, In
Figure 1 the new laws for various values of in are represented graphieallyo
We will now assign these values numerically:
m~2 m-3 m~6 rn 12 mp100 ra=ao
0.000
2.000
1.667
1.333
1.167
1.020
1.000
0.125
1.969
1.664
1.333
1.167
10020
1.000
00250
1.875
1.6L~1
1.333
1.167
1.020
1.000
0.375
'
1.789
1.610
1.331
1.167
1.020
1.000
0.00
1.0o
1..459
1.312
1.167
1.020
1.000
0.625
1.219
1.260
1.254
1.163
1.020
1.000
0.750
~
0.875
0.96'4
1.096
1.130
1.020
1.000
0.875
o.69
0.;50
0.735
0.921
1.020
1.000
0.937
0.244
0.296
0.1431
0.633
1.018
1.000
0
999
0
004
005
0
004
0
o, o14
0.096
1.000
.
.
.
.
1.000
0.000
0.000
0.000
0.000
0.000
0.000
much more uniformly over the cross
bulent flow the velocity is distributed
section than during laminar flow. In this case, however, we have not obtained
complete agreement with the measurements, especially not in the immediato
neighborhood of the pipe's wall.
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Immediately apparent from the figure is the familar fact that during tur.-
9)
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3. ThE~ most general case; Here we have four quantities A, a, b, m at
our disposal, so that we can fit very exactly the theory to observations. In
order to show this, we shall now compute an example. First we must determine
the number m, and indeed from the relation (18) we set r R and
b ga;
obtain:
and in the "irunediate neighborhood" of the pipets wall w / c = 0.Tl. Since,
.
,
-.--- ------. Or
a
crn
4
fu11y smoothed cement pine o h
C
c, t_
V\ +' .
_ 1 M
411(/
(23)
(2L)
(25)
This formula is very convenient for practical computation. For a care-
' found 10) w c 1.16?
in
u
dis
!
f r
0
according to the investi?,ations of Forchheimer 11) , a value r 39 .96 cm lies
closer for the last-named case, it follows that we have for m a value between
2000 and 3000. It is very interesting here that the number m is not too very
'tsensitivett; this number, however, cannot exceed in our case the value 5000.
We shall now compare the results of Bazin's measurements with the results of
our theory:
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a and finally
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r/R
w / c (obs.)
n i2200
w / w
III
m 3000
w
0.000
1.167
1.167
10167
0,125
1.160
1.162
1.162
0.250
1,1)47
1.1146
1.1!j6
0.375
1.126
1.120
1.120
00500
1.092
1.081
1.0811
0.625
1.017
1,037
1.037
0.750
1.001
0.980
0.980
0.875
0.922
0.912
0.912
0.937
0.8Li,6
0875
0.875
0.999
0.7lj1
0. Thi
0.792
1.000
0.000
0.000
0.000
The agreement is really an excellent one 12)1 In th
e case of the value
measured by Bazin 0.711 is valid for still larger nei hbor
~ hoods than 0.0L cm
at the pipers wall; thus we shall keep the value in 3000. In this case we
must keep in mind that we have maintained here at all times the nva.rtual,
internal f'riatian or rrturbulencetI ~
Constant throughout; nevertheless our
theory governs perfectly the turbulent flow considered.
After we have determined the number m, we can detra
rtnine also g by means
of relation (2!~), and the formula (16) will assume
the following i'orm;
VJ
-s- .,w_1 . _._ .ww
Ac 2t r- iT
T' Li+ r& 2.
I
4
(i-
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(26)
 g3 = A
C
+ a r%1 ) (1 t))
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where we must still insert: