Report on investigation of developed turbulence

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Material Information

Title:
Report on investigation of developed turbulence
Series Title:
TM
Physical Description:
7 p. : ill ; 27 cm.
Language:
English
Creator:
Prandtl, L
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Atmospheric turbulence   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Bibliography:
Includes bibliographic references (p. 6).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by L. Prandtl.
General Note:
"Report No. NACA TM 1231."
General Note:
"Report date September 19, 1949."
General Note:
"Translation of "Bericht über Untersuchungen zur ausgebildeten Turbulenz." Zeitschrift für angewandte Mathematik und Mechanik, vol. 5, no. 2, April 1925."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003759946
oclc - 43313654
sobekcm - AA00006233_00001
System ID:
AA00006233:00001

Full Text
I~A~ATh-,2~1










NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM NO. 1231


REPORT ON INVESTIGATION OF DEVELOPED TURBULENCE*

By L. Prandtl


The recent experiments by Jakob and Erk (reference 1) on the
resistance of flowing water in smooth pipes, which are in good agreement
with earlier measurements by Stanton and Pannell (reference 2), have
caused me to change my opinion that the empirical Blasius law (resist-
ance proportional to the 7/4 power of the mean velocity) was applicable
up to arbitrarily high Reynolds numbers. According to the new tests
the exponent approaches 2 with increasing Reynolds number, where it
remains an open question whether or not a specific finite limiting
value of the resistance factor X Is obtained at R = o.

With the collapse of Blasius' law the requirements which produced
the relation that the velocity in the proximity of the wall varied in
proportion to the 7th root of the wall distance must also become void
(reference 3). However, it is found that the fundamental assumption
that led to this relationship can be generalized so as t2, furnish a
velocity distribution for any empirical resistance law. These funda-
mental assumptions can be so expressed that for the law of velocity
distribution in proximity of the wall as well as for that of friction
at the wall, a form can be found in which the pipe diameter no longer
occurs, or in other words, that the processes in proximity of a wall
are not dependent upon the distance of the opposite wall.

For the velocity u (time average value) at y distance from the
wall only one nondimenslonal can then be formed, namely, (v = kine-
matic viscosity), giving for u a formula of the form


u = Cc() (1)

where C is a velocity and cp an arbitrary function. The shearing
stress at the wall then must be


T = pC2 (2)

where t is a constant.

*'Bericht Uber Untersuchungen zur ausgebildeten Turbulenz."
Zeitschrift fur angewandte Mathematik und Mechanik, vol. 5, no. 2,
April 1925, pp. 136-139.






NACA TM No. 1231


To obtain the function (p we proceed from the value
by theory would be constant = 7 and put


d In y
Sn y= f(a)
d In u


where a = Inu. With In
becomes


d In y
d In u' which


u = n and In y = a + Inv, formula (3)


S f(a) + 1,
dn

and, after integration and removal of logarithms,



u = CeP d
f(a) + 1


which, after including


vea
y = ..


gives the velocity profile in parameter representation.

The empirical law for wall friction reads T= Xpi2, with
function of = mean velocity, a = pipe diameter).
function of g (R = mean velocity, a = pipe diameter).
V


Discounting for simplicity the difference between the mean velocity
and the velocity in the center ul and assuming for it the value from
formula (4), although it is no longer exactly true in the pipe center.
ula
we put with l = Zn- -


- Ink = g(alo)


By (2) and (4) we get


T 2
pC2


T
Pl2
pul1


l2
C2


e f(i)+ 1d
_ e f(a) + 1


X a







NACA TM No. 1231


or
1 do
Znt = g(al) + 2 f(a) + 1 = const. (7)


hence after differentiation with respect to 01



2'( 2 g() (8)
g'(l) 1 + f(al) or f(a) g'(a)


With this the problem is solved and it is readily seen that f(a),
which for g'(a) = T assumes the value 7 as before, increases with
decreasing g'(a). A more accurate experimental check is awaited, but
even so it is plainly seen that at Reynolds numbers of about 200,000
the 8th root of the wall distance is definitely better than the 7th root.
For g'(a) = 1, f(a) = 1, which corresponds to the laminar boundary
zone.

Furthermore I would like to speak of a formula intended for a
hydrodynamic calculation of the distribution of the brse flow of a
turbulent motion under the most varied conditions.

After various fruitless attempts gratifying success was attained.
In addition it was found that the formula for the apparent shearing
stress T produced by the interchange of momentum, lends-itself to a
very clear explanation.

du
In the Boussinesq formula T = p~-d, is a measure for the
dy
turbulent "exchange" and in its dimension, which is the same as that
of v, it is the product of a length and a velocity. The velocity is
the transverse velocity w at which on the average the fluid bodies
advancing from both sides pass through the layer with the time average
value of the velocity = u.

The liquid bodies coming from the side of the greater velocity
entertain higher values of velocity u, those from the side of smaller
velocities, smaller values, with the result that more and more momentum
is transported in one direction than in the other (excepting the point
of umax). The desired length I is characterized by the fact that it
indicates the distance of the particular layer, in which the average u
velocities, which the liquid bodies have at their passage, are found as the
time average value of the flow velocity. Approximated these velocities







NACA TM No. 1231


du du
are u + 1 and u Z-. (Incidentally 2 is in agreement in order
dy
of magnitude with the diameter of the fluid bodies (more accurately it
is the decelerating path of the fluid bodies in the remaining fluid,
which is, however, proportional to the diameter).) As to the length 2
it can, for the present, only be stated that it must approach zero at
the wall, where only bodies of smaller diameter than the wall distance
can move as discussed. Elsewhere 2 is to have a very regular distri-
bution. If 0 is the average proportional share of the surface
du
occupied by the fluid bodies entering from one side, a momentum Ppwl-
per second passes at this side through the unit surface, and approxi-
mately the same amount from the other side. This confirms the Boussinesq
theorem, so we can put e = 2gw?.

The next problem is to find a practical formula for the mixing
speed w. This mixing speed is rapidly reduced and must be
continuously renewed. Hence the assumption that it is produced at the
concurrence of two bodies of different velocity u and therefore
proportional to the velocity difference, that is, the magnitude of -.
dy
With this, however, if all unknown numerical factors are thrown on
the not more accurately known length 1, the apparent shearing stress
becomes


T= p221uu (9)
dy df

This formula necessitates a correction for the case that du = 0.
dy

For the creation of velocity w the neighborhood in a certain
du
width cooperates; it does not become zero when = 0 it rather can
dy
be put proportional to a statistical average value of that is

proportional to (). If the velocity profile varies In flow direc-

tion, as in convergent divergent channels, the points over which the
averages are made must be shifted upstream for a certain amount, since
the process of development of the velocity w takes time.

Formula (7) has already proved itself in many respects. In a pipe,
for instance, the shearing stress is according to the equilibrium con-
ditions proportional to the distance r from the center, hence


T = 7p2( 2 = cr
kdY7 c







NACA TM No. 1231


Assuming I as constant gives


u = A Br3/2 (10)


for r > 0, the reflection for r < 0 (fig. 1). At r = 0 the radius
du
of curvature is then equal to Oj taking instead of -u the previously
d4y
discussed statistical value that can be approximately written as

2 + i 2 2u the curvature at r = 0 is finite. The actual

velocity distributions (reference 1, p. 21) exhibit the singular behavior
of the center very plainly (fig. 1). The refinement Just mentioned
needs to be applied only where increased accuracy requirements are
involved. The tctal velocity distribution in the pipe is fairly accu-
rately obtained when (in the range of the 1/7 power law) 2 is put
proportional to (a r)(a + r) 6/7, a = pipe radius.

The foregoing formulas have also been applied to the case of the
free turbulence, that is, flows without confining walls, as for
example, of a fluid jet diffusing in a chamber and to the intermingling
of a homogeneous air stream with the adjacent air at rest, which is
entrained by it. For stationary flows of this type the formula I = ex
is appropriate, x being the distance from the point where the mixing
starts. The case reproduced in figure 2 leads to the differential
equation for the stream function F(



2cF"F"' + FF" = 0


which is solved by F" = 0 or by 2cF"' + F = 0. Phe two solutions of
uniform velocity and variable velocity abut in F"' with discontinuity.
This and other calculations were numerically carried out by Tollmlen,
who is to publish an article on it. The agreement of his calculations
with experimental data is excellent.

Further experimental studies included the velocity distributions
in channels of other than circular sections with very unusual results
and for which the explanation has not yet been found. Slightly

It is approximately so that u7 plotted against the cross
section of the channel gives a sloping surface.







NACA TM No. 1231


divergent and convergent smooth-walled channels have also been investi-
gated, rough-walled channels are in preparation. It is hoped that our
formula will prove itself here also.


Translated by J. Vanier
National Advisory Committee
for Aeronautics



REFERENCES


1. Jakob and Erk: Der Druckabfall in glatten Rohren und die
Durchflussziffer von NormaldUsen. (The Pressure Drop in
Smooth Pipes and the Discharge Factor in Standard Nozzles.)
Forschungearbelten des VDI, Heft 267, 1924.

2. Stanton and Pannell: Phil. Trans. Roy. Soc. London (A), vol. 214,
1914, p. 199.

3. Von Karman: Ueber laminare und turbulente Relbung. (On Laminar
and Turbulent Friction.) Z.A.M.M. 1, 1921, p. 223.







NACA TM 1231


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